diff --git "a/9NFLT4oBgHgl3EQfty_-/content/tmp_files/2301.12153v1.pdf.txt" "b/9NFLT4oBgHgl3EQfty_-/content/tmp_files/2301.12153v1.pdf.txt"
new file mode 100644--- /dev/null
+++ "b/9NFLT4oBgHgl3EQfty_-/content/tmp_files/2301.12153v1.pdf.txt"
@@ -0,0 +1,9826 @@
+arXiv:2301.12153v1  [math.AP]  28 Jan 2023
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+EDUARDO GARC´IA-JU´AREZ˚, PO-CHUN KUO:,;, YOICHIRO MORI:,§,
+AND ROBERT M. STRAIN:,¶
+Abstract. This paper introduces the 3D Peskin problem: a two-dimensional
+elastic membrane immersed in a three-dimensional steady Stokes flow.
+We
+obtain the equations that model this free boundary problem and show that
+they admit a boundary integral reduction, providing an evolution equation
+for the elastic interface. We consider general nonlinear elastic laws, i.e., the
+fully nonlinear Peskin problem, and prove that the problem is well-posed in
+low-regularity H¨older spaces. Moreover, we prove that the elastic membrane
+becomes smooth instantly in time.
+Contents
+1.
+Introduction
+2
+2.
+Formulation and Boundary Integral Reduction
+9
+3.
+Preliminaries
+12
+4.
+Nonlinear decomposition
+18
+5.
+Calculus estimates
+24
+6.
+Frozen-coefficient Semigroup
+39
+7.
+Local well-posedness
+56
+8.
+Higher Regularity
+66
+Appendix A.
+Besov Spaces and Fourier Multiplier Theorems
+75
+Appendix B.
+Estimates for the semigroup e´tLApξq
+77
+References
+90
+˚Departamento de An´alisis Matem´atico, Universidad de Sevilla, C/Tarfia s/n, Cam-
+pus Reina Mercedes, 41012, Sevilla, Spain. egarciajuarez@ub.edu
+:Department
+of
+Mathematics,
+University
+of
+Pennsylvania,
+David
+Rittenhouse
+Lab.,
+209
+South
+33rd
+St.,
+Philadelphia,
+PA
+19104,
+USA.
+;kuopo@sas.upenn.edu
+§y1mori@math.upenn.edu ¶strain@math.upenn.edu
+Date: January 31, 2023.
+2020 Mathematics Subject Classification. 35Q35, 35C15, 35R11, 35R35, 76D07.
+Key words and phrases. Peskin problem, 3D, Fluid-Structure Interaction, immersed boundary
+problem, Stokes flow.
+˚supported by the European Union’s Horizon 2020 research and innovation programme under
+the Marie Sk�lodowska-Curie grant agreement CAMINFLOW No 101031111, and the AEI project
+PID2021-125021NAI00 (Spain).
+;partially supported by NSF grant DMS-2042144 (USA) awarded to YM.
+§partially supported by the NSF grant DMS-1907583, 2042144 (USA) and the Math+X award
+from the Simons Foundation.
+¶partially supported by the NSF grants DMS-1764177 and DMS-2055271 (USA).
+
+2
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+1. Introduction
+The immersed boundary method, introduced by Peskin [40, 41] to study the
+blood flow around heart valves, has been widely applied to numerically study fluid-
+structure interaction (FSI) problems. These FSI problems, in which a fluid interacts
+with elastic structures, appear naturally in many engineering and biophysics appli-
+cations [44,46]. Despite their importance, both the computational methods and the
+FSI problems themselves are poorly understood from an analytical standpoint. A
+major impediment has been the lack of analytical understanding of the underlying
+PDEs, which are typically nonlinear and nonlocal. Results are particularly scarce in
+the more realistic three-dimensional settings, where the coupling of nonlocal effects
+with non-trivial geometry substantially increases the complexity of the problem.
+Since the recent breakthrough works [34] and [37], which provided the strong
+solution theory for the problem of an immersed elastic string in a two-dimensional
+fluid, the so-called 2D Peskin problem has attracted a lot of attention [8,9,23,25,
+33,51,52]. In this paper, we initiate the study of its three-dimensional counterpart.
+We introduce the formulation and develop the well-posedness theory for the three-
+dimensional (fully nonlinear) Peskin problem of an elastic membrane immersed in
+a fluid.
+1.1. Description of the problem. We consider the following problem in which
+a three-dimensional incompressible Stokes fluid interacts with an elastic membrane
+in R3. A closed elastic interface Γ encloses a simply connected bounded domain
+Ω Ă R3 filled with a Stokes fluid with viscosity µ. The outside region R3zΩ is filled
+with a Stokes fluid of viscosity 1. The equations satisfied are:
+µ∆u ´ ∇p “ 0 in Ω,
+(1.1)
+∆u ´ ∇p “ 0 in R3zΩ,
+(1.2)
+∇ ¨ u “ 0 in R3zΓ.
+(1.3)
+Here u is the velocity field and p is the pressure. We impose the following condition
+in the far field:
+(1.4)
+u Ñ 0 as |x| Ñ 8.
+We supplement the above with interface conditions on the time-evolving surface Γ.
+For any quantity w defined on Ω and R3zΩ, we set:
+�w� “ w|Γi ´ w|Γe
+where w|Γi,e are the trace values of w at Γ evaluated from the Ω (interior) and
+R3zΩ (exterior) sides of Γ. Let n be the outward pointing unit normal vector on
+Γ. The interface conditions are:
+�u� “ 0,
+(1.5)
+�Σn� “ F el, Σ “
+#
+µ
+`
+∇u ` p∇uqT˘
+´ pI
+in Ω
+∇u ` p∇uqT ´ pI
+in R3zΩ ,
+(1.6)
+BX
+Bt “ upX, tq,
+(1.7)
+where I is the 3 ˆ 3 identity matrix and X : S2 ÞÑ Γptq the map that describes the
+evolving membrane. This map gives the deformation of the reference configuration
+S2, the standard embedding of the sphere of radius 1 in R3. The first condition
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+3
+is the no-slip boundary condition and the second is the stress balance condition
+where Σ is the fluid stress and F el is the elastic force exerted by the interface Γ.
+The last condition states that the membrane evolves with the fluid flow. Note that
+the elastic surface Γ “ Γptq and hence Ω “ Ωptq changes with time. Once given
+the constitutive equation for the elastic force F el, equations (1.1)-(1.7) form the
+so-called jump formulation of the 3D Peskin problem. Let pg and g denote the metric
+tensors on S2 and Γ respectively. A natural choice for the elastic stretching force
+is given by [19,28]
+(1.8)
+F el “
+a
+detppg´1gq∇S2 ¨ T p∇S2Xq,
+where
+T p∇S2Xq :“ T p|∇S2X|q
+|∇S2X|
+∇S2X “: T p|∇S2X|q∇S2X,
+∇S2 denotes the surface gradient on S2, |A| denotes the Frobenius norm of matrix
+A, and T has to satisfy T ą 0, dT {dλ ě 0 (see Section 3.1 for further notation).
+In Section 2 more details about the derivation of the elastic force are given. For a
+Hookean material, T is linear and hence the elastic force is linear in X. We will
+consider general T , i.e., the fully nonlinear Peskin problem.
+Compared to fluid interface problems, such as a drop of liquid surrounded by
+another fluid or vacuum [17,43,45,49,50], where only the shape of the interface mat-
+ters, here it is not expected that Eulerian methods on their own should suffice. Due
+to the elastic nature of the membrane, the stretching, given by the parametriza-
+tion, has a strong influence on the evolution. Thus, one needs to keep track of
+the membrane configuration. Lagrangian methods are needed, making it harder
+to work in higher dimensions. In particular, one cannot freely reparametrize the
+surface, an idea frequently used to obtain extra cancellations in the study of fluid
+interfaces [13,22,30].
+An important feature of the Peskin problem is that it admits a Boundary Integral
+formulation, whose derivation is given in Section 2. When µ “ 1, the problem (1.1)-
+(1.8) is equivalent to the following evolution equation for X:
+(1.9)
+BX
+Bt ppxq “
+ż
+S2GpXppxq ´ Xppyqq∇S2 ¨
+´
+T p|∇S2Xppyq|q ∇S2Xppyq
+|∇S2Xppyq|
+¯
+dµS2ppyq,
+Xppxq|t“0 “ X0ppxq,
+where Gpxq is the Stokeslet tensor in R3:
+(1.10)
+Gpxq “ 1
+8π
+´ 1
+|x|I3 ` x b x
+|x|3
+¯
+.
+We have suppressed the dependence of X on t to avoid cluttered notation. Hence-
+forth, we will assume µ “ 1.
+It will be sometimes convenient in the analysis
+to work with coordinates.
+Let θ “ pθ1, θ2q be a (local) coordinate system on
+S2 and let px “ x
+Xpθq P S2 Ă R3 be the point on S2 corresponding to θ.
+Let
+Xpθq “ Xpx
+Xpθqq P Γ Ă R3 be the position on Γ corresponding to the coordinate
+point θ (see Figure 1). If px “ x
+Xpθq, we will write Xppxq and Xpθq in an abuse of
+notation. Then, after integration by parts and choosing an isothermal coordinate
+system, equation (1.9) becomes
+BX
+Bt pθq “ ´p.v.
+ż
+R2
+B
+Bηi
+GpXpθq´Xpηqq ˜F el,ipXqpηqdη1dη2,
+(1.11)
+
+4
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+where we denote
+˜F el,ipXqpηq “ T pλpηqq
+λpηq
+B
+Bηi
+Xpηq,
+λpηq “
+a
+trppg´1pηqgpηqq.
+Above we use the explicit definitions of pg and g given in (2.1).
+Figure 1. Deformation map Xp¨, tq : S2 Ñ Γptq.
+Some important properties of the solutions to the Peskin problem (1.9) are easier
+to deduce from the jump formulation (1.1)-(1.7). The incompressibility condition
+(1.3), together with (1.7), implies the conservation of the volume of the enclosed
+region Ω:
+d
+dt|Ωptq| “ 0,
+|Ωptq| “ 1
+3
+ż
+S2 Xppxq ¨ nppxqdµΓptqppxq.
+Moreover, the elastic energy defined as follows
+EpXq “
+ż
+S2 AEp|∇S2Xppxq|qdµS2ppxq,
+A1
+Epλq “ T pλq,
+satisfies the balance
+d
+dtEpXq “ ´
+ż
+R3 |∇u|2dx,
+which shows that the elastic energy is dissipated due to the viscosity of the fluid.
+This relation follows from (1.7), integration by parts, and using conditions (1.6),
+(1.3), (1.1)-(1.2), and (1.5), consecutively. For a linear elasticity law, the elastic
+energy is the 9H1pS2q norm of the interface,
+EpXq “ 1
+2
+ż
+S2 |∇S2Xppxq|2dµS2ppxq.
+A third important property of the Peskin problem is that it satisfies a scaling
+invariance. We must first mention that the definition of solutions requires that the
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+5
+interface is non-degenerate and does not self-intersect. This is typically enforced
+through the arc-chord condition:
+|X|˚ :“
+inf
+px‰py
+px, pyPS2
+|X ppxq ´ X ppyq|
+|px ´ py|
+ą 0.
+If Xppx, tq solves (1.9), then, for any λ ą 0, Xλppx, tq :“ λ´1Xpλpx, λtq also solves
+the equation, and |Xλ|˚ “ |X|˚. Hence, 9C1pS2q and spaces with the same scaling,
+such as 9H2pS2q, are critical spaces for 3D Peskin problem. Notice that the energy
+balance above only gives control of the 9H1pS2q norm, hence the Peskin problem is
+supercritical.
+1.2. Main results. The formulation of the problem, both in jump and Boundary
+Integral forms, is derived in Section 2. Once the formulation is provided, the main
+objective of the paper is to show that the problem is well-posed. More specifically,
+we will first show the existence and uniqueness of strong solutions with initial data
+in little H¨older spaces, h1,γpS2q, γ P p0, 1q, defined as the completion of the set of
+smooth functions in C1,γpS2q.
+Definition 1.1 (Strong solution). Let X P Cpr0, T s; C1,γpS2qqXC1pr0, T s; CγpS2qq,
+γ P p0, 1q, and |Xptq|˚ ą 0 for t P r0, T s. Then, X is a strong solution to the
+3D Peskin problem with initial data Xp0q “ X0 if it satisfies equation (1.9) for
+t P p0, T s and Xptq Ñ X0 in C1,γpS2q as t Ñ 0.
+The choice of little H¨older spaces will be needed to obtain the convergence to
+the initial data. In Section 7 we will prove the following:
+Theorem 1.2. Consider the 3D Peskin problem (1.9) with initial data satisfying
+X0 P h1,γpS2q, |X0|˚ ą 0, and T P C3 such that T ą 0, dT {dλ ě 0. Then, there
+exists some time T ą 0 such that (1.9) has a unique strong solution X,
+X P Cpr0, T s; h1,γpS2qq X C1pr0, T s; hγpS2qq.
+It is instructive to briefly recall the idea of the proof for the 2D linear Peskin
+problem [37]. For X a non-degenerate, closed simple plane curve, the boundary
+integral formulation in 2D is given by
+BtXpθ, tq “ ´
+ż
+S1 BηGpXpθq ´ XpηqqBηXpηqdη,
+where G is the Stokeslet in R2. It turns out that one can perform a small-scale
+decomposition [30,37] to write it as follows
+BtX “ 1
+4ΛX ` RpXq,
+ΛX “ HBθX,
+with RpXq a lower order operator compared to Λ. Then, it is natural to construct
+the solution as a fixed point of the equation written in Duhamel form:
+Xptq “ etΛX0 `
+ż t
+0
+ept´τqΛRpXpτqqdτ
+We notice two important facts: the semigroup is explicit, both in space and Fourier
+variables, and the equation is semilinear. Even for nonlinear elastic law, the lead-
+ing term has a kernel not depending on the curve itself, ´ 1
+4HpT p|∇S2X|q∇S2Xq,
+making it possible to use the Λ-like structure via energy methods [8].
+
+6
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Here, we first consider the strategy adapted to nonlinear equations in [35] (see
+also [47] for 2D Peskin). Let us write equation (1.9) as follows
+BX
+Bt “ FpXq,
+X|t“0 “ X0.
+Then, at least formally, linearization around the initial data would give
+(1.12)
+Xptq “ eLpX0qtX0 `
+ż t
+0
+eLpX0qpt´τqEpXpτqqdτ,
+with LpX0q “ BXFpX0qX the Gateaux derivative of F at X0 and EpXq “
+FpXq ´ LpX0q. Hence, while EpXq is not expected to be smoother than LpX0q,
+it should be small for short time. However, one first need to make sense of the
+above expression (1.12) by showing that LpX0q generates an analytic semigroup,
+which amounts to proving that the operator is sectorial in adequate spaces. This
+is the core of the abstract Theorem 7.1, whose proof encompasses a fixed point
+argument. The application of this theorem to our problem soon becomes highly
+involved. This is done in Propositions 7.3-7.6. Since the equation is not semilinear,
+the process will require to further decompose the operator LpX0q and then freeze
+the coefficients at a given point. The decomposition must be done maintaining
+a derivative structure for the kernel that allows extra cancellations, required to
+control the singular integral operators that appear, and so that we can invert the
+frozen-coefficient operator (the study of this part is done separately in Section 6).
+Schematically, we decompose the kernel in (1.11) as follows
+B
+Bηi
+`
+GpXpθq´Xpηqq
+˘
+« ´ B
+Bxj
+G p∇Xpηqpθ ´ ηqq BXj
+Bηi
+pηq ` RpXqpθq
+« 1
+8π
+BXpηq
+Bηi
+¨ p∇Xpηqpθ´ηqq
+|∇Xpηqpθ´ηq|3
+` ¨ ¨ ¨ ` RpXqpθq,
+(1.13)
+where one expects RpXq to be lower order and the dots represent additional terms
+of high order coming from the second term in Gppxq (1.10) (we note that in 2D
+these additional high-order terms cancel each other). These leading kernels are
+not of convolution type and cannot be written as a derivative. For this purpose,
+one could be tempted to use ∇Xpθq in the approximation instead of ∇Xpηq.
+However, higher derivatives of X would appear later in the proof and the argument
+would not close. Thus, to take advantage of the derivative structure, we will be
+forced to estimate together the leading and remainder terms. In a second step,
+we approximate ∇Xpηq in the leading kernels above by its value at a given point
+(see Lemma 5.10 for more details), which requires the introduction of a partition
+of unity for the sphere. Due to the geometry of the problem, we need to work
+with charts, and due to the nonlocal character of the equation a second localization
+procedure will be needed. A fine implementation of these localization procedures
+will be crucial to avoid transition maps that would otherwise overcomplicate the
+proof.
+For the fully nonlinear Peskin problem, we must linearize and freeze the coeffi-
+cient of the elastic force as well. In Section 4.2, we show that the frozen-coefficient
+linear operator in the general force case is given by
+pLAY qkpθq “ ´
+ż
+R2
+B
+Bηi
+pGk,lpA pθ ´ ηqqqpTF pAq∇Y ql,ipηqdη1dη2,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+7
+where A is a constant matrix and TFpAq a tensor (4.31). Thus, in the general case,
+the multiplier for the frozen-coefficient linear operator becomes:
+LApξq “ I ` vpξq b vpξq
+4detpBq |B´1ξ|
+ˆT p∥A∥F q
+∥A∥F
+ˆ
+|ξ|2 I´ Aξ b Aξ
+∥A∥F
+˙
+` dT
+dλ p∥A∥F qAξ b Aξ
+∥A∥F
+˙
+,
+where || ¨ ||F denotes the Frobenius norm. It is not difficult to see that, if T ą 0
+and dT {dλ ě 0, then the above is coercive in |ξ|2. Moreover, in contrast to the
+2D case, dT {dλ “ 0 is allowed. In fact, if T satisfies T ą 0 and pdT {dλq{T ą ´1,
+then the problem is expected to be locally well-posed if the initial condition is
+sufficiently close to the uniform sphere (pg´1g is close to a multiple of the identity
+matrix). This is an interesting difference between 2D and 3D Peskin. We will use
+this operator (in conjunction with the localization procedures) to show that the full
+operator LpX0q “ BXFpX0qX is sectorial. The approximation in (1.13) is done
+on the equation written in coordinates partly to obtain a linear leading operator
+given by a Fourier multiplier.
+Next, we notice that the regularity obtained in Theorem 1.2 for the strong solu-
+tions is not enough to satisfy equation (1.9) in a classical sense. Obtaining higher
+regularity for the solutions is also important since this further regularity is needed
+for the equivalence between different formulations to hold. The abstract theory for
+nonlinear equations in [35] does not yield gain of smoothness for the solution, and
+in fact this important point is left open in the 2D results in [47]. Nevertheless, we
+are able to prove that initial data in little H¨older spaces become smooth for positive
+times.
+Theorem 1.3. Let X be the solution to the Peskin problem with initial data X0 P
+h1,γpS2q constructed in Theorem 1.2. Then, for any α P p0, 1q, it holds that X P
+C1pp0, T s; C3,αpS2qq. Moreover, for any 3 ď n P N and α P p0, 1q, assuming that
+T P Cn,α, it holds that X P C1pp0, T s; Cn`1,βpS2qq, for any β ă α.
+We use the solutions constructed in the previous theorem and Duhamel formula
+(1.12) to perform a bootstrapping argument. We build on the properties of the
+semigroup e´tLA (see Section 6 and Appendix B) to first gain regularity in mixed-
+type spaces Lpp0, T ; Cn,αpS2qq and then transfer this higher regularity in space to
+show regularity in time as well. A key point is that, while the kernels are not of
+convolution type, we find that it is possible to move derivatives in θ to derivatives
+in η at the expense of new terms of the same order, but not higher (see (8.12)). As
+explained above, we must work with the equation localized around a given point
+and later deal with the corresponding commutators and combine the estimates (see
+Section 8 for more details). However, the bootstrapping argument cannot be done
+on (1.12) directly, because the right-hand side contains terms of highest regularity.
+We combine this process with a regularization argument (see (8.14)), where the use
+of little H¨older spaces becomes crucial.
+1.3. Related results. The first analytical results for the 2D Peskin problem ap-
+peared recently in [34,37]. In [34], energy arguments are used to prove local well-
+posedness for H
+5
+2 initial data and also exponential convergence to steady states for
+sufficiently close to equilibrium initial data is shown. The authors in [37] lowered
+the required initial regularity to barely subcritical spaces, h1,γ, γ P p0, 1q, showed
+instant smoothing, and provided a blow up criterion.
+
+8
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+After these works, many improvements for the 2D Peskin problem have appeared.
+The work [25] deals with the setting in which the enclosed fluid is different to the
+exterior one, and shows asymptotic stability for small data in Wiener algebra critical
+spaces. The result [8] shows the local well-posedness and smoothing for general data
+in the critical Besov space B
+3
+2
+2,1, including the case of nonlinear elastic law. The
+sharpest result in terms of regularity appeared in [9], where the semilinear 2D Peskin
+problem is shown to be well-posed in B1
+8,8, and thus with possibly non-Lipschitz
+curves.
+In relation to the Peskin problem, the article [51] introduces a regularization of
+the problem inspired by the immersed boundary method and studies its conver-
+gence. Filaments that resist both bending and stretching are considered in [33].
+Finally, we mention two works that introduce simplified models of the 2D Peskin
+problem. The work [23] considers a model for the normal component and shows
+the existence of global solutions for Lipschitz data near the equilibrium. Very re-
+cently, [52] derives a PDE to model the tangential effects of the Peskin problem in
+the case of an infinitely long and straight string and obtains global solutions with
+initial data in the energy class. Moreover, the author presents many connections of
+the model with well-known one-dimensional PDEs.
+From a mathematical point of view, there are remarkable similarities between
+the 2D Peskin problem and the so-called Muskat problem.
+In particular, both
+problems have the same leading linear operator, they can be written in Boundary
+Integral form [13,30], they have the same scaling and satisfy an energy balance [12,
+29]. The Muskat problem, which describes the movement of the interface between
+incompressible fluids in a porous medium, has been intensively studied in the last
+two decades [1,2,4,7,12,13,18,36,39], and some of the techniques developed there
+have been successfully extended in the last years to lower the required regularity
+for the well-posedness of the 2D Peskin problem [8,9,23,25]. However, while there
+are also results for the 3D Muskat problem [3,5,10,14,24], in all these results the
+interface is a surface given by graph, hence the geometry does not play a major role.
+Even in 2D, in the recent non-graph setting [22] that considers a bubble of fluid
+surrounded by another in a porous medium, a change of parametrization becomes
+crucial, which is not allowed in the Peskin problem.
+We finally mention some results with more complex elastic interactions [6,11,16,
+32,38,42,53,54], mostly dedicated to more qualitative results and weak solutions.
+Part of the interest generated by the Peskin problem is due to its relative simplic-
+ity, which makes it possible to initiate the analytical study of the rich variety of
+behaviors in FSI problems, including longtime dynamics.
+1.4. Outline. The rest of the paper is structured as follows. In Section 2, we ob-
+tain the expression for the elastic law and show the Boundary Integral formulation
+for the 3D Peskin problem. Section 3.1 contains the notation used along the paper
+as well as some definitions and standard results concerning the stereographic pro-
+jection. Next, in Section 4, we introduce the operators that will be used later in
+the paper, we decompose the equation and compute the multiplier of the leading
+term. Section 5 is dedicated to study the operators previously defined, to show the
+needed commutators estimates, and to prove Lemma 5.10. These lemmas will be
+repeatedly used in the proof of the main theorems. In Section 6, we show that the
+frozen-coefficient operator generates an analytic semigroup (for which we need the
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+9
+multiplier results contained in Appendix A, with further properties studied in Ap-
+pendix B). Finally, Sections 7 and Section 8 contain the proofs of the main results:
+Theorems 1.2 and 1.3.
+2. Formulation and Boundary Integral Reduction
+The formulation of the problem (1.1)-(1.6) is closed once that the expression for
+F el is given. To specify the elastic force F el in (1.6), we consider the elastic energy
+of the interface Γ. We consider an elastic energy EpXq of the form:
+EpXq “
+ż
+S2 E
+ˆBX
+Bθ , θ
+˙
+dµS2,
+where µS2 is the standard measure on the unit sphere. From this, we may com-
+pute the elastic force by taking the variational derivative as follows.
+Let X “
+pX1, X2, X3qT. Define the following metric tensors pg and g on S2 and Γ respec-
+tively, whose i, j components are given by:
+(2.1)
+pgij “ Bx
+X
+Bθi
+¨ Bx
+X
+Bθj
+,
+gij “ BX
+Bθi
+¨ BX
+Bθj
+.
+We write the energy density as follows:
+(2.2)
+E “ AEpsij, θq, sij “ BXi
+Bθj
+, i “ 1, ¨ ¨ ¨ 3, j “ 1, 2.
+Let Y “ pY1, Y2, Y3qT be a perturbation of the configuration that is compactly
+supported on the open set U on which the coordinate system θ is defined. We have:
+d
+dτ EpX ` τY q
+ˇˇˇˇ
+τ“0
+“
+ż
+U
+BAE
+Bsij
+BYi
+Bθj
+a
+detpgdθ1dθ2
+“ ´
+ż
+U
+B
+Bθj
+ˆBAE
+Bsij
+a
+detpg
+˙
+Yidθ1dθ2,
+(2.3)
+where the summation convention is in effect. We set:
+Fel,i “
+1
+?detg
+B
+Bθj
+ˆBAE
+Bsij
+a
+detpg
+˙
+,
+where Fel,i are the components of the elastic force F el of equation (1.6). With this
+prescription of the elastic force, the solutions satisfy the following energy relation:
+(2.4) dE
+dt “ ´
+˜ż
+Ω
+2µ |∇Su|2 dx `
+ż
+R3zΩ
+2 |∇Su|2
+¸
+dx, ∇Su “ 1
+2
+`
+∇u ` p∇uqT˘
+.
+We will now impose symmetry conditions to determine the explicit form of AE and
+hence E. Let θ be a (local) orthogonal coordinate system on S2 so that the two
+coordinate tangent vectors are orthogonal:
+Bx
+X
+Bθ1
+¨ Bx
+X
+Bθ2
+“ 0.
+We thus have an orthonormal frame on (a neighborhood of) S2 given by the two
+vectors:
+pei “ Bx
+X
+Bθi
+�����
+Bx
+X
+Bθi
+�����
+´1
+, i “ 1, 2.
+
+10
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+The deformation map X maps the above unit orthogonal vectors to the following
+two vectors:
+ei “ BX
+Bθi
+�����
+Bx
+X
+Bθi
+�����
+´1
+, i “ 1, 2.
+Consider the matrix 3 ˆ 2 matrix B “ pe1, e2q whose column vectors are given by
+ei. We may say that the energy density E is a function of B and θ:
+E “ AEpB, θq,
+where we have continued to use the notation AE as in (2.2). By homogeneity of
+the unit sphere, we impose that AE does not have an explicit dependence on θ.
+Furthermore, the value of AE should not depend on the choice of orthonormal frame
+pei or the coordinate system in which X resides. This implies the following.
+(2.5)
+AEpBq “ AEpR3BR2q for all R3 P SOp3q and R2 P SOp2q
+where SOp2q and SOp3q are the group on rotation matrices in 2 and 3 dimensions
+respectively. Let:
+H “
+ˆ
+e1 ¨ e1
+e1 ¨ e2
+e1 ¨ e2
+e2 ¨ e2
+˙
+.
+The invariance condition (2.5) implies that AE can only be a function of the trace
+and determinants of H,
+(2.6)
+E “ AEpλ, γq, λ “
+a
+trpHq, γ “
+a
+detpHq.
+In terms of the metric tensors g and pg, we can write λ and γ as:
+(2.7)
+λ2 “ trppg´1gq, γ2 “ detppg´1gq.
+The above expressions for λ and γ are valid even when θ is not an orthogonal
+coordinate system. We may substitute (2.6) into (2.3) to obtain:
+Fel,k “ Fλ,k ` Fγ,k,
+Fλ,k “
+1
+?detg
+B
+Bθi
+ˆ 1
+λ
+BAE
+Bλ
+a
+detpg pgij BXk
+Bθj
+˙
+,
+Fγ,k “
+1
+?detg
+B
+Bθi
+ˆBAE
+Bγ
+a
+detg gij BXk
+Bθj
+˙
+,
+(2.8)
+where we use the standard notation aij to denote the inverse tensor pa´1qij. Note
+that the expressions Fλ,k and Fγ,k are similar but differ crucially in whether pg or
+g features inside the force expressions. This is most clearly seen in the following
+simple cases. If we let AE “ λ2{2, we have:
+(2.9)
+Fel,k “ Fλ,k “ γ∆S2Xk, ∆S2Xk “
+1
+a
+detpg
+B
+Bθi
+ˆa
+detpg pgij BXk
+Bθj
+˙
+,
+where ∆S2 is the Laplace-Beltrami operator on the unit sphere. If we let AE “ γ,
+we have:
+Fel,k “ Fγ,k “ ∆ΓXk “
+1
+?detg
+B
+Bθi
+ˆa
+detg gij BXk
+Bθj
+˙
+“ ´2κΓnk,
+where ∆Γ is the Laplace-Beltrami operator of the closed elastic surface Γ, κΓ is the
+mean curvature of Γ and nk is the k-th component of the outward normal vector
+n of Γ. This is just the well-known statement on the variation of surface area. We
+see from the above expressions that the Fλ,k expresses an elastic force that depends
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+11
+strongly on the stretching of the spherical reference configuration whereas Fγ,k is
+a surface tension force.
+The prescription of interfacial elastic energy density as in (2.2) or (2.6) has its
+origins the classical work of [19], and may be called the membrane neo-Hookean
+model. Specific forms for this energy have been used extensively in the modeling
+and simulation of fluid-structure interaction problems [20,28,31].
+We now rewrite our evolution equation in a form suitable for our analysis. Hence-
+forth we focus on the case when AE is only a function of λ, and the viscosity µ of
+the interior fluid is equal to 1.
+Let us rewrite the equations of motion. Let G be the Stokeslet tensor in R3:
+(2.10)
+Gi,jpxq “ 1
+8π
+˜
+δi,j
+|x| ` xixj
+|x|3
+¸
+, x “ px1, x2, x3q.
+Let
+pFel,k “ 1
+γ Fel,k “
+1
+a
+detpg
+B
+Bθi
+ˆ
+λ´1T pλq
+a
+detpg pgij BXk
+Bθj
+˙
+“ ∇S2 ¨
+`
+λ´1T pλq∇S2Xk
+˘
+,
+where T pλq “ BAE{Bλ (see (2.8)). Let pF el “ pFel,1, Fel,2, Fel,3qT. Notice that we
+can write λ in terms of X,
+(2.11)
+λppxq2 “ |∇S2Xppxq|2,
+which can be seen from their definitions
+λ2 “ trppg´1gq “ pgijgji,
+|∇S2X|2 “ ∇S2Xk ¨ ∇S2Xk “ BXk
+Bxi
+BXk
+Bxl
+pgijpglmpgjm “ gilpgil.
+When µ “ 1, we may write the evolution of X as
+BX
+Bt ppxq “
+ż
+S2 GpXppxq ´ XppyqqpF elppyqdµS2ppyq
+“
+ż
+S2GpXppxq ´ Xppyqq∇S2 ¨
+´
+T p|∇S2Xppyq|q ∇S2Xppyq
+|∇S2Xppyq|
+¯
+dµS2ppyq,
+and integrating by parts, we obtain
+(2.12)
+BX
+Bt ppxq“´p.v.
+ż
+S2∇S2GpXppxq´Xppyqq¨T p|∇S2Xppyq|q ∇S2Xppyq
+|∇S2Xppyq|dµS2ppyq.
+In the following, we will suppress the principal value notation. Introducing a smooth
+partition of unity tρnu, subordinate to a finite atlas of the sphere tUnu, we may
+write our problem as follows
+BX
+Bt ppxq“´
+ÿ
+n
+ż
+S2∇S2GpXppxq´Xppyqq¨ T p|∇S2Xppyq|q
+|∇S2Xppyq|
+∇S2`
+ρnppyqXppyq
+˘
+dµS2ppyq.
+This can be rewritten using the local charts:
+BX
+Bt ppxq “ ´
+ÿ
+n
+ż
+Un
+pgijpηq B
+Bηi
+GpXppxq´Xnpηqq
+ˆ pgjmpηqT p
+a
+pgqrgrqpηqq
+a
+pgqrgrqpηq
+pgpmpηq B
+Bηp
+`
+ρnpηqXnpηq
+˘a
+detpgpηqdη1dη2,
+
+12
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+where Xnpηq is the coordinate map on the n-th coordinate chart and ρnpηq “
+ρnpx
+Xpηqq (see Section 3.1 for details of notation). We may take an isothermal
+coordinate system (the stereographic projection gives such a system, for example)
+on each chart Un, which yields:
+BX
+Bt ppxq “ ´
+ÿ
+n
+ż
+Un
+B
+Bηi
+GpXppxq´XnpηqqT pλnpηqq
+λnpηq
+B
+Bηi
+`
+ρnpηqXnpηq
+˘
+dη1dη2,
+(2.13)
+where we denote
+(2.14)
+λnpηq “
+a
+trppg´1pηqgpηqq “
+?
+2}∇Xnpηq}F }∇x
+Xnpηq}´1
+F ,
+and }A}F :“
+a
+trpAT Aq is the Frobenius norm.
+3. Preliminaries
+In this section we introduce the notations that will be used in the rest of the
+paper and summarize some standard results about stereographic projection charts
+for the sphere.
+3.1. Notations. Einstein notation over repeated indices will be of constant use.
+Given vectors v, w and matrices A, B, C with the same size, we denote
+|v| :“ ∥v∥ “ ?vivi,
+∥A∥ :“ sup
+|v|ą0
+|Av|
+|v| “ sup
+|v|“1
+|Av| ,
+A : B :“tr
+`
+ATB
+˘
+“ AijBij,
+|A| :“ ∥A∥F :“
+?
+A : A,
+v b w :“vwT ,
+pA b Bqijkl :“AijBkl,
+ppA b Bq Cqij :“AijBklCkl “ pB : Cq Aij.
+We will denote C‚,j to the vector given by the jth column of C, and Cj,‚ to the
+one given by the jth row.
+We will denote µS2 the standard measure on the unit sphere, and for simplicity
+we will write dpy instead of dµS2ppyq.
+We will write high partial derivatives in Rk by multi-index α, where multi-index
+α is a sequence of k nonegative integers. i.e. α “ pα1, α2, ¨ ¨ ¨ , αkq P Nk
+0, where
+N0 “ t0u Ş N.
+Definition 3.1 (Multi-index). Given α, β P Nk
+0, we have the following arithmetic
+about the multi-index.
+(i)
+|α| “α1 ` α2 ` ¨ ¨ ¨ ` αk
+α! “α1!α2! . . . αk!
+α ` β “ pα1 ` β1, α2 ` β2, ¨ ¨ ¨ , αk ` βkq
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+13
+(ii) We set α ď β, which is αi ď βi for all i “ 1, 2, ¨ ¨ ¨ , k. Then, we have
+α ´ β “ pα1 ´ β1, α2 ´ β2, ¨ ¨ ¨ , αk ´ βkq
+ˆ
+α
+β
+˙
+“
+α!
+pα ´ βq!β! “
+α1!
+pα1 ´ β1q!β1! ¨ ¨ ¨
+αk!
+pαk ´ βkq!βk!
+High partial derivatives can be written as
+Bα
+x f pxq :“ Bα1
+Bxα1
+1
+Bα2
+Bxα2
+2
+¨ ¨ ¨ Bαk
+Bxαk
+k
+f pxq
+(3.1)
+where α :“ pα1, α2, ¨ ¨ ¨ , αkq and |α| “ α1 ` ¨ ¨ ¨ ` αk is the total number of deriva-
+tives.
+We will use the following set of non-singular matrices
+(3.2)
+DAσ1,σ2 :“ tA : @ξ ‰ 0, σ2 |ξ| ď |Aξ| ď σ1 |ξ|u
+where σ1 ě σ2 ą 0.
+Euclidean balls of radius R centered at px P Rn will be denoted by Bpx,R, and for
+balls centered at the origin we will also denote BpRq :“ B0,R.
+We will denote Xppx; tq : S2 ÞÑ Γptq the deformation map that describes the
+evolving membrane, and we will omit the dependence on time for simplicity of
+notation, Xppxq. We will consider a finite atlas tUn, x
+Xnu of the sphere with 0 P Un,
+such that the coordinate functions x
+Xnpθq : Un Ă R2 Ťt8u ÞÑ S2 satisfy
+(3.3)
+Bx
+Xnpθq
+Bθ1
+¨ Bx
+Xnpθq
+Bθ2
+“ 0,
+�����
+Bx
+Xnpθq
+Bθ1
+����� “
+�����
+Bx
+Xnpθq
+Bθ2
+����� .
+In particular, we will choose the standard stereographic coordinates. We set tρnu
+to be a smooth partition of unity subordinate to the coordinate patches tUnu. For
+convenience in the definition of H¨older continuity, we take our system tUn, x
+Xn, ρnu
+satisfying the following properties with some R, δ ą 0.
+Definition 3.2 (System tUn, x
+Xn, ρnu). Given R ą 2δ ą 0, we set our isothermal
+coordinate charts tUnu with the coordinate functions tx
+Xnpθqu and the partition
+tρnu to have the following properties:
+i) Set pxn “ x
+Xn p0q, then S2 Ă Ť
+n Bpxn,R, and there exists 0 ă Rn ă 8 s.t
+Bpxn,4R X S2 Ă x
+Xn pB0,Rnq Ă x
+XnpUnq
+ii) @px P S2, Dn s.t.
+x
+Xn pθq “ px
+for some θ P B0,Rn, and pBpx,2δ X S2q Ă x
+Xn pB0,Rnq .
+iii) 0 ď ρn ď 1,
+ρn P C8pS2q,
+supp pρnq Ă Bpxn,2R X S2.
+iv) @px P S2, ř
+n ρn ppxq “ 1.
+Remark 3.3. If |x
+Xn pθq´ x
+Xn pηq | ě C |θ ´ η| on Un, then Un is totally bounded.
+Given f ppxq : S2 Ñ R, we will denote fn pθq : Un Ă R2 Ñ R with fnpθq :“
+fpx
+Xnpθqq. Analogously, we will denote Xnpθq “ Xpx
+Xnpθqq.
+Remark 3.4. If x
+Xn pθq “ x
+Xm pηq, then Xn pθq “ Xm pηq.
+
+14
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Definition 3.5 (H¨older semi-norm).
+�f ppxq�Cγ
+δ pS2q :“ sup
+0ă|px´py|ăδ
+|f ppxq´f ppyq|
+|px ´ py|γ
+“ sup
+n
+sup
+0ă|x
+Xnpθq´x
+Xnpηq|ăδ
+|fn pθq ´ fn pηq|
+|x
+Xn pθq´x
+Xn pηq |γ ,
+�f ppxq�CγpS2q :“
+sup
+0ă|px´py|
+|f ppxq ´ f ppyq|
+|px ´ py|γ
+“
+sup
+x
+Xnpθq‰x
+Xmpηq
+|fn pθq ´ fm pηq|
+|x
+Xn pθq ´ x
+Xm pηq |γ .
+Definition 3.6 (Arc-chord condition).
+|f|˚ :“ inf
+px‰py
+|f ppxq ´ f ppyq|
+|px ´ py|
+“
+inf
+x
+Xnpθq‰x
+Xmpηq
+|fn pθq ´ fm pηq|
+|x
+Xn pθq ´ x
+Xm pηq |
+.
+Definition 3.7 (Locally Arc-chord condition in the Charts). Given Vn Ă Un,
+|f|˝,n :“
+inf
+θ‰η,θ,ηPVn
+|fnpθq ´ fnpηq|
+|θ ´ η|
+,
+|f|˝ :“ inf
+n |f|˝,n .
+Definition 3.8 (Lp norms).
+∥f ppxq∥p
+LppS2q :“
+ÿ
+n
+ż
+Un
+ρn pθq |fn pθq|p a
+det pgndθ
+“
+ÿ
+n
+���pρnq
+1
+p fn
+���
+p
+LppUnq ď
+ÿ
+n
+∥fn pθq∥p
+LppUnq .
+3.2. Standard Stereographic Projection. We will see the properties of the
+standard stereographic projection (i.e. the projection point is p0, 0, 1q). For the
+other projection points, because S2 is centrosymmetric, we just need to rotate the
+coordinates of S2.
+Hence, most properties among the projection charts are the
+same.
+Definition 3.9 (Standard Stereographic Projection). We set x
+X : R2 Ťt8u Ñ S2
+with
+x
+X pθq “
+˜
+2θ1
+1 ` |θ|2 ,
+2θ2
+1 ` |θ|2 , ��1 ` |θ|2
+1 ` |θ|2
+¸
+,
+x
+X p8q :“ lim
+|θ|Ñ8
+x
+X pθq “ p0, 0, 1q.
+Then,
+θ ppxq “
+ˆ
+px1
+1 ´ px3
+,
+px2
+1 ´ px3
+˙
+, θ p0, 0, 1q “ 8.
+We call the parameterization x
+X the standard stereographic projection.
+We will denote VR the coordinate balls in R2,
+(3.4)
+VR :“ B
+´
+R
+?
+4 ´ R2
+¯
+Ă R2.
+Proposition 3.10. The standard stereographic projection has the following prop-
+erties:
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+15
+i)
+Bx
+Xpθq
+Bθ1
+¨ Bx
+Xpθq
+Bθ2
+“ 0,
+�����
+Bx
+Xpθq
+Bθ1
+����� “
+�����
+Bx
+Xpθq
+Bθ2
+����� “
+2
+1 ` |θ|2 .
+ii) For all R ą 0 and all VR (3.4), x
+XpVRq “ Bp0,0,´1q,R X S2.
+iii) For θ, η P R2,
+|x
+X pθq ´ x
+X pηq | ď 2 |θ ´ η| ,
+(3.5)
+and if θ, η P VR with R ď
+?
+2,
+|x
+X pθq ´ x
+X pηq | ě 2
+π |θ ´ η| .
+(3.6)
+Proof. For iii), set pξ “
+θ´η
+|θ´η|, then
+|x
+X pθq´x
+X pηq | “
+ˇˇˇ
+ż |θ´η|
+0
+B
+Bs
+x
+Xpη`spξqds
+ˇˇˇ ď
+ż |θ´η|
+0
+|∇x
+Xpη`spξq ¨ pξ|ds
+“
+ż |θ´η|
+0
+2
+1`|η ` spξ|2 ds ď 2 |θ ´ η| .
+Set distppx, py; S2q as the length of shortest curve connecting px and py on S2. When
+θ, η P VR with R ď
+?
+2, the shortest curve ℓ for distpx
+X pθq , x
+X pηq ; S2q is on
+Bp0,0,´1q,R X S2 and
+2
+π ď
+R
+?
+4 ´ R2
+2 cos´1 2´R2
+2
+ď
+|x
+X pθq ´ x
+X pηq |
+distpx
+X pθq , x
+X pηq ; S2q
+ď 1,
+where the above function of R is a decreasing function. Then, since x
+X is isothermal
+and x
+X
+´1 pℓq is in VR,
+distpx
+X pθq , x
+X pηq ; S2q “
+ż
+ℓ
+dl ppxq “
+ż
+x
+X
+´1pℓq
+2
+1 ` |θ|2 dl pθq
+ě4 ´ R2
+2
+ż
+x
+X
+´1pℓq
+dl pθq ě 4 ´ R2
+2
+|θ ´ η| .
+Therefore
+|x
+X pθq ´ x
+X pηq | ě 2
+π dist
+´
+x
+X pθq , x
+X pηq ; S2¯
+ě 2
+π |θ ´ η| .
+□
+3.3. The Quantitative Relationships between S2 and R2 on the Standard
+Stereographic Projection Chart. Given a function f ppxq on S2, we may define
+f pθq :“ fpx
+X pθqq on R2. x
+X pθq is isothermal, but the chart is neither isometric nor
+area-preserving. Therefore, some quantities of f between S2 and R2 are different.
+We have to check their quantitative relationships.
+First, let see the H¨older continuous seminorm Cγ and the arc-chord condition.
+Proposition 3.11. Given f on S2, it holds that �f�CγpR2q ď 2γ�f�CγpS2q.
+
+16
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Proof.
+�f�CγpR2q “ sup
+θ‰η
+´|x
+X pθq ´ x
+X pηq |
+|θ ´ η|
+¯γ |fpx
+X pθqq ´ fpx
+X pηqq|
+|x
+X pθq ´ x
+X pηq |γ
+ď 2γ�f�CγpS2q.
+□
+Notice that |f|˝ for R2 is zero, and the other sided inequality between Cγ pUq
+and Cγ pUq cannot hold with some constant C ą 0, thus we only can find local
+inequalities. Set BR “ Bp0,0,´1q,R X S2, VR as in (3.4), and ρ smooth on S2 and
+supported in BR.
+Proposition 3.12. Given R ď
+?
+2, it holds that
+�ρf�CγpS2q ď
+`π
+2
+˘γ�ρf�CγpR2q,
+|f|˚ ďπ
+2 |f|˝ .
+Proof. For �ρf�CγpS2q, when px “ x
+X pθq , py “ x
+X pηq P BR, θ, η P VR, so
+|ρf ppxq ´ ρf ppyq|
+|px ´ py|γ
+“
+´
+|θ ´ η|
+|x
+X pθq ´ x
+X pηq |
+¯γ |ρf pθq ´ ρf pηq|
+|θ ´ η|γ
+ď
+`π
+2
+˘γ�ρf�CγpR2q.
+Next, when px “ x
+X pθq , py “ x
+X pηq P Bc
+R, |ρfppxq´ρfp pyq|
+|px´ py|γ
+“ 0. Finally, when px “
+x
+X pθq P BR, py “ x
+X pηq P Bc
+R, θ P VR, η P V c
+R, set pz “ x
+X pξq P BBR s.t. |px ´ pz| “
+dist ppx, BBRq. Since ρ is smooth on S2 and supported in BR, ρ ppzq “ 0 “ ρ ppyq.
+Then,
+|θ ´ ξ| ď π
+2 |px ´ pz| ď π
+2 |px ´ py| ,
+so
+|ρf ppxq ´ ρf ppyq|
+|px ´ py|γ
+“ |ρf ppxq ´ ρf ppzq|
+|px ´ py|γ
+“
+´
+|θ ´ ξ|
+|x
+X pθq ´ x
+X pηq |
+¯γ |ρf pθq ´ ρf pξq|
+|θ ´ ξ|γ
+ď
+`π
+2
+˘γ�ρf�CγpR2q.
+Now, for |f|˝,
+|f|˝ “
+inf
+θ‰η,θ,ηPV
+|x
+X pθq ´ x
+X pηq |
+|θ ´ η|
+|fpx
+X pθqq ´ fpx
+X pηqq|
+|x
+X pθq ´ x
+X pηq |
+ě 2
+π |f|˚ .
+□
+Next, let us discuss the relationship between C1,γ `
+S2˘
+and C1,γ `
+R2˘
+on the
+standard stereographic projection chart. In the standard stereographic projection
+chart px “ x
+X pθq, the surface gradient of f, ∇S2f ppxq, is
+∇S2f ppxq “
+ÿ
+i,j
+pgi,j B
+Bθi
+f pθq B
+Bθj
+x
+X pθq “
+´1 ` |θ|2
+2
+¯2 ÿ
+i
+B
+Bθi
+f pθq B
+Bθi
+x
+X pθq ,
+where pgij denotes the inverse tensor of pg. Hence,
+2
+1 ` |θ|2
+���∇S2fpx
+Xpθqq
+��� “ |∇f pθq| .
+We may use the above expressions to obtain the following proposition.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+17
+Proposition 3.13. There exist R0 ą 0 and C ě 21`γ s.t. for all R ă R0
+∥f∥C1,γpR2q ďC ∥f∥C1,γpS2q ,
+∥ρf∥C1,γpS2q ď ∥ρf∥C1,γpR2q ,
+|f|˚ ď |f|˝ ,
+where ρ is smooth on S2 and supported in BR.
+3.4. Stereographic Projection Charts x
+Xn Covering S2. We consider a finite
+cover of the sphere consisting of balls of radius R ă R0 and center pxn, tBpxn,RXS2u,
+and a smooth partition of unity subordinated to it, tρnu, such that ρnppxq “ 0 if
+|px´ pxn| ě 2R. For convenience, we set px0 “ p0, 0, ´1q. Then we take stereographic
+projection charts x
+Xn covering S2 (see Figure 2).
+Figure 2. Stereographic projection charts, x
+Xnpθq.
+Definition 3.14 (Stereographic Projection Charts covering S2). x
+Xn are standard
+stereographic projection charts with
+x
+Xn : R2 Ñ S2
+x
+Xnpθq “ Θnx
+Xpθq
++
+,
+where Θn is the rotation matrix with pxn “ Θnpx0.
+Proposition 3.15. In each chart x
+Xn, given R ă R0
+4 and R0, C from Proposition
+3.13, since ρ are supported in Bn,
+∥f∥C1,γpR2q ď C ∥f∥C1,γpS2q ,
+∥ρnf∥C1,γpS2q ď ∥ρnf∥C1,γpR2q ,
+|f|˚ ď |f|˝,n .
+
+18
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+As a consequence, |f|˚ ď |f|˝.
+4. Nonlinear decomposition
+In this section we extract the leading structure of the equation and compute
+its symbol, introducing the notation for the operators that will appear along the
+paper.
+4.1. Nonlinear decomposition. We will usually consider separately the two terms
+involved in the Stokeslet kernel (2.10),
+(4.1)
+Gk,lpxq “ G1
+k,lpxq ` G2
+k,lpxq,
+G1
+k,lpxq “ 1
+8π
+δk,l
+|x| ,
+G2
+k,lpxq “ 1
+8π
+xkxl
+|x|3 ,
+and thus we will write our equation (2.12) as follows:
+(4.2)
+BX
+Bt ppxq “ FpXqppxq “ F 1pXqppxq ` F 2pXqppxq,
+where
+(4.3)
+FpXqppxq “ ´
+ż
+S2 ∇S2GpXppxq ´ Xppyqq ¨ T p|∇S2Xppyq|q∇S2Xppyqdpy,
+F jpXqppxq “ ´
+ż
+S2 ∇S2GjpXppxq ´ Xppyqq ¨ T p|∇S2Xppyq|q∇S2Xppyqdpy.
+and we introduced the notation
+(4.4)
+T p|∇S2X|q “ T p|∇S2X|q
+|∇S2X|
+.
+Above, we use the shorter notation dpy “ dµS2ppyq. We define the following associate
+linear operators,
+(4.5)
+pNpXqZqkppxq “ ´
+ż
+S2 ∇S2GklpXppxq ´ Xppyqq ¨ Zl,‚ppyqdpy,
+pN jpXqZqkppxq “ ´
+ż
+S2 ∇S2Gj
+klpXppxq ´ Xppyqq ¨ Zl,‚ppyqdpy.
+Then, we compute the kernels:
+(4.6)
+B
+Bxi
+G1
+k,lpxq “ ´1
+8π
+xi
+|x|3 δk,l,
+B
+Bxi
+G2
+k,lpxq “ 1
+8π
+δk,ixl ` xkδi,l
+|x|3
+´ 3
+8π
+xkxlxi
+|x|5 ,
+and by the chain rule,
+(4.7)
+qj
+k,lppx, pyq : “ ∇S2Gj
+k,lpXppxq ´ Xppyqq
+“ ´ B
+Bxi
+Gj
+k,lpXppxq ´ Xppyqq∇S2Xippyq.
+so that we write
+(4.8)
+pN jpXqZqkppxq “ ´
+ż
+S2 qj
+k,lppx, pyq ¨ Zl,‚ppyqdpy.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+19
+The explicit expression for qj
+k,l is given by
+q1
+k,lppx, pyq “ 1
+8π
+∆ pyXjppxq∇S2Xjppyq
+|∆ pyXppxq|3
+δk,l
+|px ´ py|2 ,
+and
+q2
+k,lppx, pyq “ ´ 1
+8π
+∆ pyXlppxq∇S2Xkppyq ` ∆ pyXkppxq∇S2Xlppyq
+|∆ pyXppxq|3
+1
+|px ´ py|2
+` 3
+8π
+∆ pyXkppxq∆ pyXlppxq∆ pyXjppxq∇S2Xjppyq
+|∆ pyXppxq|5
+1
+|px ´ py|2 .
+Using the standard stereographic projection (see Section 3.1) and the notation
+Xpθq “ Xpx
+Xpθqq, the equation for each component of Xpθq becomes
+BXk
+Bt pθq “ pFpXqqkpθq
+“ ´
+ż
+R2
+B
+Bηi
+Gk,lpXpθq´XpηqqT pλpηqqBXl
+Bηi
+pηqdη1dη2,
+where λpηq is given in (2.14) and we denote accordingly F jpXqpθq, N jpXqZpθq. If
+we use the stereographic projection centered at pxn, then we denote N jpXqZnpθq.
+Then, we take the derivative in Gk,l (4.1) to obtain
+B
+Bηi
+Gk,lpXpθq´Xpηqq “ qi,k,lpθ, ηq
+“ q1
+i,k,lpθ, ηq ` q2
+i,k,lpθ, ηq,
+where
+(4.9)
+q1
+i,k,lpθ, ηq “
+B
+Bηi
+G1
+k,lpXpθq´Xpηqq “ 1
+8πδk,l
+δηXpθq ¨ BX
+Bηi pηq
+|δηXpθq|3
+,
+q2
+i,k,lpθ, ηq “
+B
+Bηi
+G2
+k,lpXpθq´Xpηqq
+“ ´ 1
+8π
+BXk
+Bηi pηqδηXlpθq ` δηXkpθq BXl
+Bηi
+|δηXpθq|3
+` 3
+8π
+δηXkpθqδηXlpθq
+|δηXpθq|5
+δηXpθq ¨ BXpηq
+Bηi
+,
+so that we can write
+(4.10)
+pN jpXqZqkpθq “ ´
+ż
+R2 qj
+m,k,lpθ, ηq B p
+Xi
+Bηm
+pηqZl,ipηqdη1dη2.
+We notice that the kernels in (4.9) are given by
+(4.11)
+qj
+i,k,lpθ, ηq “ ´ B
+Bxm
+Gj
+k,lpXpθq ´ XpηqqBXm
+Bηi
+pηq.
+We introduce the following notation for finite differences,
+(4.12)
+δηgpθq “ gpθq ´ gpηq,
+∆ηgpθq “ δηgpθq
+|θ ´ η|,
+and we extract the expected leading terms by replacing
+δηXpθq « ∇Xpηqpθ ´ ηq,
+p∇Xqp,qpηq “ BXn,p
+Bηq
+pηq.
+
+20
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Hence, we define the associate kernels
+(4.13)
+mi,k,lpθ, ηq “ ´ B
+Bxj
+Gk,l p∇Xpηqpθ ´ ηqq BXj
+Bηi
+pηq
+“ m1
+i,k,lpθ, ηq ` m2
+i,k,lpθ, ηq,
+and define the linear operators Mp∇Xqz as follows:
+(4.14)
+pMp∇XqZqkpθq “ ´
+ż
+R2 mm,k,lpθ, ηq B p
+Xi
+Bηm
+pηqZl,ipηqdη1dη2
+“ pM1p∇XqZqkpθq ` pM2p∇XqZqkpθq,
+with
+pMjp∇XqZqkpθq “ ´
+ż
+R2 mj
+m,k,lpθ, ηq B p
+Xi
+Bηm
+pηqZl,ipηqdη1dη2.
+We compute the explicit expression of these kernels mj
+i,k,l (4.13),
+m1
+i,k,lpθ, ηq “ 1
+8π
+BXpηq
+Bηi
+¨ p∇Xpηqpθ´ηqq
+|∇Xpηqpθ´ηq|3
+,
+m2
+i,k,lpθ, ηq “ ´ 1
+8π
+BXkpηq
+Bηi
+p∇Xpηqpθ´ηqql ` BXlpηq
+Bηi
+p∇Xpηqpθ´ηqqk
+|∇Xpηqpθ´ηq|3
+` 3
+8π
+p∇Xpηqpθ´ηqqkp∇Xpηqpθ´ηqql
+|∇Xpηqpθ´ηq|5
+p∇Xpηqpθ´ηqq ¨ BXpηq
+Bηi
+.
+We will use the notation
+∇Xpηqpθ ´ ηq “ pθ ´ ηq ¨ ∇Xpηq,
+pz “ z
+|z|,
+and we define
+(4.15)
+EηXpθq :“ p{
+θ ´ ηq ¨ ∇Xpηq ´ ∆ηXpθq,
+for which we have that,
+(4.16)
+|EηpXpθqq|
+|θ ´ η|γ
+ď �∇X�CγpR2q.
+Thus, we can write
+(4.17)
+NpXqZpθq “ Mp∇XqZpθq ` RpXqZpθq,
+where the remainder term
+RpXqZpθq “
+2ÿ
+j“1
+RjpXqZpθq
+is given by
+(4.18)
+pRjpXqZqqkpθq “ pN jpXqZqkpθq ´ pMjp∇XqpZqqkpθq
+“
+ż
+R2 Kj
+m,k,lpθ, ηq B p
+Xi
+Bηm
+pηqZl,ipηqdη1dη2,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+21
+with kernels
+K1
+i,k,lpθ, ηq “ ´q1
+i,k,l pθ, ηq ` m1
+i,k,l pθ, ηq
+“ 1
+8π
+δk,l
+|θ ´ η|2
+BXpηq
+Bηi
+¨
+ˆ EηXpθq
+|∆ηXpθq|3
+´ pp{
+θ ´ ηq ¨ ∇Xpηqq
+´
+1
+|∆ηXpθq|3 ´
+1
+|p{
+θ ´ ηq ¨ ∇Xpηq|3
+¯˙
+,
+and
+K2
+i,k,lpθ, ηq “ ´q2
+i,k,l pθ, ηq ` m2
+i,k,l pθ, ηq “ K2,1
+i,k,lpθ, ηq ` K2,2
+i,k,lpθ, ηq,
+K2,1
+i,k,lpθ, ηq “ 1
+8π
+1
+|θ ´ η|2
+ˆ ´ BXkpηq
+Bηi
+|∆ηXpθq|3 EηXlpθq
+` BXkpηq
+Bηi
+pp{
+θ ´ ηq ¨ ∇Xpηqql
+´
+1
+|∆ηXpθq|3 ´
+1
+|p{
+θ ´ ηq ¨ ∇Xpηq|3
+¯
+´
+BXlpηq
+Bηi
+|∆ηXpθq|3 EηXkpθq
+` BXlpηq
+Bηi
+pp{
+θ ´ ηq ¨ ∇Xpηqqk
+´
+1
+|∆ηXpθq|3 ´
+1
+|p{
+θ ´ ηq ¨ ∇Xpηq|3
+¯˙
+,
+K2,2
+i,k,lpθ, ηq “ 1
+8π
+3
+|θ ´ η|2
+ˆ∆ηXkpθq∆ηXlpθq
+|∆ηXpθq|5
+BXpηq
+Bηi
+¨ EηXpθq
+`
+BXpηq
+Bηi
+¨ pp{
+θ ´ ηq ¨ ∇Xpηqq
+|∆ηXpθq|5
+∆ηXkpθqEηXlpθq
+`
+BXpηq
+Bηi
+¨ pp{
+θ ´ ηq ¨ ∇Xpηqq
+|∆ηXpθq|5
+pp{
+θ ´ ηq ¨ ∇XpηqqlEηXkpθq
+´
+BXpηq
+Bηi
+¨ pp{
+θ ´ ηq ¨ ∇Xpηqq
+|∆ηXpθq|5
+pp{
+θ ´ ηq ¨ ∇Xpηqqlpp{
+θ ´ ηq ¨ ∇Xpηqqk
+ˆ
+´
+1
+|∆ηXpθq|5 ´
+1
+|p{
+θ ´ ηq ¨ ∇Xpηq|5
+¯˙
+.
+Remark 4.1. Note that for all positive odd integers k, it holds that
+1
+|u|k ´
+1
+|v|k “ pv ´ uq ¨ pu ` vq
+|u|k ` |v|k
+řk
+i“1 |u|2pi´1q|v|2pk´iq
+|u|k|v|k
+(4.19)
+and
+����
+1
+|u|k ´
+1
+|v|k
+���� “ ||v| ´ |u|| řk
+i“1 |u|i´1 |v|k´i
+|u|k |v|k
+ď |v ´ u| řk
+i“1 |u|i´1 |v|k´i
+|u|k |v|k
+(4.20)
+In particular, formula (4.19) with u “ ∆ηXpθq and v “ p{
+θ ´ ηq ¨ ∇Xpηq,
+together with (4.15)-(4.16), makes it clear that there is an extra cancellation in the
+kernels of (4.18),
+
+22
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+In summary, our equation (4.2) is given by
+BX
+Bt pθq “ FpXqpθq
+“ NpXqpT p|∇S2X|q∇S2Xqpθq
+“ Mp∇XqpT p|∇S2X|q∇S2Xqpθq ` RpXqpT p|∇S2X|q∇S2Xqpθq.
+4.2. Symbol of the leading term. As a preliminary step towards studying the
+leading term M (4.14), let us consider its frozen-coefficient counterpart, i.e., re-
+placing ∇Xpηq by a constant matrix A and letting pg “ I2. We start with the case
+T “ Id, that is, T ” 1:
+(4.21)
+pLL
+AY qkpθq “ p ˜
+MpAq∇Y qkpθq,
+where we define
+˜
+MpAq “ ˜
+M1pAq ` ˜
+M2pAq and
+(4.22)
+p ˜
+MjpAqZqkpθq “ ´
+ż
+R2
+B
+Bηi
+pGj
+k,lpA pθ ´ ηqqqZl,ipηqdη1dη2.
+Let Fθ be the 2D Fourier transform in θ and ξ “ pξ1, ξ2qT:
+Fθwpξq “
+ż
+R2 wpθq expp´iθ ¨ ξqdθ.
+We now compute the Fourier transform of the function GA:
+GApθq “ GpAθq “ 1
+8π
+˜
+I
+|Aθ| ` Aθ b Aθ
+|Aθ|3
+¸
+,
+where I3 is the 3 ˆ 3 identity matrix. Given that θ P R2 and A is a 3 ˆ 2 matrix,
+it is convenient to rewrite GA as follows. First, note that:
+|Aθ|2 “ Aθ ¨ Aθ “ θ ¨
+`
+ATAθ
+˘
+“ |Bθ|2 , B “
+?
+ATA.
+Notice that B is a 2 ˆ 2 symmetric positive definite matrix. Using this B, we have:
+(4.23)
+GApθq “ 1
+8π
+˜
+I
+|Bθ| ` Q
+˜
+Bθ b Bθ
+|Bθ|3
+¸
+QT
+¸
+, Q “ AB´1.
+We note that Q is an isometry in the sense that QTQ “ I2 where I2 is the 2 ˆ 2
+identity matrix. We are now ready to compute the Fourier transform of GA. First,
+note that:
+(4.24)
+Fθ
+ˆ 1
+|θ|
+˙
+“ 2π
+|ξ|.
+Thus, a simple change of variable yields:
+(4.25)
+Fθ
+ˆ
+1
+|Bθ|
+˙
+“
+2π
+detpBq |B´1ξ|.
+Next, note that:
+Fθ
+˜
+θiθj
+|θ|3
+¸
+“ Fθ
+ˆ
+θiθj∆θ
+ˆ 1
+|θ|
+˙˙
+“
+B2
+BξiBξj
+ˆ
+|ξ|2 Fθ
+ˆ 1
+|θ|
+˙˙
+“ 2π
+B2
+BξiBξj
+|ξ| “ 2π
+˜
+δij
+|ξ| ´ ξiξj
+|ξ|3
+¸
+,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+23
+where ∆θ is the Laplacian in R2, δij is the Kronecker delta and we used (4.24) in
+the third equality. In matrix notation, the above can be written as:
+Fθ
+˜
+θ b θ
+|θ|3
+¸
+“ 2π
+˜
+I2
+|ξ| ´ ξ b ξ
+|ξ|3
+¸
+.
+Again, by changing variables, we see that:
+(4.26)
+Fθ
+˜
+Bθ b Bθ
+|Bθ|3
+¸
+“
+2π
+detpBq
+˜
+I2
+|B´1ξ| ´ B´1ξ b B´1ξ
+|B´1ξ|3
+¸
+.
+Using (4.26), (4.24) and (4.23), we obtain:
+FθGA “
+1
+4detpBq
+˜
+I ` QQT
+|B´1ξ|
+´ QB´1ξ b QB´1ξ
+|B´1ξ|3
+¸
+“
+1
+4
+a
+detpATAq
+˜
+I ` ApATAq´1AT
+pξ ¨ pATAq´1ξq1{2 ´ ApATAq´1ξ b ApATAq´1ξ
+pξ ¨ pATAq´1ξq3{2
+¸
+This implies that the Fourier symbol of LL
+A is given by:
+LL
+AY “ ´F´1
+ξ LL
+ApξqFθY , LL
+Apξq “ |ξ|2 pFθGAq pξq.
+To better understand the properties of Fourier multiplier LL
+Apξq, we first note that:
+QQT ´ QB´1ξ b QB´1ξ
+|B´1ξ|2
+“ Q
+˜
+I2 ´ B´1ξ b B´1ξ
+|B´1ξ|2
+¸
+QT “ vpξq b vpξq,
+vpξq “ QRπ{2
+B´1ξ
+|B´1ξ|, Rπ{2 “
+ˆ
+0
+´1
+1
+0
+˙
+.
+(4.27)
+Note that vpξq P R3 is a unit vector, and hence, the above matrix 3 ˆ 3 matrix is
+an orthogonal projection on to the subspace spanned by vpξq. We see that:
+(4.28)
+LL
+Apξq “
+|ξ|2
+4detpBq |B´1ξ| pI ` vpξq b vpξqq .
+It is now immediate that LL
+Apξq is a symmetric positive definite matrix for each
+ξ ‰ 0 with eigenvalues:
+(4.29)
+λ “ µ
+4 |ξ|2 and µ
+2 |ξ|2 ,
+µ “
+1
+detpBq |B´1ξ|,
+where the eigenspace for µ{2 is spanned by vpξq and the two-dimensional eigenspace
+of µ{4 is spanned by the orthogonal complement of vpξq. We also have:
+(4.30)
+µ
+4 |ξ|2 |w|2 ď w ¨ Lpξqw ď µ
+2 |ξ|2 |w|2
+for any w P R3.
+In the case of general T , the frozen coefficient linear operator can be obtained
+by a further linearization of the force function. Consider the expression:
+T pλτq
+λτ
+BpXl ` τYlq
+Bθi
+, λτ “ λpX ` τY q
+
+24
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+where λ is here viewed as a function of X through its dependence on g (see (2.7)).
+Now,
+d
+dτ
+ˆT pλτq
+λτ
+BpXl ` τYlq
+Bθi
+˙ˇˇˇˇ
+τ“0
+“
+ˆ 1
+λ
+dT
+dλ ´ T
+λ2
+˙ dλτ
+dτ
+ˇˇˇˇ
+τ“0
+BXl
+Bθi
+` T
+λ
+BYl
+Bθi
+“
+ˆ 1
+λ
+dT
+dλ ´ T
+λ2
+˙ 1
+λppg´1qm,n
+BXq
+Bθm
+BYq
+Bθn
+BXl
+Bθi
+` T
+λ
+BYl
+Bθi
+.
+Now, the frozen coefficient approximation amounts to taking pg “ I2, BXl{Bθi “ Al,i
+and λ “ ∥A∥F . Thus,
+d
+dτ
+ˆT pλτq
+λτ
+BpXl ` τYlq
+Bθi
+˙ˇˇˇˇ
+τ“0
+« pTF pAqqi,l,m,q
+BYq
+Bθm
+,
+with
+(4.31)
+TFpAq “ T p∥A∥F q
+∥A∥F
+I2 b I2 ´
+ˆT p∥A∥F q
+∥A∥F
+´ dT
+dλ p∥A∥F q
+˙ A b A
+∥A∥2
+F
+.
+Thus, the frozen-coefficient linear operator in the general force case is given by
+(4.32)
+pLAY qkpθq “ ´
+ż
+R2
+B
+Bηi
+pGk,lpA pθ ´ ηqqqpTF pAq∇Y ql,ipηqdη1dη2.
+Let us now take the Fourier transform of the divergence of the above:
+Fp∇ ¨ pTF pAq∇Y qqpξq “ ´MApξqFY pξq,
+where
+(4.33)
+MApξq “ T p∥A∥F q
+∥A∥F
+˜
+|ξ|2 I ´ Aξ b Aξ
+∥A∥2
+F
+¸
+` dT
+dλ p∥A∥F qAξ b Aξ
+∥A∥2
+F
+.
+Note that, if we set T “ Id, then TF “ Id and the above reduces to Mpξq “ |ξ|2.
+Thus, in the general case, the multiplier in LApξq of (4.32) becomes:
+LApξq “ pFθGAq pξqMApξq
+“ I ` vpξq b vpξq
+4detpBq |B´1ξ|
+ˆT p∥A∥F
+∥A∥F
+ˆ
+|ξ|2 I ´ Aξ b Aξ
+∥A∥F
+˙
+` dT
+dλ p∥A∥F qAξ b Aξ
+∥A∥F
+˙
+.
+(4.34)
+It is not difficult to see that, if T ą 0 and dT {dλ ě 0, then the above is coercive in
+|ξ|2.
+5. Calculus estimates
+In this section we include some estimates of the operators that will be frequently
+used in later sections.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+25
+Lemma 5.1. Let X P C1pS2q, such that |X|˚ ą 0. Then, the kernels Gj
+k,lpxq (4.1)
+and qj
+k,lppx, pyq (4.7) satisfy the following bounds
+(5.1)
+���Bα
+x Gj
+k,lpxq
+��� ď
+C
+|x|1`|α| ,
+���∆y
+´
+Bα
+x Gj
+k,lpxq
+¯��� ď C M 1`|α|
+m3`2|α| ,
+|qj
+k,lppx, pyq| ď C |∇S2Xppyq|
+|∆ pyXppxq|2
+1
+|px ´ py|2 ď C }∇S2X}C0pS2q
+|X|2˚
+1
+|px ´ py|2 ,
+where |α| defined in (3.1), M “ max p|x| , |y|q, and m “ min p|x| , |y|q.
+For the sake of completeness, we include a version of the divergence theorem
+that will be used. Notice that, following standard convention, we will not explicitly
+write the principal values elsewhere.
+Lemma 5.2. Given a matrix A and a compact set D Ă R2 containing 0, then
+p.v.
+ż
+Dc ∇GpAηqdη : “ lim
+LÑ8
+ż
+DcXBpLq
+∇GpAηqdη
+“ ´
+ż
+BD
+GpAηqn pηq dl pηq ,
+where B pLq Ă R2 is the ball centered at 0 of radius L. In particular,
+p.v.
+ż
+R2 ∇GpAηqdη :“
+lim
+LÑ8,εÑ0
+ż
+BpLqzBpεq
+∇GpAηqdη “ 0.
+Proof. Since D is compact and contains 0, D Ă B pLq when L is large enough.
+Then, by integration by parts
+ż
+DcXBpLq
+∇GpAηqdη “ ´
+ż
+BD
+GpAηqn pηq dl pηq `
+ż
+BBpLq
+GpAηqn pηq dl pηq .
+Since GpAηq is even, the boundary term vanishes. Therefore,
+ż
+Dc ∇GpAηqdη “ lim
+LÑ8
+ż
+DcXBpLq
+∇GpAηqdη “ ´
+ż
+BD
+GpAηqn pηq dl pηq .
+Next, set D “ B pεq,
+ż
+R2 ∇GpAηqdη “
+lim
+LÑ8,εÑ0
+ż
+BpLqXBpεqc ∇GpAηqdη “ 0.
+□
+Lemma 5.3. Let A be a matrix in the set DAσ1,σ2. Then, the linear operator
+˜
+MpAq (4.22) maps CγpR2q X L2pR2q to CγpR2q X L2pR2q for any any γ P p0, 1q.
+Moreover,
+} ˜
+MjpAqZ}CγpR2q ď C
+σ2
+´
+1 `
+´σ1
+σ2
+¯2¯
+}Z}CγpR2qXL2pR2q.
+And given A1, A2 P DAσ1,σ2
+} ˜
+MjpA1qZ ´ ˜
+MjpA2qZ}CγpR2q ď C
+σ2
+2
+´
+1 `
+´σ1
+σ2
+¯5¯
+}Z}CγpR2qXL2pR2q ∥A1 ´ A2∥ .
+
+26
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Proof. Taking into account Lemma 5.2, we have
+p ˜
+MjpAqZqkpθq “ ´
+ż
+R2
+B
+Bηm
+Gj
+k,lpApθ ´ ηqq
+`
+Zl,mpηq ´ Cl,m
+˘
+dη1dη2,
+where Cl,m is an arbitrary constant, that we will take to be zero or Zl,mpθq. Then,
+the estimate for |p ˜
+MjpAqZqkpθq| follows by splitting the integral in two terms,
+p ˜
+MjpAqZqkpθq “ I1pθq ` I2pθq,
+with
+I1pθq “ ´
+ż
+|θ´η|ď1
+B
+Bηm
+Gj
+k,lpApθ ´ ηqq
+`
+Zl,mpηq ´ Zl,mpθq
+˘
+dη1dη2,
+I2pθq “ ´
+ż
+|θ´η|ě1
+B
+Bηm
+Gj
+k,lpApθ ´ ηqqZl,mpηqdη1dη2.
+Since
+B
+Bηm
+Gj
+k,lpApθ ´ ηqq “ ´
+BGj
+k,l
+Bxi
+pApθ ´ ηqqAi,m,
+(5.2)
+the kernel bounds (5.1) and the fact that A P DAσ1,σ2 provide that
+|I1pθq| ď C σ1
+σ2
+2
+�Z�CγpR2q,
+|I2pθq| ď C σ1
+σ2
+2
+}Z}L2pR2q,
+hence
+|p ˜
+MjpAqZqkpθq| ď C σ1
+σ2
+2
+}Z}CγpR2qXL2pR2q.
+We proceed with the seminorm. Let h P R2, |h| ď 1, and perform the following
+splitting
+p ˜
+MjpAqZqkpθq ´ p ˜
+MjpAqZqkpθ ` hq “ J1 ` J2 ` J3 ` J4,
+where
+J1 “ ´
+ż
+|θ´η|ď2|h|
+B
+Bηm
+Gj
+k,lpApθ ´ ηqqδηZl,mpθqdη,
+J2 “
+ż
+|θ´η|ď2|h|
+B
+Bηm
+Gj
+k,lpApθ ` h ´ ηqqδηZl,mpθ ` hqdη,
+J3 “ δθZl,mpθ ` hq
+ż
+|θ´η|ą2|h|
+B
+Bηm
+Gj
+k,lpθ ´ ηqdη,
+J4 “
+ż
+|θ´η|ą2|h|
+´ B
+Bηm
+Gj
+k,l pApθ`h´ηqq´
+B
+Bηm
+Gj
+k,lpApθ´ηqq
+¯
+δηZl,mpθ`hqdη.
+Absolutely,
+|J1| ` |J2| ď C σ1
+σ2
+2
+�Z�CγpR2q |h|γ .
+Then, by Lemma 5.2,
+J3 “ pZl,mpθ ` hq ´ Zl,mpθqq
+ż
+|θ´η|“2|h|
+Gj
+k,l pApθ ´ ηqq nmpηqdlpηq,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+27
+and so
+|J3| ď C�Z�CγpR2q |h|γ 1
+σ2
+ż
+|θ´η|“2|h|
+1
+|θ ´ η|dl pηq
+ď C 1
+σ2
+�Z�CγpR2q |h|γ .
+Finally, since
+B
+Bθp
+B
+Bηm
+Gj
+k,lpApθ ´ ηqq “ ´ B
+Bxq
+B
+Bxi
+Gj
+k,lpApθ ´ ηqqAi,mAq,p,
+(5.3)
+it follows that
+|J4| “
+ˇˇˇ
+ż
+|θ´η|ą2|h|
+ż 1
+0
+hp
+B
+Bθp
+B
+Bηm
+Gj
+k,l pApθ ` sh ´ ηqq δηpZl,mpθ ` hqdsdη
+ˇˇˇ
+ď C σ2
+1
+σ3
+2
+�Z�CγpR2q |h|
+ż
+|θ´η|ą2|h|
+ż 1
+0
+|θ ` h ´ η|γ
+|θ ` sh ´ η|3 dsdη.
+In the domain where |θ ´ η| ą 2 |h|, it holds that for s P r0, 1s,
+1
+2 |θ ` h ´ η| ď |θ ` sh ´ η| ď 3
+2 |θ ` h ´ η| .
+Hence,
+|J4| ď C σ2
+1
+σ3
+2
+�Z�CγpR2q |h|γ .
+Therefore, we obtain
+��� ˜
+MjpAqZ
+���
+CγpRq ď C
+σ2
+´
+1 ` σ2
+1
+σ2
+2
+¯
+}Z}CγpR2qXL2pR2q.
+For the L2 norm, since ξmFθ
+”
+Gj
+k,l pAθq
+ı
+pξq is bounded by σ1 and σ2, we have
+���p ˜
+MjpAqZqk
+���
+L2pRq “
+���ξmFθ
+”
+Gj
+k,l pAθq
+ı
+pξq FrZl,ms pξq
+���
+L2pRq
+ď C pσ1, σ2q ∥FrZs∥L2pRq “ C pσ1, σ2q ∥Z∥L2pRq
+(5.4)
+Next, through (5.1), (5.2), and (5.3), we have
+���Gj
+k,lpA1pθ ´ ηqq´Gj
+k,lpA2pθ ´ ηqq
+��� ď C σ1 |pA1´A2q pθ ´ ηq|
+σ3
+2 |θ ´ η|3
+ď C σ1
+σ3
+2
+∥pA1´A2q∥
+|θ ´ η|
+,
+ˇˇˇ B
+Bηm
+Gj
+k,lpA1pθ´ηqq´ B
+Bηm
+Gj
+k,lpA2pθ´ηqq
+ˇˇˇ
+ď
+ˇˇˇ
+BGj
+k,l
+Bxi
+pA1pθ´ηqq´
+BGj
+k,l
+Bxi
+pA2pθ´ηqq
+ˇˇˇ|A1,i,m|
+`
+ˇˇˇ
+BGj
+k,l
+Bxi
+pA2pθ´ηqq pA1,i,m´A2,i,mq
+ˇˇˇ
+ď C
+´ 1
+σ2
+2
+` σ3
+1
+σ5
+2
+¯∥pA1 ´ A2q∥
+|θ ´ η|2
+,
+
+28
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+and
+ˇˇˇ B
+Bθp
+B
+Bηm
+Gj
+k,lpA1pθ´ηqq´ B
+Bθp
+B
+Bηm
+Gj
+k,lpA2pθ´ηqq
+ˇˇˇ
+ď
+ˇˇˇ B
+Bxq
+B
+Bxi
+Gj
+k,lpA1pθ´ηqq´ B
+Bxq
+B
+Bxi
+Gj
+k,lpA2pθ´ηqq
+ˇˇˇ|A1,i,mA1,q,p|
+`
+ˇˇˇ B
+Bxq
+B
+Bxi
+Gj
+k,lpA2pθ´ηqq pA1,i,mA1,q,p´A2,i,mA2,q,pq
+ˇˇˇ
+ď C
+´σ1
+σ3
+2
+` σ5
+1
+σ7
+2
+¯∥pA1´A2q∥
+|θ ´ η|3
+.
+Hence, we obtain
+} ˜
+MjpA1qZ´ ˜
+MjpA2qZ}CγpR2q ď C
+σ2
+2
+´
+1`
+´σ1
+σ2
+¯5¯
+}Z}CγpR2qXL2pR2q ∥A1´A2∥ .
+□
+As an immediate consequence of the previous lemma with ˜Zl,m “ Bx
+Xi
+Bηm pηqZl,ipηq,
+we obtain the following lemma for MpAq:
+Lemma 5.4. Let A be a matrix in the set DAσ1,σ2. Then, the linear operators
+MjpAq (4.14) map CγpS2q to CγpR2q for any any γ P p0, 1q. Moreover,
+}MjpAqZ}CγpR2q ď C
+σ2
+´
+1 `
+´σ1
+σ2
+¯2¯
+sup
+l,m
+} Bx
+X
+Bηm
+pηq ¨ Zl,‚pηq}CγpR2qXL1pR2q
+ď Cpσ1, σ2q}Z}CγpS2q.
+Remark 5.5. In most cases, Proposition 5.4 will be used with Z compactly sup-
+ported and given by a multiple of a gradient. Notice that in that case, for Z “
+λ∇S2X, λ : R2 ÞÑ R,
+Bx
+X
+Bηm
+pηq ¨ Zl,‚pηq “ λpηq BXl
+Bηm
+pηq,
+and therefore
+}MjpAqpλ∇S2Xq}CγpR2q ď Cpσ1, σ2q}λ∇X}CγpR2q.
+Lemma 5.6. Let X P C1pS2q such that |X|˚ ą 0. Then, the linear operators
+N jpXq (4.5) map CγpS2q to CγpS2q for any γ P p0, 1q. Moreover,
+(5.5)
+}N jpXqZ}CγpS2q ď
+C
+|X|˚
+´
+1 `
+´}∇S2X}C0pS2q
+|X˚|
+¯2¯
+}Z}CγpS2q.
+Proof. We first notice that we can introduce an arbitrary constant (matrix),
+(5.6)
+NpXqZppxq “ ´
+ż
+S2 ∇S2GpXppxq ´ Xppyqq ¨ pZppyq ´ Cqdpy.
+We will usually take C “ 0 or C “ Zppxq. Recalling the kernels (4.7) and (4.8), we
+write the equation for each component
+pN jpXqZqkppxq “ ´
+ż
+S2 qj
+k,lppx, pyq ¨ pZl,‚ppyq ´ Cl,‚qdpy,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+29
+where Cl,‚ “ 0 or Cl,‚ “ Zl,‚ppxq. We first perform the estimate for |NpXqZppxq|,
+px P S2,
+(5.7)
+|NpXqZppxq| ď
+2ÿ
+j“1
+|N jpXqZppxq|.
+Using the bound (5.1) for the kernel, we have
+(5.8)
+|NpXqZppxq| ď C }∇S2X}C0pS2q
+|X|2˚
+ż
+S2
+|Zl,‚ppxq ´ Zl,‚ppyq|
+|px ´ py|2
+dpy
+ď C }∇S2X}C0pS2q
+|X|2˚
+}Z}CγpS2q.
+We proceed to estimate the H¨older seminorm. Let px, pxh P S2, and denote h “
+|px ´ pxh|. We write
+pN jpXqZqkppxq´pN jpXqZqkppxhq “
+ż
+S2qj
+k,lppx, pyq ¨ pZl,‚ppxq´Zl,‚ppyqqdpy
+´
+ż
+S2qj
+k,lppxh, pyq¨pZl,‚ppxhq´Zl,‚ppyqqdpy,
+and perform the following splitting
+(5.9)
+pN jpXqZqkppxq ´ pN jpXqZqkppxhq “ I1 ` I2 ` I3 ` I4,
+where
+I1 “
+ż
+t|px´ py|ď2huXS2 qj
+k,lppx, pyq ¨ pZl,‚ppxq ´ Zl,‚ppyqqdpy,
+I2 “ ´
+ż
+tpx´ py|ď2huXS2 qj
+k,lppxh, pyq ¨ pZl,‚ppxhq ´ Zl,‚ppyqqdpy,
+I3 “ pZl,‚ppxq ´ Zl,‚ppxhqq ¨
+ż
+t|px´ py|ě2huXS2 qj
+k,lppx, pyqdpy,
+I4 “
+ż
+t|px´ py|ě2huXS2pqj
+k,lppx, pyq ´ qj
+k,lppxh, pyqq ¨ pZl,‚ppxhq ´ Zl,‚ppyqqdpy.
+The first two terms are estimated directly
+(5.10)
+|I1| ` |I2| ď C }∇S2X}C0pS2q
+|X|2˚
+}Z}CγpS2qhγ.
+For the third, we use that the kernel is a derivative to integrate by parts and obtain
+that
+(5.11)
+|I3| “ |pZl,‚ppxq ´ Zl,‚ppxhqq ¨
+ż
+t|px´ py|“2huXS2 GjpXppxq ´ XppyqqnppyqdlS2ppyq|
+ď C }Z}CγpS2q
+|X|˚
+|h|γ.
+Finally, we use the mean-value theorem on the kernel to estimate I4. Set ℓpsq the
+shortest path function from pxh to px respect to arc-length s variable, and L “
+
+30
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+dist
+`
+pxh, px; S2˘
+. Then, we have
+Gj
+k,lpXppxq´Xppyqq ´ Gj
+k,lpXppxhq´Xppyqq “
+ż L
+0
+B
+BsGj
+k,lpXpℓ psqq´Xppyqqds
+“
+ż L
+0
+∇S2Gj
+k,lpXpℓ psqq´Xppyqq ¨ B
+Bsℓ psq ds.
+Hence, for qj
+k,l (4.7),
+|qj
+k,lppx, pyq ´ qj
+k,lppxh, pyq| “
+ˇˇˇ∇S2
+ż L
+0
+∇S2Gk,lpXpℓ psqq´Xppyqq ¨ B
+Bsℓ psq ds
+ˇˇˇ
+“
+ˇˇˇ∇S2
+ż L
+0
+B
+Bxi
+Gk,lpXpℓ psqq´Xppyqq
+ˆ
+∇S2Xi pℓ psqq ¨ B
+Bsℓ psq
+˙
+ds
+ˇˇˇ
+“
+ˇˇˇ
+ż L
+0
+B
+Bxj
+B
+Bxi
+Gk,lpXpℓ psqq´Xppyqq
+ˆ
+∇S2Xi pℓ psqq¨ B
+Bsℓ psq
+˙
+∇S2Xj ppyq ds
+ˇˇˇ,
+and recalling (5.1) we obtain the bound
+|qj
+k,lppx, pyq ´ qj
+k,lppxh, pyq| ďC
+∥∇S2X∥2
+C0pS2q
+|X|3
+˚
+ż L
+0
+1
+|ℓ psq ´ py|3 ds.
+Then, we have that
+|I4| ď C
+}∇S2X}2
+C0pS2q}Z}CγpS2q
+|X|3
+˚
+ż
+t|px´ py|ě2huXS2
+|pxh´ py|γ
+ż L
+0
+ds
+|ℓ psq´ py|3 dpy.
+We notice that since ℓpsq is the shortest path function from pxh to px on S2, it holds
+that |ℓ psq ´ px| ď |px ´ pxh|. Thus,
+|ℓ psq ´ py| ě |px ´ py| ´ |ℓ psq ´ px| ě |px ´ py| ´ |pxh ´ px| ě 1
+2 |px ´ py| .
+In addition,
+|pxh ´ py| ď |px ´ py| ` |pxh ´ px| ď 3
+2 |px ´ py| .
+Finally, since L ď Ch, we conclude that
+(5.12)
+|I4| ď C
+}∇S2X}2
+C0pS2q}Z}CγpS2q
+|X|3
+˚
+h
+ż
+tpx´ py|ě2huXS2
+dpy
+|px ´ py|3´γ
+ď C
+}∇S2X}2
+C0pS2q}Z}CγpS2q
+|X|3
+˚
+hγ.
+Joining the bounds (5.10), (5.11), and (5.12) back in (5.9), we conclude that
+(5.13)
+rNpXqZsCγpS2q ď
+C
+|X|˚
+´
+1 `
+´}∇S2X}C0pS2q
+|X˚|
+¯2¯
+}Z}CγpS2q,
+and, with (5.7), the same bound holds for the H¨older norm.
+□
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+31
+We will need to localize the operators above. For that purpose, let us define the
+cutoff function pρn,
+(5.14)
+pρnppxq “
+#
+1
+if
+|px ´ pxn| ď 3R,
+0
+if
+|px ´ pxn| ě 4R,
+and recall the partition of unity tρnu based on the points pxn (see Subsection 3.4).
+Lemma 5.7. Let X P C1pS2q such that |X|˚ ą 0, NpXq the linear operator
+defined by (4.5), and pρn the cutoff function (5.14). Then, for Z P C0pS2q compactly
+supported on Bpxn,2R X S2, it holds that,
+}p1 ´ pρnqN jpXqZ}C1pS2q ď CpR, |X|˚, }∇S2X}C0pS2qq}Z}C0pS2q.
+Proof. Let
+Ippxq “ p1 ´ pρnppxqqN jpXqZppxq.
+Since 1 ´ pρppxq “ 0 when px P Bpxn,3R, let px P S2zBpxn,3R.
+Then, recalling the
+condition on the support of Z,
+Ippxq“ppρnppxq´1q
+ż
+B pxn,2RXS2
+∇S2GpXppxq´Xppyqq¨Zppyqdpy,
+and using the bound (5.1) for the kernel,
+|Ippxq| ď C }∇S2X}C0pS2q
+|X|2˚
+}Z}C0pS2q
+ż
+B pxn,2RXS2
+|px´ py|´2dpy.
+Since we have that |px ´ py| ě R, we obtain
+(5.15)
+|Ippxq| ď Cp|X|˚, }∇S2X}C0pS2qq}Z}C0pS2q.
+To estimate the H¨older seminorm, consider two points px, pxh P S2, h “ |px ´ pxh|.
+Due to the cut-off function pρn, the only non-trivial case is px, pxh P S2zBpxn,3R:
+|Ippxhq ´ Ippxq| “ ppρnppxq ´ pρnppxhqq
+ż
+B pxn,2RXS2qk,lppx, pyq ¨ Zl,‚ppyqdpy
+` p1´pρnppxhqq
+ż
+B pxn,2RXS2
+`
+qk,lppx, pyq´qk,lppxh, pyq
+˘
+¨ Zl,‚ppyqdpy
+“ J1 ` J2.
+The first term is bounded as (5.15),
+|J1| ď C}pρn}C1pS2q
+}∇S2X}C0pS2q
+|X|2˚
+}Z}C0pS2qh.
+Recalling the expression of qk,l (4.7), we can check that, since |px´py| ě R, |pxh´py| ě
+R,
+|qk,lppx, pyq ´ qk,lppxh, pyq| ď C
+}∇S2X}2
+C0pS2q
+|X|3˚R3
+h,
+hence
+|J2| ď C
+R
+}∇S2X}2
+C0pS2q
+|X|3˚
+}Z}C0pS2qh.
+Therefore,
+(5.16)
+}I}C1pS2q ď CpR, |X|˚, }∇S2X}C0pS2q}C1q}Z}C0pS2q.
+□
+
+32
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+The previous lemma holds analogously for the operators ˜
+MpAq:
+Lemma 5.8. Let A P DAσ1,σ2,
+˜
+MjpAq the linear operator defined by (4.22), and
+pρn the cutoff function (5.14). Then, for Z P C0pR2q compactly supported on V2R,
+it holds that,
+}p1 ´ pρnq ˜
+MjpAqZ}C1pR2q ď CpR, σ1, σ2q}Z}C0pR2q.
+Lemma 5.9. Let pxn P S2, X P C1pS2q, |X|˚ ą 0, and ρn P C8pBpxn,2R X S2q,
+R ă 1{10. Then, the commutator
+rρn,NpXqsZppxq “ ´
+ż
+S2∇S2GpXppxq´Xppyqq¨ Zppyqpρnppyq´ρnppxqqdpy,
+satisfies that, for any γ P p0, 1q,
+(5.17)
+∥rρn, NpXqsZ∥CγpS2q ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q.
+Proof. Recalling the kernel bound (5.1),
+|rρn, NpXqsZppxq| ď C }∇S2ρn}C0pS2q}Z}C0pS2q}∇S2X}C0pS2q
+|X|2˚
+ż
+S2
+1
+|px ´ py|dpy
+ď Cp|X˚, }∇S2X|C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q.
+Next, we study the H¨older seminorm.
+We take two points px, pxh, denote h “
+|px ´ pxh|, and perform a splitting analogous to (5.9). Using the kernel notation
+(4.7),
+rρn, NpXqsZppxq´rρn, NpXqsZppxhq“
+ż
+S2 qk,lppx, pyq pρnppyq´ρnppxqq¨Zl,‚ppyqdpy
+´
+ż
+S2qk,lppxh, pyq pρnppyq´ρnppxhqq¨Zl,‚ppyqdpy
+“ I1
+n ` I2
+n ` I3
+n ` I4
+n,
+where
+I1
+n “
+ż
+t|px´ py|ď2huXS2 qk,lppx, pyq pρnppyq ´ ρnppxqq ¨ Zl,‚ppyqdpy
+I2
+n “ ´
+ż
+t|px´ py|ď2huXS2 qk,lppxh, pyq pρnppyq ´ ρnppxhqq ¨ Zl,‚ppyqdpy
+I3
+n “
+ż
+t|px´ py|ě2huXS2 qk,lppx, pyq pρnppxhq ´ ρnppxqq ¨ Zl,‚ppyqdpy
+and
+I4
+n “
+ż
+t|px´ py|ě2huXS2
+pqk,lppx, pyq´qk,lppxh, pyqq pρnppyq´ρnppxhqq ¨ Zl,‚ppyqdpy.
+Then, for I1
+n and I2
+n,
+|I1
+n| ` |I2
+n| ď C
+∥∇S2ρn∥C0pS2q ∥∇S2X∥C0pS2q
+|X|2
+˚
+}Z}C0pS2qh.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+33
+Next, for I3
+n, integration by parts gives that
+��I3
+n
+�� ď C |ρnppxhq´ρnppxq|
+ż
+t|px´ py|ě2huXS2
+}Z}C0pS2q ∥∇S2X∥C0pS2q
+|X|2
+˚ |px ´ py|2
+dpy
+ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2qh log h´1.
+Finally, in I4
+n, the use of the mean-value theorem provides that
+��I4
+n
+�� ď C
+∥∇S2ρn∥C0pS2q }Z}C0pS2q ∥∇S2X∥2
+C0pS2q
+|X|3
+˚
+ˆ
+ż
+t|px´ py|ď2huXS2
+ż L
+0
+|pxh ´ py|
+|ℓ psq ´ py|3 dsdpy
+ď Cp|X˚, ∥∇S2X∥C0pS2qq ∥∇S2ρn∥C0pS2q ∥Z∥C0pS2q h.
+Thus,
+∥rρn, NpXqsZ∥CγpS2q ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q.
+□
+Lemma 5.10. Let pxn P S2 and x
+Xn : R2 Y t8u Ñ S2 the stereographic projection
+centered at pxn. Let X P C1,γpS2q, |X|˚ ą 0, and pρn given by (5.14). Let the linear
+operators NpXq and MpAq be defined by in (4.5) and (4.14), with A “ ∇Xnp0q,
+where we are using the notation Xnpθq “ Xpx
+Xnpθqq. Denote
+Inpθq “ pρnpθqrMpAq ´ NpXqsZnpθq,
+Then, for Z P CγpS2q compactly supported on Bpxn,2R X S2, the following estimates
+hold:
+(5.18)
+}In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq
+`
+p1 ` }∇S2X}CγpB pxn,5RXS2qq}Z}C0pS2q
+` εpRq}Z}CγpS2q
+˘
+,
+and
+(5.19)
+}In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq
+`
+}Z}C0pS2q`}∇S2X}C
+γ
+2 pB pxn,5RXS2q}Z}C
+γ
+2 pS2q
+` εpRq}Z}CγpS2q
+˘
+,
+with εpRq Ñ 0 as R Ñ 0 given by the modulus of continuity of ∇S2X.
+Proof. First, we write
+(5.20)
+In “ I1
+n ` I2
+n
+“
+2ÿ
+j“1
+pρnpθqrMjpAq ´ N jpXqsZnpθq
+“
+2ÿ
+j“1
+pρnpθqrMjpAq ´ Mjp∇XnqsZnpθq ´ RjpXnqZnpθq,
+and focus on the term I1
+n, as the other I2
+n will follow similarly. We then write
+I1
+n “ ´pρnpθq
+ż
+R2 P 1
+m,k,lpθ, ηq B p
+Xi
+Bηm
+pηqZn,lipηqdη1dη2,
+
+34
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+where we used the notation (3.4) and
+(5.21)
+P 1
+m,k,lpθ, ηq “δk,l
+8π
+B
+Bηm
+´
+1
+|Apθ ´ ηq| ´
+1
+|Xnpθq ´ Xnpηq|
+¯
+“ ´ 1
+8π
+pBpθ´ηqqm
+|Apθ´ηq|3 δk,l ` q1
+m,k,lpθ, ηq
+“ ´ 1
+8π
+pBpθ´ηqqm
+|Apθ´ηq|3 δk,l ` m1
+m,k,lpθ, ηq ´ K1
+m,k,lpθ, ηq
+“ ´ 1
+8π
+ˆpBpθ´ηqqm
+|Apθ´ηq|3 ´ pBnpηqpθ´ηqqm
+|Anpηqpθ´ηq|3
+˙
+δk,l ´ K1
+m,k,lpθ, ηq
+:“P1
+m,k,lpθ, ηq ´ K1
+m,k,lpθ, ηq.
+Above, we are denoting B “ AT A, A “ ∇Xnp0q, Anpηq “ ∇Xnpηq, and K1
+m,k,l
+was given in (4.18). We denote
+˜Zl,mpηq “ B p
+Xi
+Bηm
+pηqZn,lipηq,
+and note that
+} ˜Z}CγpR2q ď C}Z}CγpS2q.
+We start with a bound for |I1
+n|. We split it as follows
+(5.22)
+I1
+n “ O1 ` O2,
+with
+O1 “ pρnpθq
+ż
+V5R
+P 1
+mklpθ, ηq
+` ˜Zl,mpηq ´ ˜Zl,mpθq
+˘
+dη1dη2
+O2 “ pρnpθq ˜Zl,mpθq
+ż
+V5R
+P 1
+m,k,lpθ, ηqdη1dη2.
+Then, we split O1 further
+|O1| ď O1,1 ` O1,2,
+where
+O1,1 “ } ˜Z}CγpV5Rq
+ż
+V5R
+|P1
+m,k,lpθ, ηq||θ ´ η|γdη1dη2,
+O1,2 “ pρnpθq
+ż
+V5R
+|K1
+m,k,lpθ, ηq||δη ˜Zl,mpθq|dη1dη2.
+For θ P V4R, η P V5R, we have the following bound for P1
+m,k,l (5.21),
+(5.23)
+|P1
+m,k,l| ď 1
+8π |pBnpηq ´ Bqpθ ´ ηq
+|Apθ ´ ηq|3
+|
+` 1
+8π |Bnpηqpθ ´ ηq||
+1
+|Apθ ´ ηq|3 ´
+1
+|Anpηqpθ ´ ηq|3 |
+ď C }Bnp¨q ´ B}C0pV5Rq
+|X|3˝,n|θ ´ η|2
+` C
+}Anp¨q ´ A}C0pV5Rq}∇Xn}7
+C0pV5Rq
+|X|9˝,n|θ ´ η|2
+,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+35
+where (4.19) has been used for the last term. Then, the first term O1,1 is high-order
+but with small coefficients,
+O1,1 ď C
+��}Bnp¨q ´ B}C0pV5Rq
+|X|3˝,n
+`
+}Anp¨q ´ A}C0pV5Rq}∇Xn}7
+C0pV5Rq
+|X|9˝,n
+¯
+Rγ
+ˆ } ˜Z}CγpV5Rq,
+since we have that
+}Anp¨q ´ A}C0pV5Rq ď εpRqCp}∇X}C0pV5Rqq,
+with εpRq Ñ 0 as R Ñ 0. The kernel bound, with θ P V4R, η P V5R,
+|K1
+m,k,lpθ, ηq| ď C }∇Xn}C0pV5Rq
+|X|3˝,n
+r∇XnsCγpV5Rq
+1
+|θ ´ η|2´γ ,
+(5.24)
+gives that
+O1,2 ď C }∇Xn}C0pV5Rq}∇Xn}CγpV5Rq
+|X|3˝,n
+Rγ} ˜Z}C0pV5Rq.
+But one also has the bound
+|K1
+m,k,lpθ, ηq| ď C
+}∇Xn}C0pV5Rq}∇Xn}C
+γ
+2 pV5Rq
+|X|3˝,n
+1
+|θ ´ η|2´ γ
+2 ,
+which gives that
+O1,2 ď C
+}∇Xn}C0pV5Rq}∇Xn}C
+γ
+2
+|X|3˝,n
+Rγ} ˜Z}C
+γ
+2 pV5Rq.
+Joining the two bounds, we obtain
+(5.25)
+|O1| ď Cp|X|˚, }∇S2X}C0pS2qqRγ`
+} ˜Z}C0pV5Rq}∇S2X}CγpB pxn,5RXS2q
+` εpRq} ˜Z}CγpV5Rq
+˘
+,
+and
+|O1| ď Cp|X|˚, }∇S2X}C0pS2qqRγ`
+}∇S2X}C
+γ
+2 pB pxn,5RXS2q} ˜Z}C
+γ
+2 pV5Rq
+` εpRq} ˜Z}CγpV5Rq
+˘
+.
+To estimate O2, we use (5.21) to integrate by parts,
+|O2| ď C|pρnpθq|| ˜Zl,mpθq|
+ż
+BV5R
+ˇˇˇ
+1
+|Apθ ´ ηq| ´
+1
+|Xnpθq ´ Xnpηq|
+ˇˇˇdlpηq.
+Next, we note that since θ P V4R, we have that for η P BV5R, |θ ´ η| ě R
+2 . We
+compute the difference
+(5.26)
+1
+|Apθ ´ ηq| ´
+1
+|Xnpθq ´ Xnpηq|
+“
+1
+|θ ´ η|
+´
+1
+|Ap{
+θ ´ ηq|
+´
+1
+|∆ηXnpθq|
+¯
+“
+1
+|θ´η|
+`
+∆ηXnpθq´Ap{
+θ ´ ηq
+˘
+¨
+`
+∆ηXnpθq`Apz
+θ´ηq
+˘
+|Apz
+θ´ηq||∆ηXnpθq|
+`
+|Apz
+θ´ηq|`|∆ηXnpθq|
+˘ ,
+
+36
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+and
+∆ηXnpθq´Ap{
+θ ´ ηq “ ∆ηXnpθq´∇Xnpηqp{
+θ ´ ηq`p∇Xnpηq ´ Aqp{
+θ ´ ηq
+“ ´EηXnpθq ` pAnpηq ´ Aqp{
+θ ´ ηq,
+where EηXnpθq is given in (4.15). Therefore,
+|O2| ď C} ˜Z}C0pV5Rq
+}∇Xn}C0pV5Rq
+|X|3˝,n
+ż
+BV5R
+|EηXnpθq|`}Anp¨q´A}C0pV5Rq
+|θ´η|
+dlpηq,
+hence
+|O2| ď Cp|X|˚, }∇S2X}C0pS2qq} ˜Z}C0pV5RqpR
+γ
+2 }∇S2X}C
+γ
+2 pB pxn,5RXS2q ` εpRqq,
+and
+|O2| ď Cp|X|˚, }∇S2X}C0pS2qqRγ}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq.
+Together with (5.25) back in (5.22), we conclude that
+(5.27)
+|I1
+n| ď Cp|X|˚, }∇S2X}C0pS2qqRγ`
+} ˜Z}C0pV5Rq}∇S2X}CγpB pxn,5RXS2q
+` εpRq} ˜Z}CγpV5Rq
+˘
+,
+and also
+|I1
+n| ď Cp|X|˚, }∇S2X}C0pS2qq
+`
+R
+γ
+2 }∇S2X}C
+γ
+2 pB pxn,5RXS2q} ˜Z}C
+γ
+2 pV5Rq
+` εpRq} ˜Z}CγpV5Rq
+˘
+.
+We proceed to estimate the H¨older seminorm. Take θ, θ ` h P V4R. We use the
+splitting (5.22), and start with the estimate for O1:
+(5.28)
+|O1pθ ` hq ´ O1pθq|
+“ |Q1 ` Q2 ` Q3 ` Q4 ` Q5|,
+where, using the notation (4.12),
+Q1 “ ´pρnpθ ` hq
+ż
+t|θ´η|ď2|h|uXV5R
+P 1
+m,k,lpθ ` h, ηqδη ˜Zl,mpθ ` hqdη1dη2,
+Q2 “ ´pρnpθ ` hq
+ż
+t|θ´η|ď2|h|uXV5R
+P 1
+m,k,lpθ, ηqδη ˜Zl,mpθqdη1dη2,
+Q3 “ pρnpθ ` hqδθ ˜Zl,mpθ ` hq
+ż
+t|θ´η|ě2|h|uXV5R
+P 1
+m,k,lpθ, ηqdη1dη2,
+Q4 “ pρnpθ ` hq
+ż
+t|θ´η|ě2|h|uXV5R
+pP 1
+m,k,lpθ, ηq ´ P 1
+m,k,lpθ ` h, ηqqδη ˜Zl,mpθ ` hqdη1dη2,
+Q5 “ ppρnpθq ´ pρnpθ ` hqq
+ż
+V5R
+P 1
+m,k,lpθ, ηqδη ˜Zl,mpθqdη1dη2.
+Recalling (5.21) and the bounds for P1
+m,k,l (5.23) and K1
+m,k,l (5.24), we obtain
+|Q1| ` |Q2| ď C
+´}Bnp¨q ´ B}C0pV5Rq
+|X|3˝,n
+`
+}Anp¨q ´ A}C0pV5Rq}∇Xn}7
+C0pV5Rq
+|X|9˝,n
+¯
+ˆ
+ż
+t|θ´η|ď2|h|uXV5R
+r ˜ZsCγpV5Rq
+|θ ´ η|2´γ dη1dη2
+` C }∇Xn}C0pV5Rq
+|X|3˝,n
+ż
+t|θ´η|ď2|h|uXV5R
+r∇XnsCγpV5Rq} ˜Z}C0pV5Rq
+|θ ´ η|2´γ
+dη1dη2,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+37
+thus
+(5.29)
+|Q1|`|Q2| ď Cp|X|˚, }∇S2X}C0pS2qq
+`
+}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq
+` εpRq} ˜Z}CγpV5Rq
+˘
+|h|γ.
+It is clear that we could also obtain the estimate
+|Q1|`|Q2| ď Cp|X|˚, }∇S2X}C0pS2qq|h|γ`
+}∇S2X}C
+γ
+2 pB pxn,5RXS2q} ˜Z}C
+γ
+2 pV5Rq
+` εpRq} ˜Z}CγpV5Rq
+˘
+.
+We see that the difference between those two type of estimate comes only from the
+kernel K, where we can distribute half a derivative. The same idea propagates along
+the lines below, hence we only show the first estimate (5.18). Then, integration by
+parts in Q3 gives
+|Q3| ď C|pρnpθ ` hq||δθ ˜Zpθ ` hq|
+ˆ
+ż
+t|θ´η|“2|h|uYBV5R
+ˇˇˇ
+1
+|Apθ ´ ηq| ´
+1
+|Xnpθq ´ Xnpηq|
+ˇˇˇdlpηq.
+Using (5.26), |h| ď 8R and that θ P V4R, we obtain that
+ż
+t|θ´η|“2|h|uYBV5R
+ˇˇˇ
+1
+|Apθ ´ ηq| ´
+1
+|Xnpθq ´ Xnpηq|
+ˇˇˇdlpηq
+ď C }∇X}C0pV5Rq
+|X|3˝,n
+ż
+t|θ´η|“2|h|uYBV5R
+|EηXnpθq|
+|θ´η|
+` }Anp¨q´A}C0pV5Rq
+|θ´η|
+dlpηq
+ď C }∇X}C0pV5Rq
+|X|3˝,n
+´
+}∇X}C0pV5Rqpεp|h|q ` εpRqq ` }∇X}C0pV5RqεpRq
+¯
+ď C
+}∇X}2
+C0pV5Rq
+|X|3˝,n
+εpRq,
+hence
+(5.30)
+|Q3| ď εpRqCp|X|˚, }∇S2X}C0pS2qq} ˜Z}CγpV5Rq|h|γ.
+The term Q4 in (5.28) is estimated by applying the mean-value theorem. As in the
+previous terms, we need to consider separately the two kernels in P 1
+m,k,l (5.21). We
+take a derivative on P1
+m,k,l,
+B
+Bθj
+P1
+m,k,lpθ, ηq “ δk,l
+8π
+´
+´
+Bm,j
+|Apθ ´ ηq|3 `
+pBnpηqqm,j
+|Anpηqpθ ´ ηq|3
+¯
+` δk,l
+8π
+´
+3pBpθ ´ ηqqmpBpθ ´ ηqqj
+|Apθ ´ ηq|5
+´ 3pBnpηqpθ ´ ηqqmpBnpηqpθ ´ ηqqj
+|Anpηqpθ ´ ηq|5
+¯
+,
+and thus
+| B
+Bθj
+P1
+m,k,lpθ, ηq| ď Cp}∇Xn}C0pV4Rq, |X|˝,nq
+εpRq
+|θ ´ η|3 ,
+
+38
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+while the derivative of K1
+m,k,l is computed and bounded below
+B
+Bθj
+K1,1
+m,k,lpθ, ηq “ ´δk,l
+BXnpηq
+Bηm
+¨
+ˆ
+¨
+˝3
+∆ηXnpθq ¨ BXnpθq
+Bθj
+|∆ηXnpθq|5
+EηXnpθq
+|θ ´ η|3
+`
+δη
+BXnpθq
+Bθj
+|∆ηXnpθq|3|θ ´ η|3
+˛
+‚,
+| B
+Bθj
+K1,1pθ, ηq| ď }∇Xn}C0pV4Rq
+|X|3˝,n
+r∇XnsCγpV4Rq
+|θ ´ η|3´γ
+ˆ
+1`3}∇Xn}C0pV4Rq
+|X|˝,n
+˙
+ď Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpB pxn,2RXS2q
+|θ ´ η|3´γ
+.
+Therefore, we obtain
+(5.31)
+|Q4| ď Cp|X|˚, }∇S2X}C0pS2qq
+`
+}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq
+` εpRq} ˜Z}CγpV5Rq
+˘
+|h|γ.
+Finally, the estimate for Q5 (5.28) follows from the bounds (5.23) and (5.24),
+(5.32)
+|Q5| ď Cp}∇Xn}C0pV5Rq, |X|˝,nq}pρn}CγpV5Rq|h|γ
+ˆ
+´
+εpRq} ˜Z}CγpV5Rq`}∇Xn}CγpV5Rq} ˜Z}C0pV5Rq
+¯
+ď Cp|X|˚, }∇S2X}C0pS2qq
+`
+}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq
+` εpRq} ˜Z}CγpV5Rq
+˘
+|h|γ.
+We combine the bounds (5.29), (5.30), (5.31), and (5.32) into (5.28) to conclude
+that
+(5.33)
+|O1pθ ` hq ´ O1pθq| ď Cp|X|˚, }∇S2X}C0pS2qq
+`
+}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq
+` εpRq} ˜Z}CγpV5Rq
+˘
+|h|γ.
+We continue with the H¨older seminorm for O2 in (5.22). Integrating by parts
+O2pθ ` hq ´ O2pθq
+“ δk,l
+8π pρnpθ ` hq ˜Zl,mpθ ` hq
+ˆ
+ż
+BV5R
+´
+1
+|Apθ ` h ´ ηq| ´
+1
+|Xpθ ` hq ´ Xnpηq|
+¯
+nmpηqdlpηq
+´ δk,l
+8π pρnpθq ˜Zl,mpθq
+ż
+BV5R
+´
+1
+|Apθ´ηq| ´
+1
+|Xnpθq´Xnpηq|
+¯
+nmpηqdlpηq.
+Therefore,
+(5.34)
+|O2pθ ` hq ´ O2pθq| “ |Q6 ` Q7|,
+with
+Q6 “ δk,l
+8π
+´
+pρnpθ ` hq ˜Zl,mpθ ` hq ´ pρnpθq ˜Zl,mpθq
+¯
+ˆ
+ż
+BV5R
+|
+1
+|Apθ ` h ´ ηq| ´
+1
+|Xpθ ` hq ´ Xnpηq||nmpηqdlpηq,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+39
+Q7 “ δk,l
+8π pρnpθq ˜Zl,mpθq
+ˆ
+ˆ ż
+BV5R
+´
+1
+|Apθ ` h ´ ηq| ´
+1
+|Xnpθ ` hq ´ Xnpηq|
+¯
+nmpηqdlpηq
+´
+ż
+BV5R
+´
+1
+|Apθ ´ ηq| ´
+1
+|Xnpθq ´ Xnpηq|
+¯
+nmpηqdlpηq
+˙
+.
+Using (5.26) and that θ P V4R, we obtain
+|Q6| ď C} ˜Z}CγpV5Rq|h|γ }∇Xn}C0pV5Rq
+|X|3˝,n
+ˆ
+ż
+BV5R
+|EηXnpθq|`}Anp¨q´A}C0pV5Rq
+|θ´η|
+dlpηq
+ď C} ˜Z}CγpV5Rq|h|γ }∇Xn}C0pV5Rq
+|X|3˝,n
+}∇Xn}C0pV5RqεpRq.
+Finally, we estimate Q7,
+|Q7| ď C} ˜Z}C0pV5Rq
+ż
+BV5R
+|
+1
+|Apθ ` h ´ ηq| ´
+1
+|Apθ ´ ηq||dlpηq
+` C} ˜Z}C0pV5Rq
+ż
+BV5R
+|
+1
+|Xnpθ ` hq ´ Xnpηq| ´
+1
+|Xnpθq ´ Xnpηq||dlpηq.
+Performing the differences we obtain that
+|Q7| ď Cp|X|˝,n, }∇Xn}C0pV5Rqqp|h| ` |h|1´γRγq
+R
+} ˜Z}C0pV5Rq|h|γ
+ď Cp|X|˝,n, }∇Xn}C0pV5Rqq} ˜Z}C0pV5Rq|h|γ.
+Thus, going back to (5.34), we conclude that
+(5.35)
+|O2pθ`hq´O2pθq|ďCp|X|˚, }∇S2X}C0pS2qq
+`
+} ˜Z}C0pV5Rq
+` εpRq} ˜Z}CγpV5Rq
+˘
+|h|γ,
+and together with (5.33) and (5.27) back into (5.22) we obtain the H¨older norm
+estimate for I1
+n. Since the kernel in I2
+n (5.20) satisfy the same estimates, we conclude
+that
+}In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq
+`
+p1 ` }∇S2X}CγpB pxn,5RXS2qq} ˜Z}C0pV5Rq
+` εpRq} ˜Z}CγpV5Rq
+˘
+|h|γ.
+□
+6. Frozen-coefficient Semigroup
+We will need later in the proof (see Lemma 7.6) to deal with the following kernels,
+with 0 ď α ď 1:
+GαpAθq “ G1pAθq ` αG2pAθq,
+
+40
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+where G1, G2 are given in (4.1). From Section 4.2, we have that
+pFθGα,Aq pξq “
+`
+FθG1pAθq
+˘
+pξq ` α
+`
+FθG2pAθq
+˘
+pξq,
+`
+FθG1pAθq
+˘
+pξq “
+1
+4 detpBq |Uξ|,
+`
+FθG2pAθq
+˘
+pξq “
+1
+4 detpBq |Uξ|
+ˆ
+P ´ Uξ
+|Uξ| b Uξ
+|Uξ|
+˙
+,
+where λ “ ∥A∥F , B “
+?
+ATA, P “ ApATAq´1AT, U “ ApATAq´1.
+Let us
+consider the operator defined in (4.14). In preparation for this, we consider the
+operator with ∇X given by a constant matrix and with pg “ I2, as defined in
+(4.32):
+Lα,AY :“ ˜
+Mα pAq pTFpAq∇Y q “
+´
+˜
+M1pAq ` α ˜
+M2pAq
+¯
+pTF pAq∇Y q
+where TFpAq is defined in (4.31). The parameter α is useful in Section 7. We
+will prove Lα,A is a sectorial operator first. That is to say, we have to estimate
+pz ` Lα,Aq´1 where z is in a set with some ω P R, 0 ă δ ă π
+2 in the complex plane:
+Sω,δ “ tz P C : |argpz ´ ωq| ď π ´ δu.
+Since
+˜
+MjpAq is a singular integral operator, it is difficult to compute its inverse
+operator. However, ˜
+MjpAq is a convolution with kernel Gj pAθq, so we may use its
+Fourier multiplier to obtain
+pz ` Lα,Aq´1 Y “F´1ppz ´ Lα,Apξqq´1 FY pξqqpθq
+where Lα,A is defined in (4.34) (in this section, we will write LA instead of Lα,A).
+Then, we will use the Fourier multiplier to estimate the original operator.
+In
+harmonic analysis, the Fourier multiplier theorem in Lp norms is well-known [26].
+We will use a Fourier multiplier theorem in semi-norms �¨�CγpR2q.
+Theorem 6.1. If T is a Fourier multiplier operator with multiplier
+m pξq P Cs pRnzt0uq X L8 pRnq ,
+s ą n
+2 ,
+and
+��Bα
+ξ m pξq
+�� ď Cα |ξ|´|α|
+for all |α| ď s, then for all u P Cγ pRnq X L2 pRnq where 0 ă γ ă 1,
+�T u�CγpRnq ď Cγ,s,nDm�u�CγpRnq,
+where Dm “ max|α|ďs Cα.
+Remark 6.2. The proof of Theorem 6.1 may be split into two parts. The first part
+is the well-known equivalence between the homogeneous Besov norm ∥¨∥ 9Bγ
+8,8pRnq
+and H¨older seminorm �¨�CγpRnq [27]. The second part is proving the Fourier mul-
+tiplier theorem in homogeneous Besov norms ∥¨∥ 9Bγ
+8,8pRnq. Although these results
+are classical, we include the proof of the version we need in Appendix A for the
+covenience of the reader.
+We will compute pz ´ Lα,Apξqq´1 in Section 6.1. Next, for the norm ∥¨∥C0pR2q,
+we may expand the result in [47, Section 3.1] in R2.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+41
+Lemma 6.3 ( [47, Proposition 3.1.2]). If �u�CγpR2q ă 8 for some γ P p0, 1q and
+Fu pξq “ 0 in a neighborhood of ξ “ 0, then ∥u∥C0pR2q ď C�u�CγpR2q, where C
+depends on the neighborhood.
+Therefore, we may choose a suitable cutting function ϕ pξq with ϕ pξq “ 1 in a
+neighborhood of ξ “ 0. Then, the rest of the work for
+���pz ` Lα,Aq´1 Y
+���
+C0pR2q is
+only
+���F´1ppz ´ Lα,Apξqq´1 ϕ pξq FY pξqqpθq
+���
+C0pR2q.
+6.1. Fundamental estimates. We need some elementary estimates on operators
+LA and LA. To achieve it, we first compute the estimate of TFpAq.
+Lemma 6.4. Given matrice A1, A2 in DAσ1,σ2, we have σ2 ď ∥A1∥F , ∥A2∥F ď
+?
+2σ1. Then, we have the following estimates for TF pAq:
+|TF pA1qijkl| ďzp0q
+M ,
+|TFpA1qijkl ´ TF pA2qijkl| ďC
+ˆ
+zp0q
+M
+σ1
+σ2
+2
+` zp1q
+M
+˙
+∥A1 ´ A2∥ ,
+where
+zp0q
+M “
+max
+σ2ďλď
+?
+2σ1
+|f1 pλq| ` |f2 pλq| ,
+zp1q
+M “
+max
+σ2ďλď
+?
+2σ1
+����
+df1
+dλ
+���� `
+����
+df2
+dλ
+���� ,
+f1 pλq “ T
+λ ,
+f2 pλq “ T
+λ ´ dT
+dλ .
+More specific, given Z P Cγ `
+R2˘ Ş L2 `
+R2˘
+with the size of A1,
+∥TFpA1qZ∥CγpR2q Ş L2pR2q ďCzp0q
+M ∥Z∥CγpR2q Ş L2pR2q
+(6.1)
+∥pTF pA1q ´ TFpA2qq Z∥CγpR2q Ş L2pR2q ďC
+ˆ
+zp0q
+M
+σ1
+σ2
+2
+` zp1q
+M
+˙
+∥A1 ´ A2∥ ∥Z∥CγpR2q Ş L2pR2q
+(6.2)
+Proof. Set λi “ ∥Ai∥F . Since
+TF pA1qijkl “ f1 pλ1q δikδjl ´ f2 pλ1q
+pA1qij pA1qkl
+λ2
+1
+,
+It is obvious to obtain the result of TF pAqijkl.
+Next,
+TF pA1qijkl ´ TF pA2qijkl “
+pf1 pλ1q ´ f1 pλ2qq δikδjl
+´ pf2 pλ1q ´ f2 pλ2qq
+pA1qij pA1qkl
+λ2
+1
+´ f2 pλ2q
+ˆpA1qij pA1qkl
+λ2
+1
+´
+pA2qij pA2qkl
+λ2
+2
+˙
+Since
+|λ1 ´ λ2| ď ∥A1 ´ A2∥F ď C ∥A1 ´ A2∥ ,
+we obtain
+|fi pλ1q ´ fi pλ2q| ď zp1q
+M |λ1 ´ λ2| ď Czp1q
+M ∥A1 ´ A2∥ .
+
+42
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+pA1qij pA1qkl
+λ2
+1
+´
+pA2qij pA2qkl
+λ2
+2
+“
+pA1 ´ A2qij pA1qkl ` pA2qij pA1 ´ A2qkl
+λ2
+1
+`
+pA2qij pA2qkl pλ2 ` λ1q pλ2 ´ λ1q
+λ2
+1λ2
+2
+,
+so
+����
+pA1qij pA1qkl
+λ2
+1
+´
+pA2qij pA2qkl
+λ2
+2
+���� ď C σ1
+σ2
+2
+∥A1 ´ A2∥
+Therefore,
+|TF pA1qijkl ´ TF pA2qijkl| ď C
+ˆ
+zp0q
+M
+σ1
+σ2
+2
+` zp1q
+M
+˙
+∥A1 ´ A2∥
+Since TF pA1q, TF pA2q are linear operators, we may obtain the estimates in Cγ `
+R2˘
+X
+L2 `
+R2˘
+of TFpA1qZ and pTF pA1q ´ TFpA2qq Z.
+□
+Then, because we have estimated ˜
+Mα in Thoerem 5.3, we can obtain the bounds
+of Lα,A and its difference Lα,A1 ´ Lα,A2.
+Theorem 6.5. Given a matrix A1, A2 P DAσ1,σ2, then for all Y P C1,γ `
+R2˘
+compactly supported, Lα,AY P Cγ `
+R2˘
+X L2 `
+R2˘
+, and
+∥Lα,AY ∥CγpRq ď zp0q
+M
+σ2
+´
+1 `
+´σ1
+σ2
+¯2¯
+∥∇Y ∥CγpR2q Ş L2pR2q ,
+∥Lα,A1Y ´ Lα,A2Y ∥CγpRq
+ď C
+σ2
+´zp0q
+M
+σ2
+´
+1 ` σ1
+σ2
+¯
+` zp1q
+M
+¯´
+1 `
+´σ1
+σ2
+¯5¯
+∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥
+Proof. Given Y
+P C1,γ `
+R2˘
+compactly supported, ∇Y is also in L2 `
+R2˘
+, so
+pTF pAq∇Y q P Cγ `
+R2˘
+X L2 `
+R2˘
+by Lemma 6.4. By Theorem 5.3 and Lemma
+6.4, Lα,AY P Cγ `
+R2˘
+X L2 `
+R2˘
+and
+∥Lα,AY ∥CγpRq ď C
+σ2
+´
+1 `
+´σ1
+σ2
+¯2¯
+∥TF pAq∇Y ∥CγpR2q Ş L2pR2q
+ďzp0q
+M
+σ2
+´
+1 `
+´σ1
+σ2
+¯2¯
+∥∇Y ∥CγpR2q Ş L2pR2q .
+Next, since
+Lα,A1Y ´ Lα,A2Y “
+´
+˜
+Mα pA1q ´ ˜
+Mα pA2q
+¯
+pTF pA1q∇Y q
+´ ˜
+Mα pA2q ppTF pA1q ´ TF pA2qq ∇Y q ,
+by Theorem 5.3 and Lemma 6.4,
+∥Lα,A1Y ´ Lα,A2Y ∥CγpRq
+ď
+C zp0q
+M
+σ2
+2
+´
+1 `
+´σ1
+σ2
+¯5¯
+∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥
+` C
+σ2
+ˆ
+zp0q
+M
+σ1
+σ2
+2
+` zp1q
+M
+˙ ´
+1 `
+´σ1
+σ2
+¯2¯
+∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥
+□
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+43
+Next, we compute some elementary estimates on the symbol LA. We denote
+Gα,A pθq :“ G1 pAθq ` αG2 pAθq. The pairs of eigenvalues and respective eigenvec-
+tors of pFθGα,Aq pξq are
+´
+p1 ` αq µ
+4 , v pξq
+¯
+,
+´µ
+4 , vK pξq
+¯
+,
+´µ
+4 , v0
+¯
+,
+(6.3)
+and the pairs of MApξq (4.33) are
+˜
+T
+λ
+˜
+|ξ|2 ´ |Aξ|2
+λ2
+¸
+` dT
+dλ
+|Aξ|2
+λ2
+, Aξ
+¸
+,
+ˆT
+λ |ξ|2 , URπ{2ξ
+˙
+,
+ˆT
+λ |ξ|2 , v0
+˙
+.
+(6.4)
+Since pFθGα,Aq pξq and MApξq are symmetric positive definite (s.p.d.), LA pξq is
+diagonalizable and p.d. Then, we have some estimates of LA and its derivatives.
+Lemma 6.6. Given A P DAσ1,σ2, LA and its derivatives satisfy
+(i)
+σ2
+σ2
+1
+|ξ|´1 ď µ ď σ1
+σ2
+2
+|ξ|´1 ,
+(6.5)
+zm |ξ|2 ď ∥MApξq∥ ď zp0q
+M |ξ|2 ,
+(6.6)
+σ2zm
+4σ2
+1
+|ξ| ď ∥LApξq∥ ď σ1zp0q
+M
+2σ2
+2
+|ξ| ,
+(6.7)
+where zm “ minσ1ďλď
+?
+2σ1
+´
+T
+λ ,
+` T
+λ ` dT
+dλ
+˘ σ2
+2
+λ2
+¯
+, zp0q
+M “ maxσ1ďλď
+?
+2σ1
+` T
+λ ` dT
+dλ
+˘
+.
+(ii)
+����
+BLA
+Bξk
+���� ď C1
+σ2
+1
+σ3
+2
+zp0q
+M ,
+(6.8)
+����
+B2LA
+BξkBξl
+���� ď C1
+σ3
+1
+σ4
+2
+zp0q
+M |ξ|´1 ,
+(6.9)
+where C1 is a constant that does not depend on α or A.
+(iii)
+B
+Bξj ∆ξLApξq, p∆ξq2 LApξq and
+B
+Bξj p∆ξq2 LApξq may be written as
+B
+Bξj
+∆ξLApξq “ 1
+|ξ|2 Φp3q
+A,jpˆξq,
+(6.10)
+p∆ξq2 LApξq “ 1
+|ξ|3 Φp4q
+A pˆξq,
+(6.11)
+B
+Bξj
+p∆ξq2 LApξq “ 1
+|ξ|4 Φp5q
+A,jpˆξq,
+(6.12)
+where Φp3q
+A,j, Φp4q
+A , Φp5q
+A,j are bounded on
+���ˆξ
+��� “ 1.
+Proof. (i) We first note the following inequalities:
+(6.13)
+σ2 ď ∥B∥ ď σ1,
+1
+σ1
+ď
+��B´1�� “ ∥U∥ ď 1
+σ2
+.
+We thus have:
+(6.14)
+σ2
+2 ď detpBq “ ∥B∥
+��B´1��´1 ď σ2
+1,
+
+44
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+where we used the fact that B is a 2 ˆ 2 symmetric positive definite matrix. From
+(6.13), we immediately have:
+(6.15)
+|Uξ| “
+��B´1ξ
+�� ě 1
+σ1
+|ξ| .
+Therefore,
+σ2
+σ2
+1
+|ξ|´1 ď µ “
+1
+detpBq |B´1ξ| ď σ1
+σ2
+2
+|ξ|´1 .
+Next, through (6.4), one of the eigenvalues of MA is bounded by
+ˆT
+λ ` dT
+dλ
+˙ σ2
+2
+λ2 |ξ|2 ď T
+λ
+˜
+|ξ|2 ´ |Aξ|2
+λ2
+¸
+` dT
+dλ
+|Aξ|2
+λ2
+ď
+ˆT
+λ ` dT
+dλ
+˙
+|ξ|2 ,
+so we may obtain (6.6). Finally, since LA is diagonalizable and p.d. with LA “
+FθGα,AMA, the eigenvalues of LA are between µ
+4 zm |ξ|2 and µ
+2 zp0q
+M |ξ|2. Hence, we
+get the bound (6.7).
+(ii) We now turn to (6.8). Note that:
+B
+Bξk
+ˆ
+1
+|Uξ|
+˙
+“ ´
+`
+U TUξ
+˘
+k
+|Uξ|3
+,
+(6.16)
+B
+Bξk
+B
+Bξl
+ˆ
+1
+|Uξ|
+˙
+“ ´
+`
+U TU
+˘
+k,l
+|Uξ|3
+` 3
+`
+U TUξ
+˘
+k
+`
+U TUξ
+˘
+l
+|Uξ|5
+.
+(6.17)
+Likewise, we have:
+B
+Bξk
+ˆpUξqj
+|Uξ|
+˙
+“ Ujk
+|Uξ| ´ pUξqj
+`
+U TUξ
+˘
+k
+|Uξ|3
+,
+(6.18)
+B
+Bξk
+B
+Bξl
+ˆpUξqj
+|Uξ|
+˙
+“ ´Ujk
+`
+U TUξ
+˘
+l
+|Uξ|3
+´ Ujl
+`
+U TUξ
+˘
+k
+|Uξ|3
+´
+pUξqj
+`
+U TU
+˘
+k,l
+|Uξ|3
+` 3pUξqj
+`
+U TUξ
+˘
+k
+`
+U TUξ
+˘
+l
+|Uξ|5
+.
+(6.19)
+The above relations, together with (6.13), show that:
+����
+B
+Bξk
+ˆ
+1
+|Uξ|
+˙���� ď σ2
+1
+σ2
+1
+|ξ|2 ,
+����
+B
+Bξk
+B
+Bξl
+ˆ
+1
+|Uξ|
+˙���� ď 4σ3
+1
+σ2
+2
+1
+|ξ|3 ,
+����
+B
+Bξk
+ˆpUξqj
+|Uξ|
+˙���� ď 2σ1
+σ2
+1
+|ξ|,
+����
+B
+Bξk
+B
+Bξl
+ˆpUξqj
+|Uξ|
+˙���� ď 6σ2
+1
+σ2
+2
+1
+|ξ|2 .
+Thus, we obtain
+����
+BFθGα,A
+Bξk
+���� ď C σ2
+1
+σ3
+2
+|ξ|´2 ,
+����
+B2FθGα,A
+BξkBξl
+���� ď C σ3
+1
+σ4
+2
+|ξ|´3 ,
+Next, set A “ rA1A2s, since
+����
+B
+Bξk
+Aξ
+���� “ |Ak| ď λ,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+45
+we have
+����
+B
+Bξk
+pAξ b Aξq
+���� ď Cσ1λ |ξ| ,
+����
+B
+Bξk
+B
+Bξl
+pAξ b Aξq
+���� ď Cλ2.
+Therefore,
+����
+B
+Bξk
+MA
+���� ď C
+ˆT
+λ ` dT
+dλ
+˙
+|ξ| ,
+����
+B
+Bξk
+B
+Bξl
+MA
+���� ď C
+ˆT
+λ ` dT
+dλ
+˙
+.
+The desired bound (6.8) now follows easily by combining the above estimates and
+1 ď σ1
+σ2 .
+(iii) By lemma B.1, we may obtain
+B
+Bξj
+∆ξLApξq “
+ÿ
+i“1,2
+B
+Bξjii
+LApξq “
+ÿ
+i“1,2
+1
+|ξ|2
+Pjii
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+9
+.
+Since σ2 ď
+���Uˆξ
+��� ď σ1 and Pjii
+´
+ˆξ1, ˆξ2
+¯
+is a matrix of polynomials on the domain
+���ˆξ
+��� “ 1,
+Pjiipˆξ1,ˆξ2q
+|Uˆξ|
+9
+is bounded. Therefore, we have
+Φp3q
+A,jpˆξq “
+ÿ
+i“1,2
+Pjii
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+9
+,
+B
+Bξj
+∆ξLApξq “ 1
+|ξ|2 Φp3q
+A,jpˆξq,
+p∆ξq2 LApξq “ 1
+|ξ|3 Φp4q
+A pˆξq,
+B
+Bξj
+p∆ξq2 LApξq “ 1
+|ξ|4 Φp5q
+A,jpˆξq.
+Similarly,
+Φp4q
+A pˆξq “
+ÿ
+i,k“1,2
+Pkkii
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+11
+,
+Φp5q
+A,jpˆξq “
+ÿ
+i,k“1,2
+Pjkkii
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+13
+.
+□
+
+46
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+6.2. Some estimates for z ` Lα,A and pz ` Lα,Aq´1. Since LA pξq is p.d. and
+diagonalizable, P ´1 pξq LA pξq P pξq “ D pξq where D is a positive diagonal matrix.
+Then,
+P ´1 pξq pz ` LA pξqq P pξq “ z ` D pξq ,
+P ´1 pξq pz ` LA pξqq´1 P pξq “ pz ` D pξqq´1 ,
+and the eigenvalues of pz ` LA pξqq´1 are on a curve
+#
+1
+z ` a
+ˇˇˇ a P
+”σ1zp0q
+M
+2σ2
+2
+|ξ| , σ2zm
+4σ2
+1
+|ξ|
+ı+
+.
+Remark 6.7. Let λ ą 0 and z P Sω,δ with ω ą 0, β :“ π´argpz ` λ ´ ωq ą δ.
+Then, we obtain the following inequality
+|z ` λ|2 “λ2 ` |z|2 ´ 2 |z| λ cos β
+ěλ2 ` |z|2 ´ 2 |z| λ cos δ
+“ cos δ pλ ´ |z|q2 ` p1 ´ cos δq
+´
+λ2 ` |z|2¯
+ě p1 ´ cos δq
+´
+λ2 ` |z|2¯
+.
+Now, we estimate
+���Bα
+ξ pz ` LA pξqq´1��� with |α| ď 2.
+Lemma 6.8. Given Sω,δ with ω ą 0 and A P DAσ1,σ2, we have the following
+estimates for all z P Sω,δ,
+1
+|z| ` σ1zp0q
+M
+2σ2
+2
+|ξ|
+ď
+���pz ` LA pξqq´1��� ď
+2
+b
+p1 ´ cos δq
+`
+p σ2zm
+4σ2
+1 |ξ|q2 ` |z|2˘,
+(6.20)
+����
+B
+Bξk
+pz ` LA pξqq´1
+���� ď C1
+σ2
+1
+σ3
+2
+zp0q
+M
+4
+p1 ´ cos δq
+`
+p σ2zm
+4σ2
+1 |ξ|q2 ` |z|2 ˘,
+(6.21)
+and
+����
+B2
+BξlBξk
+pz ` LA pξqq´1
+���� ď C2
+1
+σ4
+1
+σ6
+2
+zp0q
+M
+2
+16
+ˆ
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙˙ 3
+2
+` C1
+σ3
+1
+σ4
+2
+zp0q
+M
+4
+p1 ´ cos δq |ξ|
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙.
+(6.22)
+Moreover, there exists a constant Cδ,σ1,σ2,T depending on δ, σ1, σ2 and T s.t. for
+all |α| ď 2,
+���Bα
+ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T
+|z|
+|ξ|´|α| ,
+(6.23)
+and
+���Bα
+ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T
+|z|2
+|ξ|1´|α| .
+(6.24)
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+47
+Proof. Since the eigenvalues of pz ` LA pξqq´1 are between
+´
+z ` σ1zp0q
+M
+2σ2
+2
+|ξ|
+¯´1
+and
+´
+z ` σ2zm
+4σ2
+1 |ξ|
+¯´1
+, it follows that
+1
+|z| ` σ1zp0q
+M
+2σ2
+2
+|ξ|
+ď
+���pz ` LA pξqq´1��� ď
+2
+d
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙.
+Next,
+B
+Bξk
+pz ` LA pξqq´1 “ ´ pz ` LA pξqq´1
+B
+Bξk
+LA pξq pz ` LA pξqq´1 ,
+so by (6.8),
+����
+B
+Bξk
+pz ` LA pξqq´1
+���� ď C1
+σ2
+1
+σ3
+2
+zp0q
+M
+4
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙.
+Finally,
+B2
+BξlBξk
+pz ` LA pξqq´1
+“
+pz ` LA pξqq´1 B
+Bξl
+LA pξq pz ` LA pξqq´1 B
+Bξk
+LA pξq pz ` LA pξqq´1
+` pz ` LA pξqq´1 B
+Bξl
+LA pξq pz ` LA pξqq´1
+B
+Bξk
+LA pξq pz ` LA pξqq´1
+´ pz ` LA pξqq´1
+B2
+BξlBξk
+LA pξq pz ` LA pξqq´1 .
+Therefore,
+����
+B2
+BξlBξk
+pz ` LA pξqq´1
+���� ď C2
+1
+σ4
+1
+σ6
+2
+zp0q
+M
+2
+16
+ˆ
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙˙ 3
+2
+` C1
+σ3
+1
+σ4
+2
+zp0q
+M
+4
+p1 ´ cos δq |ξ|
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙.
+From the inequalities
+1
+dˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙ ď 1
+|z| and
+1
+dˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙ ď
+1
+σ2zm
+4σ2
+1 |ξ|,
+we obtain
+���Bα
+ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2
+|z|
+|ξ|´|α| ,
+���Bα
+ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2
+|z|2
+|ξ|1´|α|
+for all |α| ď 2, where the constant Cδ,σ1,σ2,T only depends on δ, σ1, σ2 and T .
+□
+Next, let us consider
+���Bα
+ξ |ξ| pz ` LA pξqq´1��� with |α| ď 2.
+
+48
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Lemma 6.9. Given Sω,δ with ω ą 0 and A P DAσ1,σ2, the following estimates hold
+for all z P Sω,δ
+���|ξ| pz ` LA pξqq´1��� ď
+2 |ξ|
+d
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙,
+(6.25)
+����
+B
+Bξk
+´
+|ξ| pz ` LA pξqq´1¯���� ď
+2
+d
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙
+` C1
+σ2
+1
+σ3
+2
+zp0q
+M
+4 |ξ|
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙,
+(6.26)
+����
+B2
+BξlBξk
+´
+|ξ| pz ` LA pξqq´1¯���� ď
+4
+|ξ|
+d
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙
+` C1
+σ3
+1
+σ4
+2
+zp0q
+M
+12
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙
+` C2
+1
+σ4
+1
+σ6
+2
+zp0q
+M
+2
+16 |ξ|
+ˆ
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙˙ 3
+2 .
+(6.27)
+Moreover, there exists a constant Cδ,σ1,σ2,T depending on δ, σ1, σ2 and T s.t. for
+all |α| ď 2,
+���Bα
+ξ |ξ| pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |ξ|´|α| .
+(6.28)
+Proof. By (6.20), it is easy to obtain
+���|ξ| pz ` LA pξqq´1��� ď
+2 |ξ|
+d
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙.
+Next,
+B
+Bξk
+´
+|ξ| pz ` LA pξqq´1¯
+“B |ξ|
+Bξk
+pz ` LA pξqq´1 ` |ξ| B
+Bξk
+pz ` LA pξqq´1 .
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+49
+Therefore, with (6.20) and (6.21),
+����
+B
+Bξk
+´
+|ξ| pz ` LA pξqq´1¯���� ď
+2
+d
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙
+` C1
+σ2
+1
+σ3
+2
+zp0q
+M
+4 |ξ|
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙.
+Finally,
+B2
+BξlBξk
+´
+|ξ| pz ` LA pξqq´1¯
+“
+B2
+BξlBξk
+|ξ| pz ` LA pξqq´1 `
+B
+Bξk
+|ξ| B
+Bξl
+pz ` LA pξqq´1
+` B
+Bξl
+|ξ| B
+Bξk
+pz ` LA pξqq´1 ` |ξ|
+B2
+BξlBξk
+pz ` LA pξqq´1 .
+Hence,
+����
+B2
+BξlBξk
+´
+|ξ| pz ` LA pξqq´1¯���� ď
+4
+|ξ|
+d
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙
+` C1
+σ3
+1
+σ4
+2
+zp0q
+M
+12
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙
+` C2
+1
+σ4
+1
+σ6
+2
+zp0q
+M
+2
+16 |ξ|
+ˆ
+p1 ´ cos δq
+ˆ´
+σ2zm
+4σ2
+1 |ξ|
+¯2
+` |z|2
+˙˙ 3
+2 .
+With
+1
+dˆˆ
+σ2zm
+4σ2
+1
+|ξ|
+˙2
+`|z|2
+˙ ď
+1
+σ2zm
+4σ2
+1
+|ξ|, we obtain
+���Bα
+ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |ξ|´|α|
+for all |α| ď 2, where the constant Cδ,σ1,σ2,T only depends on δ, σ1, σ2 and T .
+□
+Now, we may prove Lα,A is a sectorial operator and obtain the estimate of
+pz ´ Lα,Aq´1.
+Theorem 6.10. Given a matrix A satisfying the condition (3.2) and a constant
+K ą 0 , there exists Sω,δ with ω, δ ą 0 s.t. for all z P Sω,δ
+���pz ´ Lα,Aq´1 Y
+���
+CγpR2q ď Cω,δ,σ1,σ2,T
+|z|
+∥Y ∥CγpR2q ,
+and
+���pz ´ Lα,Aq´1 Y
+���
+C1,γpR2q ď Cω,δ,σ1,σ2,T ∥Y ∥CγpR2q ,
+for all Y P CγpR2q X LppR2q with 1 ď p ď 2.
+
+50
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Proof. Since
+Lα,AY pθq “ ´F´1LApξqFY ,
+the Fourier multiplier of pz ´ Lα,Aq´1 is pz ` LAq´1 and of
+B
+Bθi pz ´ Lα,Aq´1 is
+ξi pz ` LAq´1. With (6.23) and (6.28), we obtain there exists Cω,δ,σ1,σ2,T s.t.
+�pz ´ Lα,Aq´1 Y �CγpR2q ďCω,δ,σ1,σ2,T
+|z|
+�Y �CγpR2q,
+(6.29)
+�pz ´ Lα,Aq´1 Y �C1,γpR2q ďCω,δ,σ1,σ2,T �Y �CγpR2q.
+(6.30)
+Next, for
+���pz ´ Lα,Aq´1 Y
+���
+C0pR2q, set ϕpξq to be a smooth and radial cutting func-
+tion with a compact support in B p1q and ϕpξq “ 1 in a neighborhood of ξ “ 0.
+Then,
+���pz ´ Lα,Aq´1 Y
+���
+C0pR2q “
+���F´1 pz ` LAq´1 pξqFY
+���
+C0pR2q
+ď
+���F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY
+���
+C0pR2q
+`
+���F´1 pz ` LAq´1 pξqϕpξqFY
+���
+C0pR2q .
+For the first term, since 1´ϕpξq “ 0 in a neighborhood of ξ “ 0 and |1 ´ ϕpξq| ď 1,
+by Lemma 6.3, we obtain
+���F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY
+���
+C0pR2q
+ďC�F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY �CγpR2q ď Cω,δ,σ1,σ2,T
+|z|
+�Y �CγpR2q.
+For the second term, define the kernel
+K0 pθq :“ F´1 pz ` LAq´1 pξqϕpξq,
+so
+���F´1 pz ` LAq´1 pξqϕpξqFY
+���
+C0pR2q
+“ ∥K0 ˚ Y ∥C0pR2q ď ∥K0∥L1pR2q ∥Y ∥C0pR2q .
+Then, we estimate ∥K0∥L1, by Lemma B.2, we have
+∥K0 pθq∥ ďCω,δ,σ1,σ2,T
+|z|
+1
+1 ` |θ|3 ,
+so
+∥K0∥L1 ď Cω,δ,σ1,σ2,T
+|z|
+ż
+R2
+1
+1 ` |θ|3 dθ ď Cω,δ,σ1,σ2,T
+|z|
+.
+Therefore,
+���F´1 pz ` LAq´1 pξqϕpξqFY
+���
+C0pR2q ď Cω,δ,σ1,σ2,T
+|z|
+∥Y ∥C0pR2q ,
+so
+���pz ´ Lα,Aq´1 Y
+���
+CγpR2q ď Cω,δ,σ1,σ2,T
+|z|
+∥Y ∥CγpR2q .
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+51
+Similarly, for
+��� B
+Bθi pz ´ Lα,Aq´1 Y
+���
+C0pR2q, we may use the above technique with the
+kernel K1,i
+K1,i pθq :“F´1ξi pz ` LAq´1 pξqϕpξq,
+∥K1,j pθq∥ ďCω,δ,σ1,σ2,T
+1
+1 ` |θ|4
+to obtain
+����
+B
+Bθi
+pz ´ Lα,Aq´1 Y
+����
+C0pR2q
+ď Cω,δ,σ1,σ2,T ∥Y ∥C0pR2q .
+Thus,
+���pz ´ Lα,Aq´1 Y
+���
+C1,γpR2q ď Cω,δ,σ1,σ2,T ∥Y ∥CγpR2q .
+□
+Theorem 6.11. Given a matrix A in DAσ1,σ2, there exists Sω,δ with ω, δ ą 0 s.t.
+for all z P Sω,δ
+∥pz ´ Lα,Aq Y ∥CγpR2q ě Cω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q ,
+and
+∥pz ´ Lα,Aq Y ∥CγpR2q ě Cω,δ,σ1,σ2,T ∥Y ∥C1,γpR2q ,
+for all compactly supported Y P C1,γ `
+R2˘
+.
+Proof. Given Y P C1,γ with a compact support, by Theorem 6.5, W “ pz ´ Lα,Aq Y P
+Cγ `
+R2˘
+X L2 `
+R2˘
+. Through Theorem 6.10,
+∥W ∥CγpR2q ě Cp1q
+ω,δ,σ1,σ2,T |z|
+���pz ´ Lα,Aq´1 W
+���
+CγpR2q “ Cp1q
+ω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q
+(6.31)
+and
+∥W ∥CγpR2q ě Cp2q
+ω,δ,σ1,σ2,T �pz ´ Lα,Aq´1 W �C1,γpR2q “ Cp2q
+ω,δ,σ1,σ2,T ∥Y ∥C1,γpR2q
+(6.32)
+□
+In each chart x
+Xn pθq and Y n pθq, A is ∇Xn p0q and ρnY n is supported in V4R
+(3.4). Lα,A will be used in Proposition 6.13 and 7.6.
+Remark 6.12. Since
+Apξ “ lim
+hÑ0
+Xnphpξq ´ Xnp0q
+h
+,
+it is clear that
+|X|˚ ď |X|˝,n ď lim inf
+ξÑ0
+|Xn pξq ´ Xn p0q|
+|ξ|
+ď
+���Apξ
+��� ,
+C ∥X∥C1,γpS2q ě ∥Xn∥C1pV4Rq ě lim sup
+ξÑ0
+|Xn pξq ´ Xn p0q|
+|ξ|
+ě
+���Apξ
+��� .
+Thus, we may set σ1 “ C ∥X∥C1,γpS2q , σ2 “ |X|˚ .
+
+52
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Next, set A0 “ ∇x
+Xn p0q, so
+A0 “
+»
+–
+2
+0
+0
+2
+0
+0
+fi
+fl ,
+and
+L0,A0Y pθq “ ´
+ż
+R2
+B
+Bηk
+1
+4π |θ|W kdη, L0,A0 pξq “ |ξ|
+8 I
+Set Lpσq
+A
+“ p1 ´ σq L0,A0 ` σL0,A, which will be used in Proposition 7.6 , then
+Lpσq
+A Y pθq “ ´ 1
+8π
+ż
+R2
+B
+Bηk
+ˆ2 p1 ´ σq
+|θ|
+`
+σ
+|Aθ|
+˙
+W kdη,
+Lpσq
+A pξq “|ξ|
+4
+ˆ1 ´ σ
+2
+` σ
+|ξ|
+det pBq |Uξ|
+˙
+We just have to adapt the above theorems and their proofs. Finally, we obtain
+Proposition 6.13. Given X P C1,γpS2q, and our Stereoghraphic projection charts
+and the partition functions tx
+Xn, ρnu with the radius R, set σ1 “ C ∥X∥C1,γpS2q,
+σ2 “ |X|˚. There exists Sω,δ with ω, δ ą 0 s.t. in each chart x
+Xn, for Y P C1,γpS2q,
+we have the following inequalities:
+(6.33)
+∥pz ´ Lα,Aq ρnY n∥CγpR2q ě Cp1q
+ω,δ,σ1,σ2,T |z| ∥ρnY n∥CγpR2q ,
+∥pz ´ Lα,Aq ρnY n∥CγpR2q ě Cp2q
+ω,δ,σ1,σ2,T ∥ρnY n∥C1,γpR2q ,
+���
+´
+z ´ Lpσq
+A
+¯
+ρnY n
+���
+CγpR2q ě Cp3q
+ω,δ,σ1,σ2,T |z| ∥ρnY n∥CγpR2q ,
+���
+´
+z ´ Lpσq
+A
+¯
+ρnY n
+���
+CγpR2q ě Cp4q
+ω,δ,σ1,σ2,T ∥ρnY n∥C1,γpR2q ,
+where A “ ∇Xn p0q, and σ, α P p0, 1q.
+6.3. Some estimates for etLA. We have two ways of representing the semigroup
+e´tLA. One is by the Dunford integral
+etLA “
+1
+2πi
+ż
+ω`γr,η
+etz pz ` LAq´1 dz,
+(6.34)
+where r ą 0, δ ă η ă π
+2 and the curve γr,η “ tz P C : |argz| “ π ´ η, |z| ě ru X tz P
+C : |argz| ď π ´ η, |z| “ ru. The other is through Fourier transform,
+etLAf pθq :“ F´1 ”
+e´tLApξqFrfs pξq
+ı
+pθq “
+ż
+R2 KA pt, θ ´ ηq f pηq dη,
+where KA pt, θq “ F´1 “
+e´tLApξq‰
+pθq.
+Proposition 6.14. Given 0 ď t0 ď t ď T and 0 ď β´α ă 1
+2, we have the following
+estimates:
+}ept´t0qLAfpt0q}CβpR2q ď
+C
+pt ´ t0qβ´α }fpt0q}CαpR2q,
+}ept´t0qLAfpt0q}L2pt0,T ;CβpR2qq ď C}fpt0q}CαpR2q,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+53
+Proof. By (6.34), Theorem 6.10 and 0 ď t ´ t0 ď T , we have
+���ept´t0qLAfpt0q
+���
+CαpR2q “
+�����
+1
+2πi
+ż
+ω`γr,η
+ept´t0qz pz ` LAq´1 fpt0qdz
+�����
+CαpR2q
+ď Cω,δ,σ1,σ2,T ∥fpt0q∥CαpR2q
+ż
+ω`γr,η
+���ept´t0qz��� 1
+|z|d |z|
+ď Cω,δ,σ1,σ2,T ∥fpt0q∥CαpR2q
+ż
+pt´t0qpω`γr,ηq
+|ez| 1
+|z|d |z|
+ď Cω,δ,σ1,σ2,T ,T ∥fpt0q∥CαpR2q ,
+and
+���ept´t0qLAfpt0q
+���
+C1,αpR2q ď Cω,δ,σ1,σ2,T ∥fpt0q∥CαpR2q
+ż
+ω`γr,η
+���ept´t0qz��� d |z|
+ď Cω,δ,σ1,σ2,T
+t ´ t0
+∥fpt0q∥CαpR2q
+ż
+pt´t0qpω`γr,ηq
+|ez| d |z|
+ď Cω,δ,σ1,σ2,T ,T
+t ´ t0
+∥fpt0q∥CαpR2q .
+Then, by interpolation theorem, we obtain
+���ept´t0qLAfpt0q
+���
+CβpR2q ď Cω,δ,σ1,σ2,T ,T,β´α
+pt ´ t0qβ´α
+∥fpt0q∥CαpR2q ,
+so
+}ept´t0qLAfpt0q}L2pt0,T ;CβpR2qq ď Cω,δ,σ1,σ2,T ,T,β´α}fpt0q}CαpR2q.
+□
+Proposition 6.15. Given 0 ď t0 ď t ď T and 0 ď α ă 1, we have the following
+estimates:
+����
+ż t
+t0
+ept´sqLAfpsqds
+����
+C1,αpR2q
+ď C sup
+t0ďsďt }fpsq}CαpR2q,
+����
+ż t
+t0
+ept´sqLAfpsqds
+����
+L2pt0,T ;C1,αpR2qq
+ď C}f}L2pt0,T ;CαpR2qq.
+Proof. First, define u as
+upt, t0, θq :“ urfs pθq“
+ż t
+t0
+ept´sqLAfps, θqds “
+ż t
+t0
+ż
+R2KApt´s, θ´ηqfps, ηqdηds.
+Since tLA pξq “ LA ptξq,
+KApt ´ s, x ´ yq “
+1
+pt ´ sq2 KA
+ˆ
+1, θ ´ η
+t ´ s
+˙
+.
+
+54
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+By (B.48) in Lemma B.3,
+|upt, t0, θq| “
+�����
+ż t
+t0
+ż
+R2
+1
+pt ´ sq2 KA
+ˆ
+1, θ ´ η
+t ´ s
+˙
+fps, ηqdηds
+�����
+ď
+ż t
+t0
+ż
+R2
+1
+pt ´ sq2
+C
+1 `
+��� θ´η
+t´s
+���
+3 |fps, ηq| dηds
+ďC ∥f∥C0
+ż t
+t0
+ż
+R2
+1
+1 ` |θ ´ η|3 dηds ď C ∥f∥C0 T.
+Next, we assume f ps, θq “ 0 when s ă t0, so
+Bu
+Bθi
+“
+ż t
+t0
+ż
+R2
+1
+pt ´ sq3
+BKA
+Bθi
+ˆ
+1, θ ´ η
+t ´ s
+˙
+fps, ηqdηds
+“
+ż t
+´8
+ż
+R2
+1
+pt ´ sq3
+BKA
+Bθi
+ˆ
+1, θ ´ η
+t ´ s
+˙
+fps, ηqdηds.
+Through (B.49),
+ż
+R2
+1
+pt ´ sq3
+BKA
+Bθi
+ˆ
+1, θ ´ η
+t ´ s
+˙
+fps, ηqdη
+“
+ż
+R2
+1
+pt ´ sq3
+BKA
+Bθi
+ˆ
+1, θ ´ η
+t ´ s
+˙
+pfps, ηq ´ f ps, θqq dη,
+and
+�����
+ż
+R2
+1
+pt ´ sq3
+BKA
+Bθi
+ˆ
+1, θ ´ η
+t ´ s
+˙
+pfps, ηq ´ f ps, θqq dη
+�����
+ď
+1
+pt ´ sq3
+ż
+R2
+|fps, ηq ´ f ps, θq|
+1 `
+��� θ´η
+t´s
+���
+4
+dη
+ď C�f ps, ¨q�Cα
+1
+pt ´ sq3
+ż
+R2
+|θ ´ η|α
+1 `
+��� θ´η
+t´s
+���
+4 dη
+“ C�f ps, ¨q�Cα
+1
+pt ´ sq1´α
+ż
+R2
+|θ ´ η|α
+1 ` |θ ´ η|4 dη
+“ C�f ps, ¨q�Cα
+1
+pt ´ sq1´α .
+By Lemma B.5, for all m ď t, we obtain
+�����
+ż t
+m
+ż
+R2
+1
+pt ´ sq3
+BKA
+Bθi
+ˆ
+1, θ ´ η
+t ´ s
+˙
+fps, ηqdηds
+�����
+ďC
+ż t
+m
+�f ps, ¨q�Cα
+1
+pt ´ sq1´α ds ď C pt ´ mqα Ml r�f ps, ¨q�Cαs ptq ,
+so set m “ 0,
+����
+Bu
+Bθi
+pt, t0, θq
+���� ď CT αMl r�f ps, ¨q�Cαs ptq .
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+55
+For � Bu
+Bθi �Cα, we first compute
+B2
+BθiBθj
+ż M
+´8
+ż
+R2KApt ´ s, θ ´ ηqfps, ηqdηds
+“
+ż M
+8
+ż
+R2
+1
+pt ´ sq4
+B2KA
+BθiBj
+ˆ
+1, θ ´ η
+t ´ s
+˙
+fps, ηqdηds,
+where M ă t. Then, by (B.50) and Lemma B.5, we have
+�����
+ż
+R2
+1
+pt´sq4
+B2KA
+BθiBj
+´
+1, θ ´ η
+t´s
+¯
+fps, ηqdη
+����� ď C�f ps, ¨q�Cα
+1
+pt´sq2´α
+ż
+R2
+|θ´η|α
+1`|θ´η|4 dη
+ď C�f ps, ¨q�Cα
+1
+pt ´ sq2´α .
+and
+�����
+B2
+BθiBθj
+ż M
+´8
+ż
+R2 KApt ´ s, θ ´ ηqfps, ηqdηds
+����� ď C pt ´ Mqα´1 Ml r�f ps, ¨q�Cαs ptq .
+When |θ ´ η| “ 1, we set a cutting function φpsq s.t. φpsq “ 1 on r´8, ´2s and
+φpsq “ 0 on r´1, 8s, and define φptqpsq “ φps ´ tq. We obtain
+����
+Bu
+Bθi
+pt, t0, θq ´ Bu
+Bηi
+pt, t0, ηq
+���� “
+����
+Bu
+Bθi
+rfspθq ´ Bu
+Bηi
+rfspηq
+����
+ď
+����
+Bu
+Bθi
+rφptqfspθq ´ Bu
+Bηi
+rφptqfspηq
+����`
+����
+Bu
+Bθi
+rp1´φptqqfspθq
+����`
+����
+Bu
+Bηi
+rp1´φptqqfspηq
+����
+ď Cpt´Mqα´1Ml
+“
+�φptqfps, ¨q�Cα‰
+ptq` Cpt´mqαMl
+“
+�p1´φptqqfps, ¨q�Cα‰
+ptq,
+where M “ t ´ 1 and m “ t ´ 2. Therefore, since 0 ď φ ď 1 and �f ps, ¨q�Cα ě 0,
+����
+Bu
+Bθi
+pt, t0, θq ´ Bu
+Bηi
+pt, t0, ηq
+���� ď CMl r�f ps, ¨q�Cαs ptq .
+When ρ “ |θ ´ η| ‰ 1, we define uρ pt, t0, θq “ 1
+ρu pρt, ρt0, ρθq , f ρ pt, θq “ f pρt, ρθq
+and ¯θ “ θ
+ρ , ¯η “ η
+ρ , ¯t “ t
+ρ, ¯t0 “ t0
+ρ . We have
+Buρ
+Bt pt, θq “LAuρ pt, θq ` f ρ pt, θq ,
+Buρ
+Bθi
+pt, θq “ Bu
+Bθi
+pρt, ρθq ,
+so
+ˇˇˇ Bu
+Bθi
+pt, t0, θq´ Bu
+Bηi
+pt, t0, ηq
+ˇˇˇ“
+ˇˇˇBuρ
+B¯θi
+p¯t, ¯t0, ¯θq´ Buρ
+B¯ηi
+p¯t, ¯t0, ¯ηq
+ˇˇˇďCMlr�f ρ p¯s, ¨q�Cαsptq.
+Since �f ρ�Cα p¯sq “ ρα�f�Cα pρ¯sq,
+Ml
+“
+�f ρ p¯s, ¨q�Cα‰
+p¯tq “ sup
+¯rą0
+1
+¯r
+ż ¯t
+¯t´¯r
+�f ρ�Cα p¯sq d¯s “ ρα sup
+¯rą0
+1
+¯r
+ż ¯t
+¯t´¯r
+�f�Cα pρ¯sq d¯s
+“ρα sup
+rą0
+1
+r
+ż t
+t´r
+�f�Cα psq ds “ ραMl r�f ps, ¨q�Cαs ptq .
+
+56
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Hence,
+����
+Bu
+Bθi
+pt, t0, θq ´ Bu
+Bηi
+pt, t0, ηq
+���� ď CMl r�f ps, ¨q�Cαs ptq |θ ´ η|α ,
+and by the Hardy-Littlewood maximal function theorem, for all 1 ď p ď 8
+���∥u∥CγpR2q
+���
+Lppt0,T q ď
+���C ∥f∥C0pR2q ` CMl
+“
+�f ps, ¨q�CαpR2q
+‰
+ptq
+���
+Lppt0,T q
+ďC
+���∥f∥C0pR2q
+���
+Lppt0,T q ` C
+��Ml
+“
+�f ps, ¨q�CαpR2q
+‰
+ptq
+��
+Lppt0,T q
+ďC
+���∥f∥C0pR2q
+���
+Lppt0,T q ` C
+���f pt, ¨q�CαpR2q
+��
+Lppt0,T q
+ďC
+���∥f∥CγpR2q
+���
+Lppt0,T q .
+□
+7. Local well-posedness
+We write the Peskin problem as an evolution equation
+(7.1)
+BX
+Bt “ FpXq,
+t ą 0,
+Xp0q “ X0,
+where FpXq is given in (4.2). We will make use of Theorem 8.4.1 in [35]:
+Theorem 7.1. Let E1 Ă E0 Ă E be Banach spaces and let 0 ă σ ă 1. Given
+T ą 0, open set O1 Ă E1 and a function
+F : r0, T s ˆ O1 ÞÑ E0,
+pt, uq ÞÑ Fpt, uq
+such that F and Fu are continuous in r0, T s ˆ O1. If for every p¯t, ¯uq P r0, T s ˆ O1
+we have Fup¯t, ¯uq : E1 ÞÑ E0 is the part of a sectorial operator S : DpSq Ă E ÞÑ E
+with DSpσq » E0 and DSpσ ` 1q » E1, then for every ¯t P r0, ts and ¯u P O1 there
+are δ ą 0, r ą 0 such that if t0 P r0, T q, |t0 ´ ¯t| ď δ, and }u0 ´ ¯u} ď r then the
+problem
+v1ptq “ Fpt, vptqq,
+t0 ď t ď t0 ` δ,
+vpt0q “ u0,
+has a unique solution v P Cprt0, t0 ` δs; E1q X C1prt0; t0 ` δs; E0q.
+Then, our main result is the following Theorem:
+Theorem 7.2. Consider the 3D Peskin problem (7.1) with initial data satisfying
+X0 P h1,γpS2q, |X0|˚ ą 0, and T P C3 such that T ą 0, dT {dλ ě 0. Then, there
+exists some time T ą 0 such that (7.1) has a unique solution X,
+X P Cpr0, T s; h1,γpS2qq X C1pr0, T s; hγpS2qq.
+Proof. Let Om “ tY P h1,γpS2q : |Y |˚ ě m ą 0u, E1 “ h1,γpS2q, E0 “ hγpS2q, and
+E “ hαpS2q, with 0 ă α ă γ. Define the operator S as the linearization of F (4.3)
+around X0:
+SpX0qY :“ BXFpX0qY “ d
+dεFpX0 ` εY q|ε“0.
+Since X0 P Om is arbitrary, we can study the Gˆateaux derivative of F at any
+X P Om, which is given by
+(7.2)
+SpXqY “ S1pXqY ` S2pXqY ,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+57
+with
+S1pXqY “ ´
+ż
+S2 ∇S2GpXppxq ´ Xppyqq¨
+ˆ d
+dε
+´
+T p|∇S2pXppyq ` εY ppyqq|qp∇S2pX ` εY qqppyq
+¯
+|ε“0dpy,
+S2pXqY “ ´
+ż
+S2∇S2 d
+dε
+´
+GpXppxq´Xppyq`εpY ppxq ´ Y ppyqqq
+¯ˇˇˇ
+ε“0¨
+ˆ T p|∇S2Xppyq|q∇S2Xppyqdpy.
+It remains to check that the hypothesis of Theorem 7.1 are satisfied, which follow
+from Propositions 7.3, 7.4, and 7.6 below.
+□
+Proposition 7.3. If m ą 0 and γ P p0, 1q, T P C2, then F (4.3) is a continuous
+map from Om Ă h1,γpS2q to hγpS2q.
+Proof. Given that FpXq “ NpXqpT p|∇S2X|q∇S2Xq (4.5), we apply Proposition
+5.5 to obtain that
+}FpXq}CγpS2q ď C
+1
+|X|˚
+´
+1 `
+´}∇S2X}C0pS2q
+|X˚|
+¯2¯
+}T p|∇S2Xq|q∇S2Xq}CγpS2q,
+hence recalling the expression for T (4.4), the bound above yields that
+(7.3)
+}FpXq}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q.
+We have thus proved that F maps C1,γpS2q to CγpS2q. We need to show that
+it also maps h1,γpS2q to hγpS2q. Having the estimate (7.3), it suffices to show that
+if X P h1,γpS2q, then FpXq P hγpS2q. Since hk,γpS2q is the completion of Ck,γpS2q
+in any Ck,αpS2q with 0 ă γ ă α ă 1, k ě 0, let X P h1,γpS2q, and tXmum a
+sequence Xm P C1,αpS2q, α ą γ, such that Xm Ñ X in C1,γpS2q. It is clear that
+the previous estimate (7.3) also holds replacing γ by α, thus FpXmq P Cα. We
+conclude that FpXq P hγpS2q by showing that
+(7.4)
+}FpXmq ´ FpXq}CγpS2q ď C}Xm ´ X}C1,γpS2q.
+The estimate will follow from the previous ones by writing FpXmq ´ FpXq as
+follows:
+(7.5)
+FpXmqppxq ´ FpXqppxq “ ∆1ppxq ` ∆2ppxq,
+with
+∆1ppxq “ ´
+ż
+S2∇S2GpXmppxq´Xmppyqq¨
+ˆ
+`
+T p∇S2Xmppyqq∇S2Xmppyq´T p∇S2Xppyqq∇S2Xppyq
+˘
+dpy,
+∆2ppxq “
+ż
+S2∇S2
+´
+GpXmppxq´Xmppyqq´GpXppxq´Xppyqq
+¯
+¨
+ˆ T p∇S2Xppyqq∇S2Xppyqdpy,
+
+58
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+where both terms have kernels given by a derivative. The first term has thus already
+been treated,
+(7.6)
+}∆1}CγpS2q
+ď Cp}∇S2Xm}C0pS2q, |Xm|˚q}T p∇S2Xmq∇S2Xm´T p∇S2Xq∇S2X}CγpS2q
+ďCp}∇S2Xm}C0pS2q, |Xm|˚, }∇S2X}C0pS2q, |X|˚, }T }C2q
+´
+}∇S2pXm´Xq}CγpS2q
+` p}∇S2X}CγpS2q ` }∇S2Xm}CγpS2qq}∇S2pXm ´ Xq}C0pS2q
+¯
+,
+while the second one can be estimated in a similar manner by noticing that one
+can always extract Xm ´ X from the difference of kernels. Consider for example
+the kernel q1
+k,l (4.7),
+q1
+k,lppx, pyq “ ´ 1
+8π
+δ pyXippxq
+|δ pyXppxq|3 ∇S2Xippyqδk,l.
+We can write
+1
+8π
+δ pyXm
+i ppxq
+|δ pyXmppxq|3 ∇S2Xm
+i ppyqδk,l ´ 1
+8π
+δ pyXippxq
+|δ pyXppxq|3 ∇S2Xippyqδk,l
+“ 1
+8π
+δ pypXm
+i ´ Xiqppxq∇S2Xm
+i ppyq
+|δ pyXmppxq|3
+` 1
+8π
+δ pyXippxq∇S2pXm
+i ´ Xiqppyq
+|δ pyXmppxq|3
+` 1
+8π δ pyXippxq∇S2Xippyq
+´
+1
+|δ pyXmppxq|3 ´
+1
+|δ pyXppxq|3
+¯
+.
+Therefore, it holds that
+}∆2}CγpS2q ď Cp}∇S2X}C0pS2q, |X|˚, }∇S2Xm}C0pS2q, |Xm|˚, }T }C1q
+ˆ }∇S2X}CγpS2q}∇S2pXm ´ Xq}C0pS2q,
+which together with (7.6) proves (7.4).
+□
+Proposition 7.4. If m ą 0 and γ P p0, 1q, T P C3, then the Gˆateaux derivative of
+F at any X P Om Ă h1,γpS2q (7.2) is continuous and maps h1,γpS2q to hγpS2q.
+Proof. The first term S1pXqY in the Gˆateaux derivative of F (7.2) is given in
+terms of the operator NpXq (4.5),
+(7.7)
+S1pXqY ppxq “ NpXqpTSp∇S2Xq∇S2Y qppxq,
+with TS given by
+(7.8)
+TSp∇S2Xq“ T p|∇S2X|q
+|∇S2X|
+`
+´
+T 1p|∇S2X|q´ T p|∇S2X|q
+|∇S2X|
+¯∇S2X b ∇S2X
+|∇S2X|2
+,
+and, in index notation,
+pTSp∇S2Xq∇S2Y ql,i “ T p|∇S2X|q
+|∇S2X|
+p∇S2Y ql,i
+`
+´
+T 1p|∇S2X|q´ T p|∇S2X|q
+|∇S2X|
+¯p∇S2Xql,ip∇S2Xqq,m
+|∇S2X|2
+p∇S2Y qq,m.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+59
+Proposition 5.5 then gives that
+(7.9)
+}S1pXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2qq}TSp∇S2Xq∇S2Y }CγpSq
+ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2Y }CγpSq
+` Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2X}CγpS2q}∇S2Y }C0pSq.
+We proceed with S2pXqY ,
+S2pXqY “ S2,1pXqY ` S2,2pXqY ,
+pS2,jpXqY qkppxq “
+ż
+S2 Qj
+k,lppx, pyq ¨ pT p∇S2Xq∇S2Xlppyq ´ Clqdpy,
+where we define the kernels
+Qj
+k,lppx, pyq “ ∇S2 d
+dε
+´
+GjpXppxq ´ Xppyq ` εpY ppxq ´ Y ppyqqq
+¯ˇˇˇ
+ε“0.
+Taking the derivatives we see that
+Qj
+k,lppx, pyq “ ∇S2
+´ B
+Bxi
+GjpXppxq ´ XppyqqpYippxq ´ Yippyqq
+¯
+“ ´ B
+Bxl
+B
+Bxi
+GjpXppxq ´ Xppyqq∇S2XlppyqpYippxq ´ Yippyqq
+´ B
+Bxi
+GjpXppxq ´ Xppyqq∇S2Yippyq,
+hence we have the following bound, similarly as in (5.1)
+|Qj
+k,lppx, pyq| ď C
+´ |∇S2Y ppyq|
+|∆ pyXppxq|2 ` |∇S2Xppyq||∆ pyY ppxq|
+|∆ pyXppxq|3
+¯
+ď C }∇S2Y }C0pS2q
+|X|2˚
+´
+1 ` }∇S2X}C0pS2q
+|X|˚
+¯
+1
+|px ´ py|2 .
+Therefore,
+|S2,jpXqY ppxq|ďC }T p∇S2Xq∇S2X}CγpS2q
+|X|2˚
+´
+1` }∇S2X}C0pS2q
+|X|˚
+¯
+}∇S2Y }C0pS2q,
+hence
+|S2,jpXqY ppxq| ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q}∇S2Y }C0pS2q.
+Given that the kernels Qj
+k,l are also a derivative, the estimate of the H¨older semi-
+norm follows the same steps as in Proposition 5.5. In fact, performing the splitting
+as in (5.9), we find that
+rS2,jpXqY sCγpS2q
+ď C }T p∇S2Xq∇S2X}CγpS2q
+|X|2˚
+´
+1 `
+´}∇S2X}C0pS2q
+|X˚|
+¯2¯
+}∇S2Y }C0pS2q,
+thus
+(7.10)
+}S2pXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q}∇S2Y }C0pS2q.
+
+60
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Together with (7.9), this shows that SpXq maps C1,γpS2q to CγpS2q,
+}SpXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2X}CγpS2q}∇S2Y }C0pS2q
+` Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2Y }CγpSq
+ď Cp|X|˚ , }∇S2X}CγpS2q, }T }C2q ∥∇S2Y ∥CγpS2q .
+(7.11)
+We are left to show that SpXq also maps h1,γpS2q to hγpS2q and that it is
+continuous with respect to X.
+We follow the lines below (7.3).
+It suffices to
+show that if Y P h1,γpS2q, then SpXqY P hγpS2q. Let Y P h1,γpS2q, and tY mum a
+sequence Y m P C1,αpS2q, α ą γ, such that Y m Ñ Y in C1,γpS2q. Since SpXqY m P
+CαpS2q, we conclude that SpXqY P hγpS2q by showing that
+}SpXqY m ´ SpXqY }CγpS2q ď C}Y m ´ Y }C1,γpS2q.
+But since we are dealing with a linear operator, the estimate is trivially satisfied
+from (7.11). That the Gˆateaux derivative is continuous in X follows along the lines
+below (7.4). In fact,
+`
+SpX1q ´ SpX2q
+˘
+Y “ pS1pX1q ´ S1pX2qqY ` pS2pX1q ´ S2pX2qqY ,
+and we decompose each Sj as in (7.5). Then, it is not hard to see that the following
+bound holds
+}pSpX2q ´ SpX1qqY }CγpS2q
+ď Cp}∇S2X1}C0pS2q, |X1|˚, }∇S2X2}C0pS2q, |X2|˚, }T }C3q
+ˆ
+´
+}∇S2Y }CγpS2q}∇S2pX1 ´ X2q}CγpS2q
+` }∇S2Y }C0pS2q}∇S2pX1 ´ X2q}C0pS2qp}∇S2X1}CγpS2q ` }∇S2X2}CγpS2qq
+¯
+.
+□
+Proposition 7.5. Consider the linear operator SpXq : C1,γpS2q Ñ CγpS2q defined
+in (7.2) with X P C1,γpS2q, T P C2, T ą 0, dT {dλ ě 0. Then, there exists a sector
+such that for all z in the sector
+}z ´ SpXqY }CγpS2q ě Cp}Y }C1,γpS2q ` |z|}Y }CγpS2qq,
+where the constant C depends only on the sector, γ, the norms }X}C1,γpS2q and
+}T }C2, and the arc-chord condition |X|˚.
+Proof. From (7.2), we have
+(7.12)
+}pz ´ SpXqqY }CγpS2q ě }pz ´ S1pXqqY }CγpS2q ´ }S2pXqY }CγpS2q,
+and using (7.10) we obtain that
+(7.13)
+}pz ´ SpXqqY }CγpS2q ě }pz ´ S1pXqqY }CγpS2q ´ C ∥∇S2Y ∥C0pS2q .
+We use the notation (7.8). Then, we can write S1pXq (7.7) as
+(7.14)
+S1pXqY ppxq “ ´
+ż
+S2 ∇S2GpXppxq ´ Xppyqq ¨ pTSp∇S2Xppyqq∇S2Y ppyq ´ Cqdpy.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+61
+Next, we introduce the partition of unity ρn (see Section 3.4) to write
+S1pXqY ppxq “
+ÿ
+n
+S1pXq pρnY q ppxq
+“´
+ÿ
+n
+ż
+S2∇S2GpXppxq´Xppyqq¨pTSp∇S2Xppyqq∇S2pρnY qppyq´Cnqdpy,
+where now we will choose Cn “ 0 or Cn “ TSp∇S2Xppxqq∇S2pρnY qppxq. We will
+extensively use that
+(7.15)
+supp ρn Ă Bpxn,2R X S2.
+We notice that
+ρnppxqzY ppxq ´ S1pXqpρnY qppxq “ ρnppxq
+´
+zY ppxq ´ S1pXqY ppxq
+¯
+`
+´
+ρnppxqS1pXqY ´ S1pXqpρnY qppxq
+¯
+,
+hence
+2}ρn}CγpS2q}pz ´ S1pXqqY }CγpS2q ě }ρn
+´
+zY ´ S1pXqY
+¯
+}CγpS2q
+ě }zρnY ´ S1pXqpρnY q}CγpS2q ´ }ρnS1pXqY ´ S1pXqpρnY q}CγpS2q,
+and summing in n we obtain
+(7.16)
+}pz ´ S1pXqqY }CγpS2q ě C
+ÿ
+n
+p}I1
+n}CγpS2q ´ }I2
+n}CγpS2qq,
+where
+(7.17)
+I1
+n “ zρnY ´ S1pXqpρnY q,
+I2
+n “ ρnS1pXqY ´ S1pXqpρnY q.
+Recalling that S1pXq is given in terms of NpXq (7.7), we split I2
+n further:
+I2
+n “ rρn, NpXqspTSp∇S2Xq∇S2Y q ` NpXq
+`
+Tsp∇S2XqY b ∇S2ρn
+˘
+,
+and by Proposition 5.5 and Lemma 5.9,
+}I2
+n}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2qq
+´
+}∇S2ρn}C0pS2q}TSp∇S2Xq∇S2Y }C0pS2q
+` }Tsp∇S2XqY b ∇S2ρn}CγpS2q
+¯
+.
+Therefore,
+}I2
+n}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2ρn}CγpS2q
+ˆ p}∇S2X}CγpS2q}Y }CγpS2q`}∇S2Y }C0pS2qq.
+We proceed to deal with the term I1
+n (7.17). We introduce the cutoff (5.14) so that
+(7.18)
+}I1
+n}CγpS2q ě }zρnY ´pρnS1pXqpρnY q}CγpS2q´ }p1 ´ pρnqS1pXqpρnY q}CγpS2q
+“ }I1,1
+n }CγpS2q ´ }I1,2
+n }CγpS2q.
+The last term will be smoother because the integral is not singular. In fact, recalling
+again the expression of S1pXq in terms of NpXq (7.7), we use Proposition 5.7 to
+obtain
+}I1,2
+n }CγpS2q ď CpR, |X|˚, }∇S2X}C0pS2qq}TSp∇S2Xq∇S2pρnY q}C0pS2q
+ď CpR, |X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2pρnY q}C0pS2q.
+
+62
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Although the constant in the bound above (5.16) becomes large for R small, it will
+suffice since it is lower order in terms of regularity for Y .
+Next, we proceed to estimate I1,1
+n
+(7.18). We can decompose further by intro-
+ducing the frozen-coefficient linear operator. We denote by x
+Xn the stereographic
+projection centered at pxn, i.e. pxn “ x
+Xnp0q, and Xnpθq “ Xpx
+Xnpθqq (see Section
+3.4). Recalling (7.7) and (4.10), (4.17), we have
+(7.19)
+}I1,1
+n }CγpS2q ě C}zρnY n ´ pρnNpXqpTSp∇S2Xq∇S2pρnY qqn}CγpR2q,
+and
+I1,1
+n pθq “ zρnpθqY npθq ´ pρnpθqNpXqpTSp∇S2Xq∇S2pρnY qqnpθq
+“ J3 ` J4 ` J5 ` J6,
+with
+J3 “ zρnpθqY npθq ´ LApρnY nqpθq,
+J4 “ pρnpθqrMpAq ´ NpXqspTSp∇S2Xq∇S2pρnY qqnpθq
+“ pρnpθqrMpAq ´ Mp∇Xnq ´ RnpXnqspTSp∇S2Xq∇S2pρnY qqnpθq,
+J5 “ pρnpθq
+´
+LApρnY nqpθq ´ MpAqpTSp∇S2Xqq∇S2pρnY q
+˘
+npθq
+¯
+,
+J6 “ p1 ´ pρnpθqqLApρnY nqpθq,
+where we denote A the constant matrix A “ ∇Xnp0q, Mp∇Xnq is defined in
+(4.14), RpXnq in (4.18), LA in (4.32), pxn “ x
+Xnp0q. The bound for J6 follows from
+Lemma 5.8 together with Remark 6.12,
+}J6}C1pR2q ď CpR, |X|˚, }∇S2X}C0pS2qq}TFpAq∇pρnY nq}C0pR2q,
+thus,
+(7.20)
+}J6}C1pR2q ď CpR, |X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2pρnY q}C0pS2q.
+Next, Lemma 5.10 with Z “ TSp∇S2Xq∇S2pρnY q provides the following bound for
+J4:
+}J4}CγpR2q
+ď Cp|X|˚, }∇S2X}C0pS2qq
+`
+p1 ` }∇S2X}CγpS2qq}TSp∇S2Xq∇S2pρnY q}C0pS2q
+` εpRq}TSp∇S2Xq∇S2pρnY q}CγpS2q
+˘
+,
+where εpRq Ñ 0 as R Ñ 0. Thus,
+}J4}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q
+´
+εpRq}∇S2pρnY q}CγpS2q
+` p1 ` }∇S2X}CγpS2qq}∇S2pρnY q}C0pS2q
+¯
+.
+We proceed with J5. Recalling the expression for TS (7.8), we have
+pTSp∇S2Xq∇S2pρnY qqn,lipηq “ pTSp∇S2Xq∇S2pρnY q ˝ x
+Xnql,ipηq
+“ pdetppgpηqqq´ 1
+2 B
+Bηr
+pρnYn,qqpηqB p
+Xm
+Bηr
+pηq
+´T pλnpηqq
+λnpηq
+δlqδim
+`
+`
+T 1pλnpηqq ´ T pλnpηq
+λnpηq
+˘
+BXn,l
+Bηj pηq Bx
+Xi
+Bηj pηq BXn,q
+Bηp pηq Bx
+Xm
+Bηp pηq
+pλnpηqq2detppgpηqq
+¯
+,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+63
+with λnpηq given in (2.14). Substituting into (4.14),
+(7.21)
+pMpAqpTSp∇S2Xq∇S2pρnY qqnqkpθq
+“ ´
+ż
+R2 mm,k,lpθ, ηq B p
+Xi
+Bηm
+pηqpTSp∇S2Xpηqq∇S2pρnY nqpηqql,idη1dη2
+“ ´
+ż
+R2 mi,k,lpθ, ηqp ˜TSqipqlpηq B
+Bηp
+pρnYn,qqpηqdη1dη2
+“ ˜
+MpAqp ˜TS∇pρnY nqqpθq,
+where we denote
+(7.22)
+p ˜TSp∇Xqqipqlpηq“ T pλnpηqq
+λnpηq
+δpiδql`
+`
+T 1pλnpηqq´ T pλnpηq
+λnpηqq
+˘
+BXn,l
+Bηi pηq BXl,q
+Bηp pηq
+pλnpηqq2a
+detppgpηqq
+.
+Thus we can write
+J5 “ pρnpθq ˜
+MpAqppTF pAq ´ ˜TSp∇Xqq∇pρnY nqqpθq.
+Proposition 5.3 with Lemma 6.12 gives that
+}J5}CγpR2q ďCp|X˚, }∇S2X}C0pS2qq}pTF pAq ´ ˜TSp∇Xqq∇pρnY nq}CγpR2q,
+and since TF pAq “ ˜TSp∇Xp0qqp0q, we obtain
+}J5}CγpR2q ď Cp|X˚, }∇S2X}C0pS2q, }T }C2q
+´
+εpRq}∇S2pρnY q}CγpS2q
+` }∇S2X}CγpS2q}∇S2pρnY q}C0pS2q
+¯
+.
+Then, we continue from (7.19),
+}I1,1
+n }CγpS2q ě }J3
+n}CγpR2q ´ }J4
+n}CγpR2q ´ }J5
+n}CγpR2q ´ }J6
+n}CγpR2q,
+so inserting back the bounds for J4
+n, J5
+n, and J6
+n, we have that
+}I1,1
+n }CγpS2q ě }J3
+n}CγpR2q
+´ Cp|X|˚, }∇S2X}C0pS2qqεpRq}∇S2pρnY q}CγpS2q
+´ Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpS2q}∇S2pρnY q}C0pS2q
+´ CpR, |X|˚, }∇S2X}C0pS2qq}∇S2pρnY q}C0pS2q.
+Then, we use the frozen-coefficient estimate in Proposition 6.13 for J3
+n (7.19). We
+first interpolate the inequalities in Theorem 6.11 to control the lower-order terms,
+J3
+n “ pz ´ LAqpρnY nqpθq,
+(7.23)
+}J3
+n}CγpR2q ě C|z|}ρnY n}CγpR2q ` C}ρnY n}C1,γpR2q ` C|z|1´σ}ρnY n}Cγ`σpR2q,
+where σ P r0, 1s is chosen so that 1 ă γ ` σ ă 1 ` γ. Therefore, we have
+}I1,1
+n }CγpR2q ě C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q
+´ Cp|X|˚, }∇S2X}C0pS2qqεpRq}∇S2pρnY q}CγpS2q
+´ Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpS2q}∇S2pρnY q}C0pS2q
+´ CpR, |X|˚, }∇S2X}C0pS2qq}∇S2pρnY q}C0pS2q.
+
+64
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+and taking R small enough,
+(7.24)
+}I1,1
+n }CγpR2q ě C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q
+´ CpR, |X|˚, }∇S2X}CγpS2qq}∇S2pρnY q}C0pS2q.
+Next, we go back to (7.16) and substitute the above bound together with (5.17)
+and (7.24),
+}pz ´ S1pXqqY }CγpS2q
+ě
+ÿ
+n
+´
+C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q
+¯
+´ CpR, |X|˚, }∇S2X}CγpS2qq}∇S2Y }C0pS2q
+´ Cp|X|˚, }∇S2X}CγpS2qq}∇S2ρn}CγpS2qp}Y }CγpS2q ` }∇S2Y }C0pS2qq.
+Plugging this inequality in (7.13), and then using the triangle inequality and the
+fact that Cα ãÑ Cβ for α ě β, we obtain
+}pz ´ SpXqqY }CγpS2q ě C|z|}Y }CγpS2q ` C}Y }C1,γpS2q ` C|z|1´σ}Y }Cγ`σpS2q
+´ CpR, |X|˚, }∇S2X}CγpS2qq}Y }Cγ`σpS2q.
+Finally, by moving the sector if necessary to make |z| big, we conclude the result
+}pz ´ SpXqqY }CγpS2q ě C|z|}Y }CγpS2q ` C}Y }C1,γpS2q.
+□
+Proposition 7.6. The Gˆateaux derivative of F at any X P Om, SpXq (7.2),
+generates an analytic semigroup on the space h0,γpS2q.
+Proof. We need to prove that the operator SpXq is sectorial, i.e., that there exists
+a sector such that for any z in the sector
+}pz ´ SpXqq´1Y }hγpS2q ď C
+|z|}Y }hγpS2q.
+Since the norm on little H¨older spaces hγpS2q is the same as in the usual H¨older
+spaces CγpS2q, from the previous Proposition 7.5 we are left to prove that the
+operator pz ´ SpXqq is invertible from hγpS2q to h1,γpS2q for any z in the sector.
+Similarly as we did in Section 6, define the following family of operators SαpXq,
+α P r0, 1s,
+SαpXqY ppxq
+“ ´α
+ż
+S2∇S2
+´
+G1pXppxq´Xppyqq`αG2pXppxq´Xppyqq
+¯
+¨pTSp∇S2Xq∇S2Y ppyqqdpy
+´p1 ´ αq
+ż
+S2∇S2
+´
+G1pXppxq´Xppyqq`αG2pXppxq´Xppyqq
+¯
+¨ ∇S2Y ppyqdpy
+` αS2pXqY ppxq,
+with
+Gαpxq “ 1
+8π pG1pxq ` αG2pxqq , x “ px1, x2, x3q,
+pG1qi,jpxq “ δij
+|x|,
+pG2qi,jpxq “ xixj
+|x|3 ,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+65
+and TS given in (7.8). In particular, SpXq “ S1pXq. Propositions 7.4 and 7.5
+hold analogously for SαpXq, as all the remainder estimates were always done inde-
+pendently for each part of the kernel G and Proposition 6.13 already included the
+parameter α. In particular, for all α P r0, 1s, it holds that
+1
+C }Y }C1,γpS2q ě }pz ´ SαpXqqY }CγpS2q ě C}Y }C1,γpS2q.
+Then, by the method of continuity, it suffices to show that the inverse of pz´S0pXqq
+exists. Additionally, define a new family of operators S0,σpXq, σ P r0, 1s, as follows
+S0,σpXqY ppxq
+“ ´
+ż
+S2∇S2
+´
+p1´σqG1ppx´ pyq`σG1pXppxq´Xppyqq
+¯
+¨ ∇S2Y ppyqdpy,
+so that S0,1pXq “ S0pXq. Then, taking into account (6.33), it is clear that the
+following bound holds for all σ P r0, 1s,
+1
+C }Y }C1,γpS2q ě }pz ´ S0,σpXqqY }CγpS2q ě C}Y }C1,γpS2q.
+Hence, by the method of continuity again we just need to show that pz ´ S0,0pXqq
+is invertible. Since the range is closed, it suffices to show that it is also dense. The
+operator S0,0pXq is linear and explicit, so we can compute its eigenspace. Since
+S0,0pXqY “ 1
+8π
+ż
+S2
+1
+|px ´ py|∆S2Y ppyqdpy,
+we only have to check a component. From [21], a single layer potential u pxq of
+g ppyq with
+u pxq “ 1
+4π
+ż
+S2
+1
+|x ´ py|g ppyq dpy
+can be transformed into a harmonic problem with
+∆u “ 0
+in R3zS2
+�u� “ 0,
+�∇u ¨ n� “ g
+on S2
+(7.25)
+If we denote the standard spherical coordinate system pr, θ, ϕq, where r is the
+radial coordinate, θ is the polar angle, and ϕ is the azimuthal angle, then for
+the harmonic equation on R3zS2, by separation of variables [15], we obtain some
+solutions uℓm pr, θ, ϕq with l ě 0 and |m| ď ℓ :
+uℓm pr, θ, ϕq “
+"
+ArℓYℓm,
+|r| ă 1
+Br´pℓ`1qYℓm,
+|r| ą 1
+where Yl,m pθ, ϕq is the usual spherical harmonic function of degree l and order m,
+which satisfies the following equation:
+(7.26)
+∆S2Yℓm “
+1
+sin θ
+B
+Bθ
+ˆ
+sin θBYℓm
+Bθ
+˙
+`
+1
+sin2 θ
+B2Yℓm
+Bϕ2
+“ ´ℓpℓ ` 1qYℓm.
+By plugging uℓm into (7.25), we obtain
+uℓm “
+1
+2ℓ ` 1Yℓm.
+Therefore, combining (7.26),
+S0,0pXqYℓ,m “ ´ ℓpℓ ` 1q
+2p2ℓ ` 1qYℓ,m,
+ℓ ě 0,
+|m| ď ℓ.
+
+66
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Finally, since finite linear combinations of Yl,m are dense in C8pS2q, we conclude
+the existence of the inverse pz ´ S0,0pXqq´1 : hγpS2q Ñ h1,γpS2q.
+□
+8. Higher Regularity
+Following the notation in Section 4.1, we recall that
+BX
+Bt ppxq “ NpXqpT p∇S2Xqqppxq,
+where we denote, with T given in (4.4),
+T p∇S2Xq “ T p|∇S2X|q∇S2X.
+We localize using the partition tρnu (see Section 3.4), and linearize T at Xppxnq,
+B
+BtpρnXqppxq “ ρnppxqNpXqpTSp∇S2Xppxnqq∇S2Xqqppxq
+` ρnppxqNpXq
+´
+T p∇S2Xq´TSp∇S2Xppxnqq∇S2X
+¯
+ppxq,
+where we recall that TSp∇S2Xq∇S2Y “
+d
+dsT p∇S2pX `sY qq|s“0 was given in (7.8).
+Next, we introduce the commutators,
+(8.1)
+B
+BtpρnXqppxq “ NpXqpTSp∇S2Xppxnqq∇S2pρnXqqppxq
+` NpXq
+´
+ρn
+`
+T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X
+˘¯
+ppxq
+` rρn, NpXqspTSp∇S2Xppxnqq∇S2Xqppxq
+´ NpXqpTSp∇S2XppxnqqX∇S2ρnqppxq
+` rρn, NpXqs
+´
+T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X
+¯
+ppxq,
+and we move to stereographic coordinates to introduce the frozen-coefficient (at
+t “ 0, px “ pxn) operator and the cutoff pρn (5.14),
+(8.2)
+B
+BtpρnXnqpθq “ LA0pρnXnqpθq `
+7ÿ
+j“1
+f jpXqpθq,
+with
+f 1pXqpθq “ NpXq
+`
+ρn
+`
+T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X
+˘˘
+npθq,
+f 2pXqpθq “ pρnpθq
+“
+NpXq ´ MpAq
+‰
+pTSp∇S2Xppxnqq∇S2pρnXqqnpθq,
+f 3pXqpθq “ pρnpθq
+`
+MpAqpTSp∇S2Xppxnqq∇S2pρnXqqnpθq´LApρnXnqpθq
+˘
+,
+f 4pXqpθq “ p1 ´ pρnpθqqNpXqpTSp∇S2Xppxnqq∇S2pρnXqqnpθq,
+f 5pXqpθq “ ´p1 ´ pρnpθqqLApρnXnqpθq,
+f 6pXqpθq “ rLA ´ LA0spρnXnqpθq,
+and
+f 7pXqpθq “ ´NpXqpTSp∇S2XppxnqqX∇S2ρnqpx
+Xnpθqq
+` rρn, NpXqsT p∇S2Xqpx
+Xnpθqq,
+where A0 “ ∇X0,np0q and A “ ∇Xnp0, tq.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+67
+Proposition 8.1. Let X be the solution to the Peskin problem with initial data
+X0 P h1,γpS2q constructed in Theorem 7.2. Then, for any α P p0, 1q, it holds that
+X P C1pp0, T s; C3,αpS2qq. Moreover, for any 3 ď n P N and α P p0, 1q, assuming
+that T P Cn,α, it holds that X P C1pp0, T s; Cn`1,βpS2qq, for any β ă α.
+Proof. The main difficulty is to show the smoothing in space. In fact, assume we
+have the higher regularity information X P L8p0, T ; Cn`1,αpS2qq. Then, Theorem
+7.2 states that BtX P C0pr0, T s; CγpS2qq, and using the equation together with X P
+L8p0, T ; Cn`1,αpS2qq, it is straightforward to see that BtX P L8p0, T ; Cn,αpS2qq.
+Finally, to get the continuity in time for the higher regularity, it suffices to inter-
+polate taking into account the higher regularity bounds and the continuity in the
+lower norm.
+We proceed to show the smoothing in space.
+We will consider the following
+mollified version of the system (8.2),
+(8.3)
+B
+BtpρnXδ
+nqpθq “ LA0pρnXδ
+nqpθq `
+7ÿ
+j“1
+Jδf jpXδqpθq,
+with mollified initial data Xδ
+0,npθq “ JδX0,npθq, where Jδ is the standard mollifier
+by convolution with a Gaussian.
+Our main goal is to obtain uniform in δ bounds for Xδ in L8p0, T ; Cn,αpS2qq.
+In fact, by construction, Xδ is smooth, and it is not hard to show that the
+limit of tXδu in L8p0, T ; C1,γpS2qq is given by the solution X in Theorem 7.2.
+Hence, by interpolation and using the uniform bounds, we would conclude that
+X P L8p0, T ; Cn,βpS2qq for any β ă α. We thus proceed to obtain the uniform
+bounds first, and show the convergence Xδ Ñ X at the end.
+We use the semigroup etLA0 to write ρnXδ
+nptq in Duhamel form:
+(8.4)
+ρnXδ
+nptq “ ept´t0qLA0 pρnXδ
+npt0qq `
+7ÿ
+j“1
+ż t
+t0
+ept´τqLA0Jδf jpXδqpτqdτ.
+In the following, we will repeatedly use the estimates in Propositions 6.14-6.15. For
+simplicity of notation, we will drop the index δ and the mollifier Jδ.
+Improving regularity to C1,αpS2q: We proceed to obtain bounds in Cα, α P p0, 1q,
+for the terms f j. We will be denoting C “ Cp|X|˚, }X}C1pS2q, }T }C2q, CpRq “
+Cp|X|˚, }X}C1pS2q, }T }C2, Rq in the bounds that follow. Lemma 5.6 gives that
+}f1}CαpR2q ďC}ρnpT p∇S2Xq´TSp∇S2Xppxnqq∇S2Xq}CαpS2q.
+We note that
+I :“ T p∇S2Xppx1qq´TSp∇S2Xppxnqq∇S2Xppx1q
+´ T p∇S2Xppx2qq`TSp∇S2Xppxnqq∇S2Xppx2q
+“ T p∇S2Xppx1qq´Tp∇S2Xppx2qq´TSp∇S2Xppxnqqp∇S2Xppx1q´∇S2Xppx2qq,
+thus
+|I| “ |
+ż 1
+0
+`
+TSps∇S2Xppx1q ` p1 ´ sq∇S2Xppx2qq ´ TSp∇S2Xppxnqq
+˘
+ds
+ˆ p∇S2Xppx1q´∇S2Xppx2qq|
+ď C maxt|∇S2Xppx1q´∇S2Xppxnq|, |∇S2Xppx2q´∇S2Xppxnqu
+ˆ |∇S2Xppx1q´∇S2Xppx2q|.
+
+68
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Hence, thanks to the presence of ρn, we obtain
+(8.5)
+}f 1}CαpR2q ďCεpRq}ρn}CαpS2q}∇S2X}CαpB pxn,2RXS2q.
+Using Lemma 5.10,
+(8.6)
+}f 2}CαpS2q ď C
+´
+εpRq}∇S2pρnXq}CαpS2q
+` }∇S2X}C
+α
+2 pB5RppxnqXS2q}∇S2pρnXq}C
+α
+2 pS2q
+¯
+,
+while f 3 is identically zero (see (7.21) and (4.32)). Lemma 5.7 provides the estimate
+for f 4,
+(8.7)
+}f 4}CαpS2q ď CpRq}∇S2pρnXq}C0pS2q.
+Then, by Lemmas 5.6 and 5.9,
+(8.8)
+}f 7}CαpS2q ď C}∇S2ρn}CαpS2q,
+and Lemma 5.8,
+(8.9)
+}f 5}CαpS2q ď C}∇S2pρnXq}C0pS2q.
+Finally, by writing
+rLA ´ LA0spρnXnqpθq “ r ˜
+MpAq ´ ˜
+MpA0qspTF pAq∇pρnXnqqpθq
+` ˜
+MpA0qppTF pAq ´ TF pA0qq∇pρnXnqqpθq,
+Lemma 5.4 yields that
+(8.10)
+}f 6}CαpS2q ď C}A ´ A0}}∇S2pρnXq}CαpS2q.
+We thus see from (8.4) and Propositions 6.14-6.15 that we can bootstrap to get
+that X P L8p0, T ; C1,αpS2qq for all α P p0, 1q. In fact, consider the case γ ă 1
+2 and
+take α such that γ ă α ` 1
+2 ď 2γ. Then, with 0 ă ǫ ď γ ´ α arbitrarily small, and
+substituting the bounds for f j, we obtain
+}ρnX}L2p0,T ;C
+3
+2 `αpS2qq ď C}ρnp0qX0}C1`α`ǫpS2qq ` C
+7ÿ
+j“1
+}f j}L2p0,T ;C
+1
+2 `αpR2qq
+ď C}ρnp0qX0}C1`γpS2q ` CpR, T q ` }X}2
+L4p0,T ;C1` 1
+4 ` α
+2 pS2qq
+` CεpR, T q
+`
+}X}L2p0,T ;C
+3
+2 `αpB5RppxnqXS2qq ` }ρnX}L2p0,T ;C
+3
+2 `αpS2q
+˘
+,
+and so
+(8.11)
+}ρnX}L2p0,T ;C
+3
+2 `αpS2qq ď C}ρnp0qX0}C1`γpS2q ` CpR, T q
+` CεpR, T q}X}L2p0,T ;C
+3
+2 `αpB5RppxnqXS2qq.
+Now, we can write
+}X}L2p0,T ;C
+3
+2 `αpB5RppxnqXS2qq ď }
+ÿ
+mPMn
+ρmX}L2p0,T ;C
+3
+2 `αpB5RppxnqXS2qq,
+where the cardinal number |Mn| can be picked independent of R and n, since the
+radius of the support of ρn and B5Rppxnq X S2 are comparable. Therefore, adding
+in n in (8.11) we obtain
+ÿ
+n
+}ρnX}L2p0,T ;C
+3
+2 `αpS2qq ď CpR, T q ` CεpR, T q|Mn|
+ÿ
+n
+}ρnX}L2p0,T ;C
+3
+2 `αpS2qq,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+69
+hence we conclude that
+}X}L2p0,T ;C
+3
+2 `αpS2qq ď
+ÿ
+n
+}ρnX}L2p0,T ;C
+3
+2 `αpS2qq ď CpR, T q.
+In particular, choosing α “ 2γ ´ 1
+2, this uniform bound allows us to conclude that
+X P L2p0, T ; C1`2γpS2qq, and thus Xptq P C1`2γpS2q for a.e. t P p0, T q. Now,
+pick t0 P p0, T q arbitrarily close to 0 and such that Xpt0q P C1`2γpS2q. It is clear
+that we can repeat the process to find t1 ą t0 such that Xpt1q P C1,αpS2q for any
+α P p0, 1q (the case γ ą 1
+2 follows in one step). Starting at t1, we find that
+}ρnXptq}C1,αpS2q ď }ρnXpt1q}C1,αpS2q ` C
+sup
+t1ďτďt
+7ÿ
+m“1
+}fm}CαpS2q.
+We can thus take the supremum in t P pt1, T q and use the previous estimates on
+f m to conclude that X P L8pt1, T ; C1,αpS2qq for any t1 ą 0 and any α P p0, 1q.
+Higher regularity: To study further smoothing, we first show that we can move
+derivatives in px to derivatives in py. In fact, denoting ∆X “ Xppxq ´ Xppyq,
+∇S2NpXqY ppxq “
+“ ´
+ż
+S2 ∇S2,px∇S2, pyGp∆Xq ¨ ∆∇S2Y dpy
+“ ´
+ż
+S2
+´
+´ ∇S2, py∇S2, pyGp∆Xq`∇S2, py
+`
+∇S2,pxGp∆Xq`∇S2, pyGp∆Xq
+˘¯
+¨∆∇S2Y dpy,
+so further integration by parts gives that
+(8.12)
+∇S2NpXqY ppxq “ NpXq∇S2Y ppxq
+´
+ż
+S2 ∇S2, py
+`
+∇S2,pxGp∆Xq`∇S2, pyGp∆Xq
+˘
+¨ ∆∇S2Y ppyqdpy
+“ NpXq∇S2Y ppxq
+´
+ż
+S2 ∇S2, py
+´ B
+Bxi
+Gp∆Xq
+`
+∇S2Xippxq ´ ∇S2Xippyq
+˘¯
+¨ ∆∇S2Y ppyqdpy.
+Therefore, we take a derivative in (8.1) to get
+B
+Bt∇S2pρnXqppxq “ NpXq
+`
+∇S2
+`
+TSp∇S2Xppxnqq∇S2pρnXq
+˘˘
+ppxq
+` NpXq
+´
+∇S2`
+ρn
+`
+T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X
+˘˘¯
+ppxq
+´
+ż
+S2 ∇S2, py
+´ B
+Bxi
+Gp∆Xq
+`
+∇S2Xippxq ´ ∇S2Xippyq
+˘¯
+¨
+ˆ ∆
+`
+TSp∇S2Xppxnqq∇S2pρnXq
+˘
+ppyqdpy
+´
+ż
+S2 ∇S2, py
+´ B
+Bxi
+Gp∆Xq
+`
+∇S2Xippxq ´ ∇S2Xippyq
+˘¯
+¨
+ˆ ∆
+`
+ρn
+`
+T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X
+˘
+ppyqdpy
+` ∇S2rρn, NpXqsT p∇S2Xqppxq ´ ∇S2NpXqpTSp∇S2XppxnqqX∇S2ρnqppxqq.
+
+70
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+We introduce the frozen-coefficient operator and the cutoff pρn,
+B
+Bt∇S2pρnXnqpθq “ LA1p∇S2pρnXq
+˘
+npθq `
+8ÿ
+j“1
+f jpθq,
+f 1pθq “ NpXq
+´
+∇S2`
+ρn
+`
+T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X
+˘˘¯
+npθq,
+f 2pθq “ pρnpθqrNpXq ´ MpAqs
+``
+TSp∇S2Xppxnqq∇S2∇S2pρnXq
+˘˘
+npθq,
+f 3pθq“ pρnpθq
+´
+MpAqpTSp∇S2Xppxnqq∇S2∇S2pρnXqqnpθq´LAp∇S2pρnXqqnpθq
+¯
+,
+f 4pθq “ p1´pρnpθqqNpXq
+`
+TSp∇S2Xppxnqq∇S2∇S2pρnXq
+˘
+npθq,
+f 5pθq “ p1´pρnpθqqLAp∇S2pρnXqqnpθq,
+f 6pθq “ rLA ´ LA1s
+`
+∇S2pρnXq
+˘
+npθq,
+f 7pθq “ ∇S2rρn, NpXqsT p∇S2Xqpx
+Xnpθqq
+´ ∇S2NpXqpTSp∇S2XppxnqqX∇S2ρnqpx
+Xnpθqq,
+and
+f 8pθq “ f 8,1pθq ` f 8,2pθq,
+with
+f 8,1pθq “ ´
+ż
+S2 ∇S2, py
+´ B
+Bxi
+GpXpx
+Xpθq ´ Xppyqq
+`
+∇S2Xipx
+Xpθqq ´ ∇S2Xippyq
+˘¯
+¨
+ˆ ∆
+`
+TSp∇S2Xppxnqq∇S2pρnXq
+˘
+ppyqdpy,
+f 8,2pθq “ ´
+ż
+S2 ∇S2, py
+´ B
+Bxi
+GpXpx
+Xpθqq ´ Xppyqq
+`
+∇S2Xipx
+Xpθqq ´ ∇S2Xippyq
+˘¯
+¨
+ˆ ∆
+`
+ρn
+`
+T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X
+˘
+ppyqdpy,
+and A1 “ ∇Xnp0, t1q, A “ ∇Xnp0, tq. Thus, we proceed as we previously did in
+(8.4),
+(8.13) ∇S2pρnXnqptq “ ept´t0qLA1p∇S2pρnXnqpt0qq `
+8ÿ
+j“1
+ż t
+t0
+ept´τqLA1f jpτqdτ.
+Therefore, to bootstrap and get C2,α regularity we need to use Propositions 6.14-
+6.15 and obtain Cα estimates for the forced terms above. The estimate (5.18) in
+Lemma 5.10 gives that
+}f 2}CαpS2q ď C
+´
+εpRq}∇2
+S2pρnXq}CαpS2q
+` }∇S2X}CαpS2q}∇2
+S2pρnXq}C0pS2q ` }∇2
+S2pρnXq}C0pS2q
+¯
+,
+while f 3 ” 0, and Lemmas 5.7 and 5.8 provide that
+}f 4}CαpS2q ` }f 5}CαpS2q ď CpRq}∇2
+S2X}C0pS2q.
+As done before in (8.10), we have that
+}f 6}CαpS2q ď C}A ´ A1}}∇2
+S2pρnXq}CαpS2q.
+We interpolate the C2pS2q norm followed by Young’s inequality to get a small
+coefficient for the higher regularity part:
+}∇2
+S2pρnXq}C0pS2q ď Cpεq}∇S2pρnXq}C1´αpS2q ` ε}∇2
+S2pρnXq}CαpS2q,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+71
+so that
+6ÿ
+j“2
+}f j}CαpS2q ď CεpR, ∆tq}∇2
+S2pρnXq}CαpS2q`CpRq}∇S2pρnXq}C1´αpS2q,
+where from now on the constants C and CpR, ∆tq, ∆t “ t ´ t1, also depend on the
+controlled norm }X}L8p0,T ;C1,maxtα,1´αupS2qq. Next,
+}f 1}CαpS2q ď Cε}∇S2X}CαpS2q
+` C}ρn
+`
+TSp∇S2Xq∇2
+S2X ´ TSp∇S2ppxnqq∇2
+S2X
+˘
+}CαpS2q
+ď C ` C}∇S2X}CαpS2q}∇2
+S2X}C0pB pxn,2RXS2q
+` CεpRq}∇2
+S2X}CαpB pxn,2RXS2q,
+so by interpolation again
+}f 1}CαpS2q ď CpRq ` CεpRq}∇2
+S2X}CαpB pxn,2RXS2q.
+The term f 8 is lower order, and thus we can control it using interpolation once
+more. In fact, taking the derivative in the kernel, we have for f 8,1
+f 8,1ppxq “
+ż
+S2
+B
+Bxj
+B
+Bxi
+Gp∆Xq∇S2Xjppyq∆∇S2Xi ¨ ∆
+`
+TSp∇S2Xppxnqq∇S2pρnXq
+˘
+dpy
+´
+ż
+S2
+B
+Bxi
+Gp∆Xq∇2
+S2Xippyq ¨ ∆
+`
+TSp∇S2Xppxnqq∇S2pρnXq
+˘
+dpy,
+and therefore, proceeding as in Lemma 5.9, we obtain
+}f 8,1}CαpS2q ď C ` C}∇2
+S2X}C0pB pxn,2RXS2q
+ď CpRq ` CεpRq}∇2
+S2X}CαpB2RppxqXS2qq.
+The estimate for f 8,2 follows in the same manner. Next, we estimate the commu-
+tator terms, f 7. Using (8.12), we write
+f 7ppxq “ f 7,1ppxq ` f 7,2ppxq ` f 7,3ppxq,
+with
+f 7,1ppxq “ ´rNpXq∇S2, ρnsT p∇S2Xqppxq,
+f 7,2ppxq “
+ż
+S2 ∇S2, py
+´ B
+Bxi
+Gp∆Xq
+`
+∇S2Xippxq ´ ∇S2Xippyq
+˘¯
+¨ ∆
+`
+ρnT p∇S2Xqppyq
+˘
+dpy
+´ ρnppxq
+ż
+S2 ∇S2, py
+´ B
+Bxi
+Gp∆Xq
+`
+∇S2Xippxq ´ ∇S2Xippyq
+˘¯
+¨ ∆
+`
+T p∇S2Xqppyq
+˘
+dpy
+`
+ż
+S2∇S2, py
+´ B
+Bxi
+Gp∆Xq
+`
+∇S2Xippxq´∇S2Xippyq
+˘¯
+¨∆
+`
+TSp∇S2XppxnqqX∇S2ρnppyq
+˘
+dpy,
+and
+f 7,3ppxq “ ´NpXq
+`
+TSp∇S2Xppxnqq∇S2pX∇S2ρnq
+˘
+ppxq
+` ∇S2ρnNpXqpT p∇S2Xqqppxq.
+The term f 7,3 is lower order and it only requires C1,αpS2q regularity for X, while
+the estimate for f 7,2 follows taking the derivative of the kernel, as done for f 8. We
+get that
+}f 7,2}CαpS2q ď C ` C}∇2
+S2X}C0pB pxn,2RXS2q ` C}∇2
+S2X}C0pS2q.
+By interpolation,
+}f 7,2}CαpS2q ` }f 7,3}CαpS2q ď Cp˜εq ` C˜ε}∇2
+S2X}CαpS2q,
+
+72
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+with ˜ε ą 0 to be chosen. The term f 7,1 is written as follows:
+f 7,1ppxq “ ´
+ż
+S2 ∇S2, pyGpXppxq´Xppyqq
+¨
+`
+ρnppxq∇S2T p∇S2Xppyqq ´ ∇S2
+`
+ρnppyqT p∇S2Xppyqq
+˘˘
+dpy
+“ rρn, NpXqs∇S2T p∇S2Xqppxq
+`
+ż
+S2 ∇S2, pyGpXppxq´Xppyqq ¨ ∇S2ρnppyqT p∇S2Xppyqqdpy,
+hence, by Lemmas 5.9 and 5.6, we have that
+}f 7,1}CαpS2q ď C}∇S2ρn}C1,αpS2q ` C}∇S2ρn}C0pS2q}∇2
+S2X}C0pS2q,
+and, by interpolation,
+}f 7,1}CαpS2q ď Cp˜εq ` C˜ε}∇2
+S2X}CαpS2q.
+Then, we have that for any t1 ą 0 and α P p0, 1
+2q,
+}∇S2pρnXqptq}L2pt1,T ;C1,αpS2qq ď C}ρnpt1qXpt1q}C
+3
+2 `α´ǫ
+`
+7ÿ
+m“1
+}f mpτq}L2pt1,T ;CαpS2qq.
+Hence, introducing the estimates above for f j and summing in n, we take the
+partition so that εpRq is small enough and then choose ˜ε small enough (depending on
+the partition ρn), to obtain that X P L2pt1, T ; C2,αpS2qq. Finally, we can take t2 ą
+t1 so that Xpt2q P C2,αpS2q and use (8.4) to conclude that X P L8pt2, T ; C2,αpS2qq
+for any t2 ą 0, α P p0, 1
+2q. Now, starting with the upgraded regularity and repeating
+the same steps with no changes, we conclude that X P L8pt2, T ; C2,αpS2qq for any
+t2 ą 0, α P p0, 1q.
+It is not difficult to show by induction that an analogous formula to (8.12) holds
+for higher derivatives. Then, by repeating the steps above one can continue the
+bootstrapping argument, concluding that for any n P N, X P L8p0, T ; Cn,αpS2qq.
+Xδ Ñ X in L8p0, T ; C1,γpS2qq: We write the difference ∆δX :“ Xδ´X as follows:
+ρn∆δXnptq “ etLA0 `
+ρn∆δX0,n
+˘
+`
+7ÿ
+j“1
+ż t
+0
+ept´τqLA0
+´
+pJδ ´ 1qf jpXδqpτq ` f jpXδqpτq ´ f jpXqpτq
+¯
+dτ,
+with f jpXq given in (8.2). Thus,
+(8.14)
+}ρn∆δXn}L8p0,T ;C1,γpR2qq ď C}ρn∆δX0,n}C1,γpR2q
+` C
+7ÿ
+j“1
+}pJδ ´ 1qf jpXδq}L8p0,T ;CγpR2qq
+` C sup
+tPr0,T s
+7ÿ
+j“1
+}
+ż t
+0
+ept´τqLA0pf jpXδq ´ f jpXqqpτq}CγpR2q.
+Since X0 P h1,γpS2q and f jpXδq P hγpS2q, the first two terms converge to zero as
+δ Ñ 0. For the third term, we need to show that it can be absorbed by the left-hand
+side. As in the previous arguments, we will show that for the quasilinear terms we
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+73
+can find a small coefficient, while for the lower order ones we will take advantage
+of the extra regularity via (6.14) to get a small coefficient for T small enough.
+From previous estimates we immediately get that for j “ 4, 5, 7 and 0 ă ǫ ă 1´γ,
+}f jpXδq ´ f jpXq}Cγ`ǫpS2q ď C}∆δX}C1pS2q,
+hence (6.14) gives that
+sup
+tPr0,T s
+ÿ
+j“4,5,7
+}
+ż t
+0
+ept´τqLA0pf jpXδq ´ f jpXqqpτq}CγpR2q ď C T ǫ}∆δX}C1pS2q.
+Also, for f 6,
+}f 6pXδq ´ f 6pXq}CγpS2q ď C}A ´ A0}}∇S2pρn∆δXq}CγpS2q
+` C}∆δX}C1pS2q}ρnXδ}C1,γpS2q,
+so that (6.14)-(6.15) give
+}
+ż t
+0
+ept´τqLA0pf 6pXδq ´ f 6pXqqpτq}L8p0,T ;CγpR2qq ď CεpT q}ρn∆δX}C1,γpS2q
+` CT ε}∆δX}C1pS2q.
+Next, we proceed with the term f 1:
+pf 1pXδq ´ f 1pXqqpθq “ I1 ` I2,
+with
+I1pθq “ pNpXδq ´ NpXqq
+`
+ρn
+`
+T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ˘˘
+npθq
+I2pθq “ NpXq
+´
+ρn
+´
+T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ
+´ T p∇S2Xq ` TSp∇S2Xppxnqq∇S2X
+¯¯
+npθq.
+The first term is estimated easily from Proposition 5.6 by noticing that one can
+always extract ∆δX “ Xδ ´X from the difference of the kernels (similarly as done
+in the proof of Proposition 7.3),
+(8.15)
+}I1}CγpR2q ďC}∇S2∆δX}C0pS2q}ρnpT p∇S2Xδq´TSp∇S2Xδppxnqq∇S2Xδq}CγpS2q
+ď CεpRq}∇S2∆δX}C0pS2q}∇S2X}CγpB pxn,2RXS2q,
+where we have used (8.5) in the second step. Proposition 5.6 gives that
+(8.16)
+}I2}CγpR2q “ C}ρn
+´
+T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ
+´ T p∇S2Xq ` TSp∇S2Xppxnqq∇S2X
+¯
+}CγpR2q.
+Denote
+JpY qppxq “ T p∇S2Y ppxqq ´ TSp∇S2Y ppxnqq∇S2Y ppxq,
+so that we can write
+JpXδppx1qq ´ JpXδppx2qq “ p∇S2Xδppx1q ´ ∇S2Xδppx2qq
+ˆ
+ż 1
+0
+`
+TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq
+˘
+ds1.
+
+74
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Then,
+JpXδppx1qq ´ JpXδppx2qq ´ JpXppx1qq ` JpXppx2qq “ J1 ` J2,
+with
+J1 “ p∇S2∆δXppx1q ´ ∇S2∆δXppx2qq
+ˆ
+ż 1
+0
+`
+TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq
+˘
+ds1,
+and
+J2 “ p∇S2Xppx1q ´ ∇S2Xppx2qq
+ˆ
+´ ż 1
+0
+`
+TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq
+˘
+ds1
+´
+ż 1
+0
+`
+TSps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ´ TSp∇S2Xppxnqq
+˘
+ds1
+¯
+.
+It follows that
+|J1| ď C maxt|∇S2Xδppx1q´∇S2Xδppxnq|, |∇S2Xδppx2q´∇S2Xδppxnqu
+ˆ |∇S2∆δXppx1q´∇S2∆δXppx2q|.
+We apply the mean-value theorem again in J2:
+ż 1
+0
+`
+TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq
+˘
+ds1
+“
+ż 1
+0
+ż 1
+0
+DTS
+`
+s2ps1∇S2Xδppx1q`p1´s1q∇S2Xδppx2qq`p1´s2q∇S2Xδppxnq
+˘
+ds1ds2
+ˆ
+`
+s1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2q ´ ∇S2Xδppxnq
+˘
+,
+hence adding and subtracting we obtain
+|J2| ď |J2,1| ` |J2,2|,
+with
+|J2,1| ď |∇S2Xppx1q ´ ∇S2Xppx2q|
+ˆ maxt|∇S2Xδppx1q´∇S2Xδppxnq|, |∇S2Xδppx2q´∇S2Xδppxnq|u
+ˆ
+ˇˇˇDTSps2ps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ` p1 ´ s2q∇S2Xδppxnqq
+´ DTSps2ps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ` p1 ´ s2q∇S2Xppxnqq
+ˇˇˇ,
+|J2,2| ď |∇S2Xppx1q ´ ∇S2Xppx2q|
+ˆ |DTSps2ps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ` p1 ´ s2q∇S2Xppxnqq|
+ˆ maxt|∇S2∆δXppx1q´∇S2∆δXppxnq|, |∇S2∆δXppx2q´∇S2∆δXppxnqu.
+Going back to (8.16), we thus conclude that
+}I2}CγpR2q ď CεpRq}∇S2∆δX}CγpB pxn,2RXS2q.
+Together with the bound for I1 (8.15), we obtain the following estimate for f 1:
+}
+ż t
+0
+ept´τqLA0pf 1pXδq ´ f 1pXqqpτq}L8p0,T ;CγpR2qq ď C T ǫ}∆δX}C1pS2q
+` CεpRq}∆δX}C1pS2q}X}C1,γpB pxn,2RXS2q ` CεpRq}∆δX}C1,γpB pxn,2RXS2q.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+75
+The estimate for f 2 follows in the same way than those for f 1 and f 6, from which
+we conclude that
+}ρn∆δXn}L8p0,T ;C1,γpR2qq ď C}ρn∆δX0,n}C1,γpR2q
+` C
+7ÿ
+j“1
+}pJδ ´ 1qf jpXδq}L8p0,T ;CγpR2qq
+` CT ε}∆δX}C1pS2q ` CεpRq}∆δX}C1pS2q}X}C1,γpB pxn,2RXS2q
+` CpεpT q ` εpRqq}∆δX}C1,γpB px,2RXS2q.
+Taking R and T small enough, the last term is absorbed by the left-hand side. Then,
+adding in n, we conclude that, for T and R small enough, the desired estimate holds
+}∆δX}L8p0,T ;C1,γpS2qq ďC}∆δX0}C1,γpS2q`C
+7ÿ
+j“1
+}pJδ´1qfjpXδq}L8p0,T ;CγpR2qq.
+□
+Appendix A. Besov Spaces and Fourier Multiplier Theorems
+In this section, we will proof Theorem 6.1. First, define Besov Spaces Bγ
+p,q by a
+dyadic decomposition. Set a function ψ pξq P C8 pRnq s.t.
+ψ pξq “
+"
+1
+|ξ| ď 1
+0
+|ξ| ě 2
+and define φ pξq :“ ψ pξq ´ ψ p2ξq. Hence, φ pξq P C8 pRnq and
+φ pξq “ 0,
+|ξ| ď 1
+2, |ξ| ě 2,
+8
+ÿ
+j“´8
+φ
+`
+2´jξ
+˘
+“ 1,
+|ξ| ‰ 0.
+Next, the homogeneous dyadic blocks 9∆j are defined by
+9∆jf pθq :“ F´1 `
+φ
+`
+2´jξ
+˘
+Ff pξq
+˘
+pθq “ Kj ˚ f
+(A.1)
+where K pθq :“ F´1 pφ pξqq pθq and Kj pθq :“ F´1 `
+φ
+`
+2´jξ
+˘˘
+pθq “ 2jnK
+`
+2jθ
+˘
+.
+Now, given γ a real number and p, q ě 1, we may define homogeneous Besov spaces
+9Bγ
+p,q pRnq with its seminorm ∥¨∥ 9Bγ
+p,qpRnq by
+∥f∥ 9Bγ
+p,qpRnq :“
+˜
+8
+ÿ
+j“´8
+ˆ
+2jγ ��� 9∆jf
+���
+LppRnq
+˙q¸ 1
+q
+,
+(A.2)
+∥f∥ 9Bγ
+p,8pRnq :“ sup
+jPZ
+ˆ
+2jγ ��� 9∆jf
+���
+LppRnq
+˙
+.
+(A.3)
+According to [27, Remark 2.2.2] and [48, Lemma 8.4.2], we know for all 0 ă γ ă
+1, ∥¨∥ 9Bγ
+p,qpRnq and �¨�CγpRnq are equivalent, so we only need to prove the Fourier
+multiplier theorem on 9Bγ
+p,q pRnq. The proof is from [48, Theorem 8.4.3].
+Given T a Fourier multiplier operator with multiplier m pξq P Cs pRnzt0uq X
+L8 pRnq, for s ą n
+2 and for all |α| ď s, such that
+��Bα
+ξ m pξq
+�� ď Cα |ξ|´|α| ,
+(A.4)
+
+76
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+we first define a related kernel with λ ą 0 by Kλ pθq :“ F´1 pφ pξq m pλξqq pθq.
+Lemma A.1. Given m pξq satisfying (A.4), then Kλ pθq is bounded by
+ż
+Rn |Kλ pθq| dθ ď Cs,n,φDm,
+(A.5)
+where Dm “ max|α|ďs Cα
+Proof. Since there exist Cs s.t. for all θ P Rn
+´
+1 ` |θ|2¯s
+ď Cs
+ÿ
+|α|ďs
+|θα|2 ,
+(A.6)
+we obtain
+ż
+Rn |Kλ pθq|2 ´
+1 ` |θ|2¯s
+dθ
+ďCs
+ÿ
+|α|ďs
+ż
+Rn |θαKλ pθq|2 dθ
+“Cn,s
+ÿ
+|α|ďs
+ż
+Rn
+��Bα
+ξ pφ pξq m pλξqq
+��2 dξ
+“Cn,s
+ÿ
+|α|ďs
+ż
+Rn
+�����
+ÿ
+βďα
+ˆ
+α
+β
+˙ ´
+Bβ
+ξ φ
+¯
+pξq λ|α´β| ´
+Bα´β
+ξ
+m
+¯
+pλξq
+�����
+2
+dξ
+ďCn,sD2
+m
+ÿ
+|α|ďs
+ż
+Rn
+�����
+ÿ
+βďα
+ˆ
+α
+β
+˙ ´
+Bβ
+ξ φ
+¯
+pξq λ|α´β| |λξ|´|α´β|
+�����
+2
+dξ.
+(A.7)
+supp pφq Ă
+␣
+ξ| 1
+2 ď |ξ| ď 2
+(
+, so
+ż
+Rn
+�����
+ÿ
+βďα
+ˆ
+α
+β
+˙ ´
+Bβ
+ξ φ
+¯
+pξq λ|α´β| |λξ|´|α´β|
+�����
+2
+dξ
+“
+ż
+1
+2 ď|ξ|ď2
+�����
+ÿ
+βďα
+ˆ
+α
+β
+˙ ´
+Bβ
+ξ φ
+¯
+pξq |ξ|´|α´β|
+�����
+2
+dξ ď Cφ,α.
+(A.8)
+Thus,
+ż
+Rn |Kλ pθq|2 ´
+1 ` |θ|2¯s
+dθ ď Cn,sD2
+m
+ÿ
+|α|ďs
+Cφ,α ď Cs,n,φD2
+m,
+(A.9)
+and by Holder inequality,
+ż
+Rn |Kλ pθq| dθ ď
+ˆż
+Rn |Kλ pθq|2 ´
+1 ` |θ|2¯s
+dθ
+˙ 1
+2 ˆż
+Rn
+´
+1 ` |θ|2¯´s
+dθ
+˙ 1
+2
+ď
+a
+Cs,n,φDm
+ˆż
+Rn
+´
+1 ` |θ|2¯´s
+dθ
+˙ 1
+2
+ď Cs,n,φDm.
+(A.10)
+□
+Next, we may use Kλ pθq and homogeneous Besov semi norm ∥¨∥ 9Bγ
+p,qpRnq to prove
+the Fourier multiplier theorem.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+77
+Proof of Theorem 6.1. First, set
+T j
+mf pθq :“ 9∆jTmf pθq “ F´1 `
+φ
+`
+2´jξ
+˘
+m pξq Ff pξq
+˘
+pθq .
+(A.11)
+Since
+F´1 `
+φ
+`
+2´jξ
+˘
+m pξq
+˘
+pθq “ 2njK2j
+`
+2jθ
+˘
+,
+(A.12)
+��T j
+mf pθq
+�� “
+��F´1 `
+φ
+`
+2´jξ
+˘
+m pξq
+˘
+˚ f pθq
+��
+ď
+��F´1 `
+φ
+`
+2´jξ
+˘
+m pξq
+˘
+pθq
+��
+L1pRnq ∥f∥L8pRnq
+ď
+��2njK2j
+`
+2jθ
+˘��
+L1pRnq ∥f∥L8pRnq
+ď ∥K2j pθq∥L1pRnq ∥f∥L8pRnq
+ďCs,n,φDm ∥f∥L8pRnq .
+(A.13)
+Next, supp
+`
+φ
+`
+2´jξ
+˘˘
+Ă
+␣
+ξ
+ˇˇ2j´1 ď |ξ| ď 2j`1 (
+, so for all j ` 1 ď k ´ 1 or j ´ 1 ě
+k ` 1
+φ
+`
+2´jξ
+˘
+φ
+`
+2´kξ
+˘
+“ 0.
+(A.14)
+Therefore,
+φ
+`
+2´jξ
+˘
+Ff pξq “ φ
+`
+2´jξ
+˘ ´
+F
+´
+9∆j´1f
+¯
+pξq ` F
+´
+9∆jf
+¯
+pξq ` F
+´
+9∆j`1f
+��
+pξq
+¯
+,
+(A.15)
+so we obtain
+T j
+mf pθq “ T j
+m
+´
+9∆j´1f
+¯
+pθq ` T j
+m
+´
+9∆jf
+¯
+pθq ` T j
+m
+´
+9∆j`1f
+¯
+pθq .
+(A.16)
+Finally,
+∥Tmf∥ 9Bγ
+8,8pRnq “ sup
+jPZ
+2jγ ��T j
+mf
+��
+L8pRnq
+ď sup
+jPZ
+2jγ ���T j
+m
+´
+9∆j´1f
+¯���
+L8pRnq ` sup
+jPZ
+2jγ ���T j
+m
+´
+9∆jf
+¯���
+L8pRnq ` sup
+jPZ
+2jγ ���T j
+m
+´
+9∆j`1f
+¯���
+L8pRnq
+ďCs,nDm
+ˆ
+sup
+jPZ
+2jγ ��� 9∆j´1f
+���
+L8pRnq ` sup
+jPZ
+2jγ ��� 9∆jf
+���
+L8pRnq ` sup
+jPZ
+2jγ ��� 9∆j`1f
+���
+L8pRnq
+˙
+“Cs,nDm
+`
+2γ ` 1 ` 2´γ˘
+∥f∥ 9Bγ
+8,8pRnq .
+(A.17)
+Since ∥¨∥ 9Bγ
+p,qpRnq and �¨�CγpRnq are equivalent,
+�Tmu�CγpRnq ď Cγ,s,nDm�u�CγpRnq.
+(A.18)
+□
+Appendix B. Estimates for the semigroup e´tLApξq
+Lemma B.1. For all β “ β1β2 ¨ ¨ ¨ βk, there exists a matrix Pβ
+´
+ˆξ1, ˆξ2
+¯
+of polyno-
+mials with degree deg pPβq ď 3 |β| ` 4 s.t.
+BβLA pξq “
+1
+|ξ||β|´1
+Pβ
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+2|β|`3 .
+(B.1)
+
+78
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+More specifically, Pβ
+´
+ˆξ1, ˆξ2
+¯
+can be written as
+pPβqi1i2
+´
+ˆξ1, ˆξ2
+¯
+“
+ÿ
+j1,j2ě0,j1`j2ď3|β|`4
+cpβ,i1,i2q
+j1,j2
+ˆ
+A, U, P,
+1
+detpBq, T , dT
+dλ , 1
+λ2
+˙
+ˆξj1
+1 ˆξj2
+2 ,
+(B.2)
+where
+cpβ,i1,i2q
+j1,j2
+ˆ
+A, U, P,
+1
+detpBq, T , dT
+dλ , 1
+λ2
+˙
+“cpβ,i1,i2q
+j1,j2
+ˆ
+A11, ¨ ¨ ¨ , A32, U11, ¨ ¨ ¨ , U32, P11, ¨ ¨ ¨ , P33,
+1
+detpBq, T
+λ , dT
+dλ , 1
+λ2
+˙
+(B.3)
+is a polynomial function.
+Moreover,
+���Pβ
+´
+ˆξ1, ˆξ2
+¯���
+C1pDAσ1,σ2q and
+���Uˆξ
+���
+C1pDAσ1,σ2q are uniformly bounded,
+i.e. there exists Cpβq
+σ1,σ2,T and Cpβq
+σ1,σ2 s.t. for all ˆξ P S1,
+���Pβ
+´
+ˆξ1, ˆξ2
+¯���
+C1pDAσ1,σ2q ď Cpβq
+σ1,σ2,T ,
+(B.4)
+���Uˆξ
+���
+C1pDAσ1,σ2q ď Cpβq
+σ1,σ2.
+(B.5)
+Proof. Since
+pFθGα,Aq pξq “ 1
+|ξ| pFθGα,Aq pˆξq
+(B.6)
+“ 1
+|ξ|
+pI ` αPq
+���Uˆξ
+���
+2
+´ αUˆξ b Uˆξ
+4 detpBq
+���Uˆξ
+���
+3
+,
+(B.7)
+and Zpξq “ |ξ|2 Zpˆξq where Zpˆξq is a matrix of polynomials with degree 2,
+LA pξq “ |ξ|
+P0
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+3
+,
+(B.8)
+where
+P0 “
+pI ` αPq
+���Uˆξ
+���
+2
+´ αUˆξ b Uˆξ
+4 detpBq
+˜
+T
+λ
+˜
+I ´ Aˆξ b Aˆξ
+λ2
+¸
+` dT
+dλ
+Aˆξ b Aˆξ
+λ2
+¸
+,
+(B.9)
+where the degree of P0 is 4. Obviously, P0 can be written as (B.2).
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+79
+When |β| “ 1,
+BLA
+Bξi
+pξq “ ξi
+|ξ|
+P0
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+3
+` |ξ|
+ÿ
+j“1,2
+B
+Bˆξj
+P0
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+3
+B
+Bξi
+ξj
+|ξ|
+“ ξi
+|ξ|
+P0
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+3
+` |ξ|
+ÿ
+j“1,2
+BP0
+Bˆξj
+���Uˆξ
+���
+2
+´ 3P0
+´
+U T Uˆξ
+¯
+j
+���Uˆξ
+���
+5
+δij ´ ˆξi ˆξj
+|ξ|
+“
+Pi
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+5
+,
+(B.10)
+where
+Pi
+´
+ˆξ1, ˆξ2
+¯
+“ ˆξiP0
+���Uˆξ
+���
+2
+`
+ÿ
+j“1,2
+˜
+BP0
+Bˆξj
+���Uˆξ
+���
+2
+´ 3P0
+´
+U T Uˆξ
+¯
+j
+¸ ´
+δij ´ ˆξi ˆξj
+¯
+.
+(B.11)
+The degrees of all terms are at most 1 ` 4 ` 2 “ 3 ` 2 ` 2 “ 4 ` 1 ` 2 “ 7. Since
+���Uˆξ
+���
+2
+“
+2ÿ
+j1,j2“1
+3ÿ
+k“1
+Ukj1Ukj2 ˆξj1 ˆξj2,
+(B.12)
+´
+U T Uˆξ
+¯
+j “
+„ř3
+k“1 UkjUk1 ˆξj
+ř3
+k“1 UkjUk2 ˆξj
+
+,
+(B.13)
+and BP0
+Bˆξj can be written as the form of (B.2), Pi
+´
+ˆξ1, ˆξ2
+¯
+can be written as (B.2).
+Thus, the case |β| “ 1 holds.
+Suppose |β| ď k´1 holds, for
+��¯β
+�� “ k, we may rewrite ¯β as ββk where |β| “ k´1.
+Then,
+B ¯βLA pξq “ BβkBβLA pξq “
+B
+Bξβk
+¨
+˚
+˝
+1
+|ξ||β|´1
+Pβ
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+2|β|`3
+˛
+‹‚
+“ ´ p|β| ´ 1q
+1
+|ξ||β| ˆξβk
+Pβ
+���Uˆξ
+���
+2|β|`3
+`
+1
+|ξ||β|´1
+ÿ
+j“1,2
+BPβ
+Bˆξj
+���Uˆξ
+���
+2
+´ p2 |β| ` 3q Pβ
+´
+U T Uˆξ
+¯
+j
+���Uˆξ
+���
+2|β|`5
+δβkj ´ ˆξβk ˆξj
+|ξ|
+“
+1
+|ξ|| ¯β|
+P¯β
+´
+ˆξ1, ˆξ2
+¯
+���Uˆξ
+���
+¯β`3 .
+(B.14)
+
+80
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+where
+Pˆβ “ ´ p|β| ´ 1q ˆξiPβ
+���Uˆξ
+���
+2
+`
+ÿ
+j“1,2
+˜
+BPβ
+Bˆξj
+���Uˆξ
+���
+2
+´
+`
+2
+��¯β
+�� ` 1
+˘
+Pβ
+´
+U T Uˆξ
+¯
+j
+¸ ´
+δβkj ´ ˆξβk ˆξj
+¯
+.
+(B.15)
+The degrees of all terms are at most 1 ` p3 |β| ` 4q ` 2 “ p3 |β| ` 3q ` 2 ` 2 “
+p3 |β| ` 4q ` 1 ` 2 “ 3
+��¯β
+�� ` 4. Again, BPβ
+Bˆξj is still able to written as the form of
+(B.2), so the case
+��¯β
+�� “ k holds. By Induction, for all β, the formulas (B.1) and
+(B.2) hold.
+Next, in Pβ, since for all square matrix M, ∥M∥ ď ř |Mij|, we just need to
+estimate each element pPβqi1i2,
+���pPβqi1i2
+´
+ˆξ1, ˆξ2
+¯��� ď
+ÿ ����cpβ,i1,i2q
+j1,j2
+ˆ
+A, U, P,
+1
+detpBq, T
+λ , dT
+dλ , 1
+λ2
+˙����
+(B.16)
+and cpβ,i1,i2q
+j1,j2
+´
+A, U, P,
+1
+detpBq, T
+λ , dT
+dλ , 1
+λ2
+¯
+is a form of a polynomial, so we may only
+check each variable. Given A P DA,
+���
+`
+AT A
+˘´1��� ,
+1
+detpBq ď
+1
+σ2
+2 and σ1 ď λ ď
+?
+2σ1,
+so all variables are bounded by σ1 and σ2.
+����
+BAT A
+BAij
+���� ď 2λ ď 2
+?
+2σ1,
+(B.17)
+�����
+B
+`
+AT A
+˘´1
+BAij
+����� “
+����
+`
+AT A
+˘´1 BAT A
+BAij
+`
+AT A
+˘´1
+���� ď 2
+?
+2σ1
+σ4
+2
+,
+(B.18)
+so on DA, U, P are C1 functions and their derivatives are bounded by σ1 and σ2.
+Absolutely,
+���Uˆξ
+���
+C1pDAσ1,σ2q ď Cpβq
+σ1,σ2.
+(B.19)
+Since
+�����
+B det
+`
+AT A
+˘
+BAij
+����� ď λ2 ď 8σ2
+1,
+(B.20)
+����
+B
+BAij
+1
+detpBq
+���� “
+�����
+1
+2 detpBq3
+B det
+`
+AT A
+˘
+BAij
+����� ď 8σ2
+1
+σ8
+2
+,
+(B.21)
+and
+����
+Bλ
+BAij
+���� “
+����
+Aij
+λ
+���� ď 1
+(B.22)
+on DA,
+1
+detpBq, T
+λ , dT
+dλ , 1
+λ2 are also C1 functions and their derivatives are bounded
+by σ1 and σ2. Therefore, we obtain
+���Pβ
+´
+ˆξ1, ˆξ2
+¯���
+C1pDAσ1,σ2q ď Cpβq
+σ1,σ2,T .
+(B.23)
+□
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+81
+Lemma B.2. Given A P DAσ1,σ2 and ϕ pξq “ ϕ p|ξ|q, a cutting and decreasing
+respect |ξ| and supported in B p1q, and set
+K0 pθq :“ F´1 ”
+pz ` LAq´1 pξqϕpξq
+ı
+,
+(B.24)
+K1,j pθq :“ F´1 ”
+ξj pz ` LAq´1 pξqϕpξq
+ı
+.
+(B.25)
+Then, for all z P Sω,δ, we have the following estimates
+∥K0 pθq∥ ďCω,δ,σ1,σ2,T
+|z|
+1
+1 ` |θ|3 ,
+(B.26)
+∥K1,j pθq∥ ďCω,δ,σ1,σ2,T
+1
+1 ` |θ|4 .
+(B.27)
+Proof. For convenience, we define Hpξq :“ pz ` LAq´1 pξq First, since
+p1 ` |θ|3q
+ż
+R2 eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ
+“
+ż
+R2p1 ´ i θ
+|θ| ¨ iθ|θ|2qeiθ¨ξ pz ` LAq´1 pξqϕpξqdξ
+“
+ż
+R2
+„
+p1 ´ i θ
+|θ| ¨ ∇ξ|∇ξ|2qeiθ¨ξ
+
+pz ` LAq´1 pξqϕpξqdξ
+“
+ż
+R2 eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ
+|θ| ¨ ∇ξ∆ξ
+”
+pz ` LAq´1 pξqϕpξq
+ı
+dξ
+“
+ż
+Bp1q
+eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ
+|θ| ¨ ∇ξ∆ξ
+”
+pz ` LAq´1 pξqϕpξq
+ı
+dξ,
+(B.28)
+By (6.20),
+�����
+ż
+Bp1q
+eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ
+����� ď Cδ,σ1,σ2,T ∥ϕ∥C0
+|z|
+.
+(B.29)
+Next, we compute
+B
+Bξj
+∆ξ
+”
+pz ` LAq´1 ϕ
+ı
+“ B
+Bξj
+∆ξ pz ` LAq´1 ϕ ` ∆ξ pz ` LAq´1 B
+Bξj
+ϕ ` 2 B
+Bξj
+∇ξ pz ` LAq´1 ¨ ∇ξϕ
+`2∇ξ pz ` LAq´1 ¨ B
+Bξj
+∇ξϕ ` B
+Bξj
+pz ` LAq´1 ∆ξϕ ` pz ` LAq´1 B
+Bξj
+∆ξϕ.
+(B.30)
+Since ϕ is smooth, we may estimate all of the terms except the first by (6.20) and
+(6.24). Obviously, for the last term,
+�����i θj
+|θ|
+ż
+Bp1q
+eiθ¨ξ pz ` LAq´1
+ˆ B
+Bξj
+∆ξϕ
+˙
+dξ
+����� ď Cδ,σ1,σ2,T ∥ϕ∥C3
+|z|
+.
+(B.31)
+
+82
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Then, for all |α| “ 1, 2, |α| ` |β| “ 3
+�����i θj
+|θ|
+ż
+Bp1q
+eiθ¨ξBα
+ξ pz ` LAq´1 pξqBβ
+ξ ϕpξqdξ
+����� ď ∥ϕ∥C2
+ż
+Bp1q
+���Bα
+ξ pz ` LAq´1 pξq
+��� dξ
+ďCδ,σ1,σ2,T
+|z|2
+∥ϕ∥C2
+ż
+Bp1q
+|ξ|1´|α| dξ ď Cδ,σ1,σ2,T
+|z|2
+∥ϕ∥C2 .
+(B.32)
+Now, for the first term,
+B
+Bξj
+∆ξH “2
+ÿ
+k“1,2
+“
+´ pEjkk ` Ekjk ` Ekkjq `
+`
+Ekpjkq ` Epjkqk
+˘‰
+`
+`
+Ejpkkq ` Epkkqj
+˘
+´ H B
+Bξj
+∆ξLAH,
+(B.33)
+where
+Eijk “H BLA
+Bξi
+H BLA
+Bξj
+H BLA
+Bξk
+H,
+Ekpjkq “H BLA
+Bξk
+H B2LA
+BξjBξk
+H,
+Ejpkkq “H BLA
+Bξj
+H∆ξLAH.
+Since
+��� BLA
+Bξj
+��� ≲ 1 and
+��� B2LA
+BξjBξk
+��� ≲ |ξ|´1, ∥Eijk∥ ≲ 1 and
+��Ekpjkq
+�� ,
+��Ejpkkq
+�� ≲ |ξ|´1.
+Therefore, we only have to check pz ` LAq´1
+B
+Bξj ∆ξLA pz ` LAq´1 ϕ. LA is an even
+function, so the term is an odd function. By LA pξq “ |ξ| LA
+´
+ˆξ
+¯
+and (6.10),
+�����i θj
+|θ|
+ż
+Bp1q
+eiθ¨ξ pz ` LA pξqq´1 B
+Bξj
+∆ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ
+�����
+“
+�����´ θj
+|θ|
+ż
+Bp1q
+sin pθ ¨ ξq pz ` LA pξqq´1 B
+Bξj
+∆ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ
+�����
+ď
+������
+ż
+Bp1q
+sin
+´
+θ ¨ ˆξ |ξ|
+¯ ´
+z ` |ξ| LA
+´
+ˆξ
+¯¯´1 Φp3q
+A,j
+´
+ˆξ
+¯
+|ξ|2
+´
+z ` |ξ| LA
+´
+ˆξ
+¯¯´1
+ϕ p|ξ|q dξ
+������
+“
+������
+ż
+S1
+ż 1
+0
+´
+z ` rLA
+´
+ˆξ
+¯¯´1
+Φp3q
+A,j
+´
+ˆξ
+¯ ´
+z ` rLA
+´
+ˆξ
+¯¯´1
+ϕ prq
+sin
+´
+θ ¨ ˆξr
+¯
+r
+drdˆξ
+������
+.
+(B.34)
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+83
+By Lemma B.4, we obtain for all ˆξ P S1
+������
+ż 1
+0
+´
+z ` rLA
+´
+ˆξ
+¯¯´1
+Φp3q
+A,j
+´
+ˆξ
+¯ ´
+z ` rLA
+´
+ˆξ
+¯¯´1
+ϕ prq
+sin
+´
+θ ¨ ˆξr
+¯
+r
+dr
+������
+ď2
+����
+´
+z ` rLA
+´
+ˆξ
+¯¯´1
+Φp3q
+A,j
+´
+ˆξ
+¯ ´
+z ` rLA
+´
+ˆξ
+¯¯´1
+ϕ prq
+����
+C1pr0,1sq
+ď4
+���Φp3q
+A,j
+´
+ˆξ
+¯��� ∥ϕ prq∥C1pr0,1sq
+����
+´
+z ` rLA
+´
+ˆξ
+¯¯´1����
+2
+C1pr0,1sq
+ďCδ,σ1,σ2,T ∥ϕ prq∥C1pr0,1sq
+˜
+1
+|z| `
+1
+|z|2
+¸2
+.
+(B.35)
+Therefore,
+������
+ż
+S1
+ż 1
+0
+´
+z ` rLA
+´
+ˆξ
+¯¯´1
+Φp3q
+A,j
+´
+ˆξ
+¯ ´
+z ` rLA
+´
+ˆξ
+¯¯´1
+ϕ prq
+sin
+´
+θ ¨ ˆξr
+¯
+r
+drdˆξ
+������
+ďCδ,σ1,σ2,T ∥ϕ prq∥C1
+˜
+1
+|z| `
+1
+|z|2
+¸2
+.
+(B.36)
+Since
+1
+|z| ď Cω,δ if z P Sω,δ,
+�����
+ż
+Bp1q
+eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ
+|θ| ¨ ∇ξ∆ξ
+”
+pz ` LAq´1 pξqϕpξq
+ı
+dξ
+�����
+ď ∥ϕ prq∥C3
+4ÿ
+k“1
+Cpkq
+δ,σ1,σ2,T
+1
+|z|k
+ďCω,δ,σ1,σ2,T ∥ϕ∥C3
+|z|
+,
+(B.37)
+and
+����
+ż
+R2 eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ
+���� ďCω,δ,σ1,σ2,T ∥ϕ∥C3
+|z|
+1
+1 ` |θ|3 .
+(B.38)
+Next, for K1,j, we use the same technique. p1 ` |θ|4qK1,j pθq becomes
+p1 ` |θ|4q
+ż
+R2 eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ
+“
+ż
+R2
+“
+p1 ` |∇ξ|4qeiθ¨ξ‰
+ξj pz ` LAq´1 pξqϕpξqdξ
+“
+ż
+Bp1q
+eiθ¨ξξj pz ` LAq´1 pξqϕpξq ` eiθ¨ξ∆2
+ξ
+”
+ξj pz ` LAq´1 pξqϕpξq
+ı
+dξ,
+(B.39)
+By (6.20),
+�����
+ż
+Bp1q
+eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ
+����� ď Cδ,σ1,σ2,T ∥ϕ∥C0
+|z|
+.
+(B.40)
+
+84
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Next,
+∆2
+ξ
+”
+ξj pz ` LAq´1 pξqϕpξq
+ı
+“ 4 B
+Bξj
+∆ξ
+”
+pz ` LAq´1 ϕ
+ı
+` ξj∆2
+ξ
+”
+pz ` LAq´1 pξqϕpξq
+ı
+(B.41)
+We have estimated the first term, so let us compute the second,
+∆2
+ξ
+”
+pz ` LAq´1 ϕ
+ı
+“
+∆2
+ξ pz ` LAq´1 ϕ ` 4∇ξ∆ξ pz ` LAq´1 ¨ ∇ξϕ ` 4
+”
+∇2
+ξ pz ` LAq´1 : ∇2
+ξϕ
+ı
+` 2∆ξ pz ` LAq´1 ∆ξϕ ` 4∇ξ pz ` LAq´1 ¨ ∇ξ∆ξϕ ` pz ` LAq´1 ∆2
+ξϕ.
+(B.42)
+For the last four terms, we may estimate them by (6.20) and (6.24) again. For the
+second term, since Bα
+ξ LA ≲ |ξ|1´|α|, by (B.33), we may obtain
+�����
+ż
+Bp1q
+eiθ¨ξξj∇ξ∆ξ pz ` LAq´1 pξq ¨ ∇ξϕ pξq dξ
+�����
+ď ∥ϕ∥C1
+4ÿ
+k“1
+Cpkq
+δ,σ1,σ2,T
+1
+|z|k .
+(B.43)
+In the first term, we have
+∆2
+ξH “
+ÿ
+j,k“1,2
+r
+8 pEjjkk ` Ejkjk ` Ejkkjq
+´8
+`
+Ejkpjkq ` Ejpjkqk ` Epjkqjk
+˘
+` 4H B2LA
+BξjBξk
+H B2LA
+BξjBξk
+H
+
+`
+ÿ
+j“1,2
+“
+´4
+`
+Ejjpkkq ` Ejpkkqj ` Epkkqjj
+˘
+` 4
+`
+Ejpjkkq ` Epjkkqj
+˘‰
+` 2H∆ξLAH∆ξLAH ´ H∆2
+ξLAH,
+(B.44)
+where
+Ejjkk “H BLA
+Bξj
+H BLA
+Bξj
+H BLA
+Bξk
+H BLA
+Bξk
+H,
+Ejkpjkq “H BLA
+Bξj
+H BLA
+Bξk
+H B2LA
+BξjBξk
+H,
+Ejjpkkq “H BLA
+Bξj
+H BLA
+Bξj
+H∆ξLAH,
+Ejpjkkq “H BLA
+Bξk
+H B∆ξLA
+Bξj
+H,
+and we only have to compute the ´ pz ` LAq´1 ∆2
+ξLA pz ` LAq´1 term. Since LA
+is an even function, ´ξj pz ` LAq´1 ∆2
+ξLA pz ` LAq´1 pξq ϕpξq is odd, again, by
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+85
+Lemma 6.6 and B.4
+�����´
+ż
+Bp1q
+eiθ¨ξξj pz ` LAq´1 ∆2
+ξLA pz ` LAq´1 pξq ϕpξqdξ
+�����
+“
+�����
+ż
+Bp1q
+ξj sin pθ ¨ ξq pz ` LA pξqq´1 ∆2
+ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ
+�����
+ď
+������
+ż
+Bp1q
+sin
+´
+θ ¨ ˆξ |ξ|
+¯ ´
+z ` |ξ| LA
+´
+ˆξ
+¯¯´1 ξjΦp4q
+A
+´
+ˆξ
+¯
+|ξ|3
+´
+z ` |ξ| LA
+´
+ˆξ
+¯¯´1
+ϕ p|ξ|q dξ
+������
+“
+������
+ż
+S1
+ˆξj
+ż 1
+0
+´
+z ` rLA
+´
+ˆξ
+¯¯´1
+Φp4q
+A
+´
+ˆξ
+¯ ´
+z ` rLA
+´
+ˆξ
+¯¯´1
+ϕ prq
+sin
+´
+θ ¨ ˆξr
+¯
+r
+drdˆξ
+������
+ď4
+ż
+S1
+���ˆξj
+���
+���Φp4q
+A
+´
+ˆξ
+¯��� ∥ϕ prq∥C1pr0,1sq
+����
+´
+z ` rLA
+´
+ˆξ
+¯¯´1����
+2
+C1pr0,1sq
+dˆξ
+ďCδ,σ1,σ2,T ∥ϕ prq∥C1pr0,1sq
+˜
+1
+|z| `
+1
+|z|2
+¸2
+.
+(B.45)
+Hence,
+�����
+ż
+Bp1q
+eiθ¨ξξj pz ` LAq´1 pξqϕpξq ` eiθ¨ξ∆2
+ξ
+”
+ξj pz ` LAq´1 pξqϕpξq
+ı
+dξ
+�����
+ď ∥ϕ prq∥C4
+5ÿ
+k“1
+Cpkq
+δ,σ1,σ2,T
+1
+|z|k
+ďCω,δ,σ1,σ2,T ∥ϕ∥C4 ,
+(B.46)
+and
+����
+ż
+R2 eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ
+���� ďCω,δ,σ1,σ2,T ∥ϕ∥C4
+1
+1 ` |θ|4 .
+(B.47)
+□
+Lemma B.3. Given kpxq “ F´1re´LApξqs, then we have the following estimates
+∥kpxq∥ ď C
+1
+1 ` |x|3 ,
+(B.48)
+����
+B
+Bxi
+kpxq
+���� ď C
+1
+1 ` |x|4 ,
+(B.49)
+����
+B
+Bxi
+B
+Bxj
+kpxq
+���� ď C
+1
+1 ` |x|5 .
+(B.50)
+
+86
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Proof. First,
+p1 ` |x|3q
+ż
+R2 eix¨ξe´LApξqdξ “
+ż
+R2p1 ´ i x
+|x| ¨ ix|x|2qeix¨ξe´LApξqdξ
+“
+ż
+R2p1 ´ i x
+|x| ¨ ∇ξ|∇ξ|2qeix¨ξe´LApξqdξ
+“
+ż
+R2 eix¨ξe´LApξq ` i x
+|x| ¨ ∇ξ∆ξe´LApξqdξ.
+Since e´LApξq ≲ e´|ξ|, we obtain
+(B.51)
+p1 ` |x|3q ∥kpxq∥ ≲
+´
+1 `
+�����
+ż
+B1p0q
+eix¨ξ x
+|x| ¨ ∇ξ∆ξe´LApξqdξ
+�����
+¯
+.
+Next, since
+B
+Bξi
+e´LApξq “ ´
+ż 1
+0
+e´p1´tqLApξq B
+Bξi
+LApξqe´tLApξqdt,
+Bjkk
+ξ
+e´LApξq
+“ ´
+ż 1
+0
+e´p1´t1qLApξqBjkk
+ξ
+LApξqe´t1LApξqdt1
+` H21 pj, kkq ` H21 pk, jkq ` H21 pjk, kq ` H22 pkk, jq ` H22 pjk, kq ` H22 pk, jkq
+´ H3 pp1 ´ t1q p1 ´ t2q p1 ´ t3q , j, p1 ´ t1q p1 ´ t2q t3, k, p1 ´ t1q t2, k, t1q
+´ H3 pp1 ´ t1q p1 ´ t2q , k, p1 ´ t1q t2 p1 ´ t3q , j, p1 ´ t1q t2t3, k, t1q
+´ H3 pp1 ´ t1q p1 ´ t2q , k, p1 ´ t1q t2, k, t1 p1 ´ t3q , j, t1t3q
+´ H3 pp1 ´ t1q p1 ´ t3q , j, p1 ´ t1q t3, k, t1 p1 ´ t2q , k, t1t2q
+´ H3 pp1 ´ t1q , k, t1 p1 ´ t2q p1 ´ t3q , j, t1 p1 ´ t2q t3, k, t1t2q
+´ H3 pp1 ´ t1q , k, t1 p1 ´ t2q t3, k, t1t2 p1 ´ t3q , j, t1t2t3q ,
+where
+H21 pα, βq “
+ż 1
+0
+ż 1
+0
+e´p1´t1qp1´t2qLApξqBα
+ξ LApξqe´p1´t1qt2LApξqBβ
+ξ LApξqe´t1LApξqdt1dt2,
+H22 pα, βq “
+ż 1
+0
+ż 1
+0
+e´t1LApξqBα
+ξ LApξqe´p1´t1qt2LApξqBβ
+ξ LApξqe´p1´t1qp1´t2qLApξqdt1dt2.
+H3 ps1, α, s2, β, s3, γ, s4q
+“
+ż 1
+0
+ż 1
+0
+ż 1
+0
+e´s1LApξqBα
+ξ LApξqe´t2LApξqBβ
+ξ LApξqe´s3LApξqBγ
+ξ LApξqe´s4LApξqdt1dt2dt3.
+Since LApξq is even and homogeneous of degree one, we have that the third deriva-
+tives of LApξq are odd and homogeneous of degree minus two. The other terms are
+less singular and thus the corresponding integrals in (B.51) are bounded directly.
+That is to say,
+���Bα
+ξ LApξq
+��� ≲ |ξ|1´|ξ| and
+��e´sLApξq�� ≲ e´sC|ξ| for some C ą 0,
+so ∥H21∥ , ∥H22∥ ≲ |ξ|´1 e´C|ξ|,∥H3∥ ≲ e´C|ξ|, and all of them are integrable.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+87
+Therefore, we only have to estimate the first, i.e.
+ż
+B1p0q
+ż 1
+0
+e´p1´t1qLApξqeix¨ξ x
+|x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ.
+We can write
+e´p1´t1qLApξq B
+Bξj
+∆ξLApξqe´t1LApξq “
+1
+|ξ|2 e´p1´t1qLApξqΦp3q
+A,jpˆξqe´t1LApξq,
+where ˆξ “ ξ{|ξ| and Φp3q
+A,jpˆξq is even and bounded from below and above, thanks
+to the arc-chord condition and the C1 regularity (see (3.2) and Remark 6.12). We
+then have that
+ż
+B1p0q
+ż 1
+0
+e´p1´t1qLApξqeix¨ξ x
+|x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ
+“
+ÿ
+j“1,2
+ixj
+|x|
+ż
+S1
+ż 1
+0
+ż 1
+0
+sin px ¨ ˆξ rq
+r
+e´p1´t1qrLApˆξqΦp3q
+A,jpˆξqe´t1rLApˆξqdrdt1dˆξ.
+By lemma B.4, we obtain for all x P R2, ˆξ P S1 and t1 P r0, 1s
+�����
+ż 1
+0
+sin px ¨ ˆξ rq
+r
+e´p1´t1qrLApˆξqΦp3q
+A,jpˆξqe´t1rLApˆξqdr
+�����
+ď2
+���e´p1´t1qrLApˆξqΦp3q
+A,jpˆξqe´t1rLApˆξq���
+C1pr0,1s;rq .
+(B.52)
+LApˆξq is positive definite and diagonalizable and Φp3q
+A,jpˆξq is boundned, so for all
+ˆξ P S1 and t1 P r0, 1s,
+���e´p1´t1qrLApˆξqΦp3q
+A,jpˆξqe´t1rLApˆξq���
+C1pr0,1s;rq ď C
+´
+1 `
+���LApˆξq
+���
+¯
+.
+(B.53)
+Therefore, since LApˆξq is bounded on S1,
+ż
+B1p0q
+ż 1
+0
+e´p1´t1qLApξqeix¨ξ x
+|x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ
+(B.54)
+is bounded, and we may conclude that
+∥kpxq∥ ≲ p1 ` |x|3q´1.
+Next, since
+B
+Bxi
+kpxq “
+ż
+R2 iξie´LApξqeix¨ξdξ,
+we have
+����
+B
+Bxi
+kpxq
+���� “
+ż
+R2 |ξi|
+���e´LApξq��� dξ ď C,
+where C only depends on A. Then,
+|x|4 B
+Bxi
+kpxq “
+ż
+R2 p∆ξq2 ´
+iξie´LApξq¯
+eix¨ξdξ,
+
+88
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+where p∆ξq2 `
+ξie´LApξq˘
+“ 4 B
+Bξi ∆ξe´LApξq ` ξi p∆ξq2 e´LApξq. The first term is the
+i component in ∇ξ∆ξe´LApξq, so we may claim the term is integratable by the
+previous techniques. For the second term, again,
+Bjjkk
+ξ
+LApξq
+“ ´
+ż 1
+0
+e´p1´t1qLApξqBjjkk
+ξ
+LApξqe´t1LApξqdt1
+` H2 pp1 ´ t1q p1 ´ t2q , jjk, p1 ´ t1q t2, k, t1q ` ¨ ¨ ¨
+´ H3 pp1 ´ t1q p1 ´ t2q p1 ´ t3q , jj, p1 ´ t1q p1 ´ t2q t3, k, p1 ´ t1q t2, k, t1q ´ ¨ ¨ ¨
+` H4 pp1 ´ t1q p1 ´ t2q p1 ´ t3q p1 ´ t4q , p1 ´ t1q p1 ´ t2q p1 ´ t3q t4,
+j, p1 ´ t1q p1 ´ t2q t3, k, p1 ´ t1q t2, k, t1q ` ¨ ¨ ¨
+where
+H2 ps1, α, s2, β, s3q
+“
+ż 1
+0
+ż 1
+0
+e´s1LApξqBα
+ξ LApξqe´s2LApξqBβ
+ξ LApξqe´s3LApξqdt1dt2
+H3 ps1, α, s2, β, s3, γ, s4q
+“
+ż 1
+0
+ż 1
+0
+ż 1
+0
+e´s1LApξqBα
+ξ LApξqe´s2LApξqBβ
+ξ LApξqe´s3LApξqBγ
+ξ LApξqe´s4LApξq
+dt1dt2dt3
+H4 ps1, α, s2, β, s3, γ, s4, δ, s5q
+“
+ż 1
+0
+ż 1
+0
+ż 1
+0
+ż 1
+0
+e´s1LApξqBα
+ξ LApξqe´s2LApξqBβ
+ξ LApξqe´s3LApξqBγ
+ξ LApξqe´s4LApξq
+Bδ
+ξLApξqe´s5LApξqdt1dt2dt3dt4.
+There are 14 H2-type terms, 36 H3-type terms, 24 H4-type terms in Bjjkk
+ξ
+LA.
+Since
+���Bα
+ξ LApξq
+��� ≲ |ξ|1´|ξ| and
+��e´sLApξq�� ≲ e´sC|ξ| for some C ą 0, ∥ξiH2∥ ≲
+|ξ|´1 e´C|ξ|,∥ξiH3∥ ≲ e´C|ξ| and ∥ξiH4∥ ≲ |ξ| e´C|ξ|. Hence, we only have to check
+ż
+R2
+ż 1
+0
+e´p1´t1qLApξqξi p∆ξq2 LApξqe´t1LApξqdt1dξ.
+Since
+ş1
+0 e´p1´t1qLApξqξi p∆ξq2 LApξqe´t1LApξqdt1 is odd, we may use the same tech-
+nique in kpxq term to obtain
+|x|4
+����
+B
+Bxi
+kpxq
+���� ď C,
+so
+����
+B
+Bxi
+kpxq
+���� ≲
+1
+1 ` |x|4 .
+Finally, since
+B
+Bxi
+kpxq “
+ż
+R2 ξie´LApξqeix¨ξdξ,
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+89
+we have
+����
+B
+Bxi
+kpxq
+���� “
+ż
+R2 |ξi|
+���e´LApξq��� dξ ď C,
+where C only depends on A.
+□
+Lemma B.4. Given a vector function M prq in C1 pr0, 1sq, we have the following
+inequality: for all A ě 0,
+����
+ż 1
+0
+M prq sin pArq
+r
+dr
+���� ď 2 ∥Mp0q∥ `
+����
+dM prq
+dr
+����
+C0pr0,1sq
+ď 2 ∥M∥C1pr0,1sq .
+(B.55)
+Proof.
+ż 1
+0
+M prq sin pArq
+r
+dr “
+ż 1
+0
+M p0q sin pArq
+r
+dr `
+ż 1
+0
+M prq ´ M p0q
+r
+sin pArqdr.
+(B.56)
+For the first term,
+����
+ż 1
+0
+M p0q sin pArq
+r
+dr
+���� “
+����M p0q
+ż 1
+0
+sin pArq
+r
+dr
+���� “ ∥Mp0q∥
+�����
+ż A
+0
+sin prq
+r
+dr
+�����
+ď ∥Mp0q∥
+ż π
+0
+sin prq
+r
+dr « 1.852 ∥Mp0q∥ ď 2 ∥Mp0q∥ .
+(B.57)
+For the second term, since
+����
+M prq ´ M p0q
+r
+���� “
+��şr
+0
+dM
+ds psq ds
+��
+r
+ď
+����
+dM prq
+dr
+����
+C0pr0,1sq
+,
+(B.58)
+����
+ż 1
+0
+M prq ´ M p0q
+r
+sin pArqdr
+���� ď
+ż 1
+0
+����
+M prq ´ M p0q
+r
+���� |sin pArq| dr
+ď
+����
+dM prq
+dr
+����
+C0pr0,1sq
+.
+(B.59)
+Therefore, we obtain the result.
+□
+Lemma B.5. Given fptq ě 0 is a locally integrable function on R, we have the
+following estimates:
+(i) If α ą ´1, for all m ă t,
+����
+ż t
+m
+pt ´ sqα fds
+���� ď C pt ´ mqα`1 Mlrfsptq.
+(B.60)
+(ii) If α ă ´1, for all M ă t,
+�����
+ż M
+´8
+pt ´ sqα fds
+����� ď C pt ´ Mqα`1 Mlrfsptq.
+(B.61)
+
+90
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+Proof. (i) By integration by part theorem,
+ż t
+m
+pt ´ sqα fds “ ´
+ż t
+m
+pt ´ sqα
+ˆ B
+Bs
+ż t
+s
+f prq dr
+˙
+ds
+“ pt ´ sqα
+ż t
+s
+f prq dr
+ˇˇˇˇ
+m
+s“t
+´ α
+ż t
+m
+pt ´ sqα
+ˆ
+1
+t ´ s
+ż t
+s
+f prq dr
+˙
+ds.
+Since
+lim sup
+sÑt´
+����pt ´ sqα
+ż t
+s
+f prq dr
+���� “ lim sup
+sÑt´
+����pt ´ sqα`1
+1
+t ´ s
+ż t
+s
+f prq dr
+����
+ď lim sup
+sÑt´ pt ´ sqα`1 Mlrfsptq “ 0,
+�����pt ´ sqα
+ż t
+s
+f prq dr
+ˇˇˇˇ
+m
+s“t
+����� ď pt ´ mqα`1 Mlrfsptq.
+Next,
+����
+ż t
+m
+pt ´ sqα
+ˆ
+1
+t ´ s
+ż t
+s
+f prq dr
+˙
+ds
+����
+ďMlrfsptq
+ż t
+m
+pt ´ sqα ds “
+1
+α ` 1 pt ´ mqα`1 Mlrfsptq.
+Therefore,
+����
+ż t
+m
+pt ´ sqα fds
+���� ď C pt ´ mqα`1 Mlrfsptq.
+(ii) The proof is basically the same. It will be
+�����
+ż M
+´8
+pt ´ sqα fds
+�����
+“
+�����pt ´ sqα
+ż t
+s
+f prq dr
+ˇˇˇˇ
+M
+s“´8
+´ α
+ż M
+´8
+pt ´ sqα
+ˆ
+1
+t ´ s
+ż t
+s
+f prq dr
+˙
+ds
+�����
+ď pt ´ Mqα`1 Mlrfsptq ` lim sup
+sÑ´8 pt ´ sqα`1 Mlrfsptq ` Mlrfsptq
+ż M
+´8
+pt ´ sqα ds
+“ pt ´ Mqα`1 Mlrfsptq ´
+1
+α ` 1 pt ´ Mqα`1 Mlrfsptq.
+since α ` 1 ă 0. Therefore,
+�����
+ż M
+´8
+pt ´ sqα fds
+����� ď C pt ´ Mqα`1 Mlrfsptq.
+□
+References
+[1] Thomas Alazard and Omar Lazar, Paralinearization of the Muskat equation and appli-
+cation to the Cauchy problem, Arch. Ration. Mech. Anal. 237 (2020), no. 2, 545–583,
+doi:10.1007/s00205-020-01514-6.
+[2] Thomas Alazard and Quoc-Hung Nguyen, Endpoint sobolev theory for the Muskat equation,
+Commun. Math. Phys. (2022).
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+91
+[3]
+,
+Quasilinearization
+of
+the
+3D
+Muskat
+equation,
+and
+applications
+to
+the
+critical
+Cauchy
+problem,
+Adv.
+Math.
+399
+(2022),
+Paper
+No.
+108278,
+52,
+doi:10.1016/j.aim.2022.108278.
+[4] David M. Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension,
+European J. Appl. Math. 15 (2004), no. 5, 597–607, doi:10.1017/S0956792504005662.
+[5]
+, Well-posedness of two-phase Darcy flow in 3D, Quart. Appl. Math. 65 (2007), no. 1,
+189–203, doi:10.1090/S0033-569X-07-01055-3.
+[6] David
+M.
+Ambrose
+and
+Michael
+Siegel,
+Well-posedness
+of
+two-dimensional
+hydroe-
+lastic
+waves,
+Proc.
+Roy.
+Soc.
+Edinburgh
+Sect.
+A
+147
+(2017),
+no.
+3,
+529–570,
+doi:10.1017/S0308210516000238.
+[7] Stephen Cameron, Global well-posedness for the two-dimensional Muskat problem with slope
+less than 1, Anal. PDE 12 (2019), no. 4, 997–1022, doi:10.2140/apde.2019.12.997.
+[8] Stephen Cameron and Robert M Strain, Critical local well-posedness for the fully nonlinear
+Peskin problem, Comm. Pure Appl. Math., arXiv:2112.00692 (2021), Accepted.
+[9] Ke
+Chen
+and
+Quoc-Hung
+Nguyen,
+The
+Peskin
+problem
+with
+9B1
+8,8
+initial
+data,
+arXiv:2107.13854 (2021).
+[10] Ke Chen, Quoc-Hung Nguyen, and Yiran Xu, The Muskat problem with C1 data, Trans. Am.
+Math. Soc. 375 (2021), 3039–3060.
+[11] C. H. Arthur Cheng and Steve Shkoller, The interaction of the 3D Navier-Stokes equations
+with a moving nonlinear Koiter elastic shell, SIAM J. Math. Anal. 42 (2010), no. 3, 1094–
+1155, doi:10.1137/080741628.
+[12] Peter Constantin, Diego C´ordoba, Francisco Gancedo, and Robert M. Strain, On the global
+existence for the Muskat problem, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 201–227,
+doi:10.4171/JEMS/360.
+[13] Antonio C´ordoba,
+Diego C´ordoba,
+and Francisco Gancedo,
+Interface evolution:
+the
+Hele-Shaw and
+Muskat
+problems,
+Ann.
+of
+Math.
+(2)
+173 (2011),
+no.
+1,
+477–542,
+doi:10.4007/annals.2011.173.1.10.
+[14] Antonio C´ordoba, Diego C´ordoba, and Francisco Gancedo, Porous media: the Muskat prob-
+lem in three dimensions, Anal. PDE 6 (2013), no. 2, 447–497, doi:10.2140/apde.2013.6.447.
+[15] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers,
+Inc., New York, N.Y., 1953.
+[16] Daniel Coutand and Steve Shkoller, The interaction between quasilinear elastodynamics
+and the Navier-Stokes equations, Arch. Ration. Mech. Anal. 179 (2006), no. 3, 303–352,
+doi:10.1007/s00205-005-0385-2.
+[17] I. V. Denisova, Problem of the motion of two viscous incompressible fluids separated by a
+closed free interface, Acta Appl. Math. 37 (1994), 31–40, doi:10.1007/BF00995127.
+[18] Joachim Escher and Bogdan-Vasile Matioc, On the parabolicity of the Muskat problem: well-
+posedness, fingering, and stability results, Z. Anal. Anwend. 30 (2011), no. 2, 193–218,
+doi:10.4171/ZAA/1431.
+[19] Evan A Evans and Richard Skalak, Mechanics and thermodynamics of biomembranes, CRC
+Press, 1980, doi:10.1201/9781351074339.
+[20] Thomas G Fai, Boyce E Griffith, Yoichiro Mori, and Charles S Peskin, Immersed boundary
+method for variable viscosity and variable density problems using fast constant-coefficient
+linear solvers i: Numerical method and results, SIAM Journal on Scientific Computing 35
+(2013), no. 5, B1132–B1161.
+[21] Gerald B. Folland, Introduction to partial differential equations, second ed., Princeton Uni-
+versity Press, Princeton, NJ, 1995.
+[22] F. Gancedo, E. Garc´ıa-Ju´arez, N. Patel, and R. M. Strain, Global regularity for gravity
+unstable muskat bubbles, Mem. Amer. Math. Soc., arXiv:1902.02318 (2021), To appear.
+[23] Francisco Gancedo, Rafael Granero-Belinch´on, and Stefano Scrobogna, Global existence in
+the lipschitz class for the N-peskin problem, Indiana Univ. Math. J., arXiv:2006.01787 (2021),
+To appear.
+[24] Francisco Gancedo and Omar Lazar, Global well-posedness for the three dimensional Muskat
+problem in the critical Sobolev space, Arch. Ration. Mech. Anal. 246 (2022), no. 1, 141–207,
+doi:10.1007/s00205-022-01808-x.
+[25] Eduardo Garc´ıa-Ju´arez, Yoichiro Mori, and Robert M. Strain, The Peskin Problem with
+Viscosity Contrast, Anal. PDE, arXiv:2009.03360 (2020), To appear.
+
+92
+E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN
+[26] Loukas Grafakos, Classical fourier analysis, third ed., Graduate Texts in Mathematics, vol.
+249, Springer, 2014, doi:10.1007/978-1-4939-1194-3.
+[27]
+, Modern fourier analysis, third ed., Graduate Texts in Mathematics, vol. 250,
+Springer, 2014, doi:10.1007/978-0-387-09434-2.
+[28] Boyce E Griffith and Neelesh A Patankar, Immersed methods for fluid–structure interaction,
+Annual review of fluid mechanics 52 (2020), 421.
+[29] Susanna V. Haziot and Benoˆıt Pausader, Note on the dissipation for the general muskat
+problem, Preprint, arXiv:2210.04395 (2022).
+[30] Thomas Y. Hou, John S. Lowengrub, and Michael J. Shelley, Removing the stiffness
+from interfacial flows with surface tension, J. Comput. Phys. 114 (1994), no. 2, 312–338,
+doi:10.1006/jcph.1994.1170.
+[31] William Ko and John M Stockie, Parametric resonance in spherical immersed elastic shells,
+SIAM Journal on Applied Mathematics 76 (2016), no. 1, 58–86.
+[32] Daniel Lengeler and Michael Ruˇziˇcka, Weak solutions for an incompressible Newtonian fluid
+interacting with a Koiter type shell, Arch. Ration. Mech. Anal. 211 (2014), no. 1, 205–255,
+doi:10.1007/s00205-013-0686-9.
+[33] Hui Li, Stability of the Stokes immersed boundary problem with bending and stretching energy,
+J. Funct. Anal. 281 (2021), no. 9, Paper No. 109204, 65, doi:10.1016/j.jfa.2021.109204.
+[34] Fang-Hua Lin and Jiajun Tong, Solvability of the Stokes immersed boundary problem in two
+dimensions, Comm. Pure Appl. Math. 72 (2019), no. 1, 159–226, doi:10.1002/cpa.21764.
+[35] Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Mod-
+ern Birkh¨auser Classics, Birkh¨auser/Springer Basel AG, Basel, 1995, [2013 reprint of the 1995
+original] [MR1329547].
+[36] Bogdan-Vasile Matioc, The Muskat problem in two dimensions:
+equivalence of formu-
+lations, well-posedness, and regularity results, Anal. PDE 12 (2019), no. 2, 281–332,
+doi:10.2140/apde.2019.12.281.
+[37] Yoichiro Mori, Analise Rodenberg, and Daniel Spirn, Well-posedness and global behavior of
+the Peskin problem of an immersed elastic filament in Stokes flow, Comm. Pure Appl. Math.
+72 (2019), no. 5, 887–980, doi:10.1002/cpa.21802.
+[38] Boris Muha and Suncica Cani´c,
+Existence of a weak solution to a nonlinear fluid-
+structure interaction problem modeling the flow of an incompressible, viscous fluid in a
+cylinder with deformable walls, Arch. Ration. Mech. Anal. 207 (2013), no. 3, 919–968,
+doi:10.1007/s00205-012-0585-5.
+[39] Huy Q. Nguyen and Benoˆıt Pausader, A paradifferential approach for well-posedness
+of
+the
+Muskat
+problem,
+Arch.
+Ration.
+Mech.
+Anal.
+237
+(2020),
+no.
+1,
+35–100,
+doi:10.1007/s00205-020-01494-7.
+[40] Charles S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys. 25 (1977),
+no. 3, 220–252, doi:10.1016/0021-9991(77)90100-0.
+[41]
+,
+The
+immersed
+boundary
+method,
+Acta
+Numer.
+11
+(2002),
+479–517,
+doi:10.1017/S0962492902000077.
+[42] P.
+I.
+Plotnikov
+and
+J.
+F.
+Toland,
+Modelling
+nonlinear
+hydroelastic
+waves,
+Philos.
+Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 369 (2011), no. 1947, 2942–2956,
+doi:10.1098/rsta.2011.0104.
+[43] C. Pozrikidis, Boundary integral and singularity methods for linearized viscous flow, Cam-
+bridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992,
+doi:10.1017/CBO9780511624124.
+[44]
+, Modeling and simulation of capsules and biological cells, Chapman & Hall/CRC
+Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL, 2003,
+doi:10.1201/9780203503959.
+[45] Jan Pr¨uss and Gieri Simonett,
+Moving interfaces and quasilinear parabolic evolution
+equations, Monographs in Mathematics, vol. 105, Birkh¨auser/Springer, [Cham], 2016,
+doi:10.1007/978-3-319-27698-4.
+[46] Thomas Richter, Fluid-structure interactions: models, analysis and finite elements, Vol. 118,
+Springer, 2017.
+[47] Analise Rodenberg, 2D Peskin Problems of an Immersed Elastic Filament in Stokes Flow,
+ProQuest LLC, Ann Arbor, MI, 2018, Thesis (Ph.D.)–University of Minnesota.
+[48] Yoshihiro Shibata, Theory of the lebesgue integral, Uchida Rokakuho, 2006.
+
+WELL-POSEDNESS OF THE 3D PESKIN PROBLEM
+93
+[49] Senjo Shimizu, Maximal regularity and its application to free boundary problems for the
+Navier-Stokes equations [translation of mr2656036], Sugaku Expositions 25 (2012), no. 1,
+105–130.
+[50] V. A. Solonnikov, Solvability of the problem of the motion of a viscous incompressible fluid
+that is bounded by a free surface, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 6, 1388–1424,
+1448.
+[51] Jiajun Tong, Regularized Stokes immersed boundary problems in two dimensions:
+Well-
+posedness, singular limit, and error estimates, Comm. Pure Appl. Math. 74(2):366–449
+(2021).
+[52]
+, Global solutions to the tangential Peskin problem in 2-D, Preprint arXiv:2205.14723
+(2022).
+[53] Sunˇcica ˇCani´c, Marija Gali´c, and Boris Muha, Analysis of a 3D nonlinear, moving boundary
+problem describing fluid-mesh-shell interaction, Trans. Amer. Math. Soc. 373 (2020), no. 9,
+6621–6681, doi:10.1090/tran/8125.
+[54] Wei Wang,
+Pingwen Zhang,
+and Zhifei Zhang,
+Well-posedness of hydrodynamics on
+the moving elastic surface,
+Arch. Ration. Mech. Anal. 206 (2012),
+no. 3,
+953–995,
+doi:10.1007/s00205-012-0548-x.
+