diff --git "a/QNFRT4oBgHgl3EQfJje8/content/tmp_files/load_file.txt" "b/QNFRT4oBgHgl3EQfJje8/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/QNFRT4oBgHgl3EQfJje8/content/tmp_files/load_file.txt" @@ -0,0 +1,597 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf,len=596 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='13496v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='AP] 31 Jan 2023 Conditional regularity for the Navier–Stokes–Fourier system with Dirichlet boundary conditions Danica Basari´c ∗ Eduard Feireisl ∗ Hana Mizerov´a ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='† ∗ Institute of Mathematics of the Czech Academy of Sciences ˇZitn´a 25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' CZ-115 67 Praha 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Czech Republic † Department of Mathematical Analysis and Numerical Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Comenius University Mlynsk´a dolina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' 842 48 Bratislava,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Slovakia Abstract We consider the Navier–Stokes–Fourier system with the inhomogeneous boundary condi- tions for the velocity and the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' We show that solutions emanating from sufficiently regular data remain regular as long as the density ̺, the absolute temperature ϑ, and the modulus of the fluid velocity |u| remain bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Keywords: Navier–Stokes–Fourier system, conditional regularity, blow–up criterion, regular solution 1 Introduction Standard systems of equations in fluid mechanics including the Navier–Stokes–Fourier system governing the motion of a compressible, viscous, and heat conducting fluid are well posed in the class of strong solutions on a possibly short time interval [0, Tmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The recent results of Merle at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' [16], [17] strongly indicate that Tmax may be finite, at least in the idealized case of “isentropic” viscous flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Conditional regularity results guarantee that a blow up will not occur as soon as some lower order norms of solutions are controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' We consider the Navier–Stokes–Fourier system governing the time evolution of the mass density ̺ = ̺(t, x), the (absolute) temperature ϑ = ϑ(t, x), and the velocity u = u(t, x) of a compressible, viscous, and heat conducting fluid: ∗The work of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=', and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' was supported by the Czech Sciences Foundation (GAˇCR), Grant Agreement 21–02411S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO:67985840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' 1 ∂t̺ + divx(̺u) = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) ∂t(̺u) + divx(̺u ⊗ u) + ∇xp(̺, ϑ) = divxS(Dxu) + ̺f, Dxu = 1 2 � ∇xu + ∇t xu � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) ∂t(̺e(̺, ϑ)) + divx(̺e(̺, ϑ)u) + divxq(∇xϑ) = S(Dxu) : Dxu − p(̺, ϑ)divxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) The fluid is Newtonian, the viscous stress S is given by Newton’s rheological law S(Dxu) = 2µ � Dxu − 1 3divxuI � + ηdivxuI, µ > 0, η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4) The heat flux obeys Fourier’s law q(∇xϑ) = −κ∇xϑ, κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5) The equation of state for the pressure p and the internal energy e is given by the standard Boyle– Mariotte law of perfect gas, p(̺, ϑ) = ̺ϑ, e(̺, ϑ) = cvϑ, cv > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6) For the sake of simplicity, we suppose that the viscosity coefficients µ, η, the heat conductivity coefficient κ as well as the specific heat at constant volume cv are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' There is a large number of recent results concerning conditional regularity for the Navier– Stokes–Fourier system in terms of various norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Fan, Jiang, and Ou [4] consider a bounded fluid domain Ω ⊂ R3 with the conservative boundary conditions u|∂Ω = 0, ∇xϑ · n|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='7) The same problem is studied by Sun, Wang, and Zhang [19] and later by Huang, Li, Wang [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' There are results for the Cauchy problem Ω = R3 by Huang and Li [13], and Jiu, Wang and Ye [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Possibly the best result so far has been established in [11], where the blow up criterion for both the Cauchy problem and the boundary value problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='7) is formulated in terms of the maximum of the density and a Serrin type regularity for the temperature: lim sup t→Tmax− � ∥̺(t, ·)∥L∞ + ∥ϑ − ϑ∞∥Ls(0,t)(Lr) � = ∞, 3 2 < r ≤ ∞, 1 ≤ s ≤ ∞, 2 s + 3 r ≤ 2, where ϑ∞ denotes the far field temperature in the Cauchy problem, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' also the previous results by Wen and Zhu [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Much less is known in the case of the Dirichlet boundary conditions u|∂Ω = uB, ϑ|∂Ω = ϑB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='8) 2 Fan, Zhi, and Zhang [5] showed that a strong solution of the Navier–Stokes–Fourier system remains regular up to a time T > 0 if (i) Ω ⊂ R2 is a bounded domain, (ii) uB = 0, ϑB = 0, and (iii) lim sup t→T− (∥̺∥L∞ + ∥ϑ∥L∞) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='9) All results mentioned above describe fluids in a conservative regime, meaning solutions are close to equilibrium in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' However, many real world applications concern fluids out of equilibrium driven by possibly large driving forces f and/or inhomogeneous boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The iconic examples are the Rayleigh–B´enard and Taylor–Couette flows where the fluid is driven to a turbulent regime by a large temperature gradient and large boundary velocity, respectively, see Davidson [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Motivated by these physically relevant examples, we consider a fluid confined to a bounded domain Ω ⊂ R3 with impermeable boundary, where the temperature and the (tangential) velocity are given on ∂Ω, ϑ|∂Ω = ϑB, ϑB = ϑB(x), ϑB > 0 on ∂Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='10) u|∂Ω = uB, uB = uB(x), uB · n = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='11) The initial state of the fluid is prescribed: ̺(0, ·) = ̺0, ̺0 > 0 in Ω, ϑ(0, ·) = ϑ0, ϑ0 > 0 in Ω, u(0, ·) = u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='12) The initial and boundary data are supposed to satisfy suitable compatibility conditions specified below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The existence of local in time strong solutions for the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6), endowed with the inhomogeneous boundary conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='10), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='11) was established by Valli [20], [21] , see also Valli and Zajaczkowski [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The solution exists on a maximal time interval [0, Tmax), Tmax > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Our goal is to show that if Tmax < ∞, then necessarily lim sup t→Tmax− � ∥̺(t, ·)∥L∞(Ω) + ∥ϑ(t, ·)∥L∞(Ω) + ∥u(t, ·)∥L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) � = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='13) The proof is based on deriving suitable a priori bounds assuming boundedness of all norms involved in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='13) as well as the norm of the initial/boundary data in a suitable function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Although approach shares some similarity with Fang, Zi, and Zhang [5], essential modifications must be made to accommodate the inhomogeneous boundary data as well as the driving force f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The importance of conditional regularity results in numerical analysis of flows with uncertain initial data was discussed recently in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' 3 The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' In Section 2, we introduce the class of strong solutions to the Navier–Stokes–Fourier system and state our main result concerning conditional regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The remaining part of the paper is devoted to the proof of the main result – deriving suitable a priori bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' In Section 3 we recall the standard energy estimates that hold even in the class of weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Section 4 is the heart of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' We establish the necessary estimates on the velocity gradient by means of the celebrated Gagliardo–Nirenberg interpolation inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' In Section 5, higher order estimates on the velocity gradient are derived, and, finally, the estimates are closed by proving bounds on the temperature time derivative in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' This last part borrows the main ideas from [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' 2 Strong solutions, main result We start the analysis by recalling the concept of strong solution introduced by Valli [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Similarly to the boundary data uB, ϑB we suppose that the driving force f = f(x) is independent of time, meaning we deal with an autonomous problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Following [21], we suppose that Ω ⊂ R3 is a bounded domain with ∂Ω of class C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' We assume the data belong to the following class: ̺0 ∈ W 3,2(Ω), 0 < ̺0 ≤ min x∈Ω ̺0(x), ϑ0 ∈ W 3,2(Ω), 0 < ϑ0 ≤ min x∈Ω ϑ0(x), u0 ∈ W 3,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' R3), ϑB ∈ W 7 2(∂Ω), 0 < ϑB ≤ min x∈∂Ω ϑB(x), uB ∈ W 7 2(∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' R3), uB · n = 0, f ∈ W 2,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) In addition, the data must satisfy the compatibility conditions ϑ0 = ϑB, u0 = uB on ∂Ω, ̺0u0 · ∇xu0 + ∇xp(̺0, ϑ0) = divxS(Dxu0) + ̺0f on ∂Ω, ̺0u0 · ∇xϑ0 + divxq(ϑ0) = S(Dxu0) : Dxu0 − p(̺0, ϑ0)divxu0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) We set D0 = max � ∥(̺0, ϑ0, u0)∥W 3,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R5), 1 ̺0 , 1 ϑ0 , 1 ϑB , ∥ϑB∥W 7 2 (∂Ω), ∥uB∥W 7 2 (∂Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3), ∥f∥W 2,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1 Local existence The following result was proved by Valli [21, Theorem A] (see also [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (Local existence of strong solutions) Let Ω ⊂ R3 be a bounded domain of class C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Suppose that the data (̺0, ϑ0, u0), (ϑB, uB) and f belong to the class (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) and satisfy the compatibility conditions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Then there exists a maximal time Tmax > 0 such that the Navier–Stokes–Fourier system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1)– (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6), with the boundary conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='10), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='11), and the initial conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='12) admits a solu- tion (̺, ϑ, u) in [0, Tmax) × Ω unique in the class ̺, ϑ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' W 3,2(Ω)), u ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' W 3,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' R3)), ϑ ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' W 4,2(Ω)), u ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' W 4,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' R3)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4) for any 0 < T < Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The existence time Tmax is bounded below by a quantity c(D0) depending solely on the norms of the data specified in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' In particular, lim τ→Tmax− ∥(̺, ϑ, u)(τ, ·)∥W 3,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R5) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2 Blow up criterion, conditional regularity Our goal is to show the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (Blow up criterion) Under the hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1, suppose that the maximal existence time Tmax < ∞ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Then lim sup τ→Tmax− ∥(̺, ϑ, u)(τ, ·)∥L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R5) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2 is in the spirit of the blow up criteria for general parabolic systems – the solution remains regular as long as it is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Of course, our problem in question is of mixed hyperbolic– parabolic type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2 follows from suitable a priori bounds applied on a compact time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (Conditional regularity) Under the hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1, let (̺, ϑ, u) be the strong solution of the Navier–Stokes– Fourier system belonging to the class (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4) and satisfying sup (τ,x)∈[0,T)×Ω ̺(τ, x) ≤ ̺, sup (τ,x)∈[0,T)×Ω ϑ(τ, x) ≤ ϑ, sup (τ,x)∈[0,T)×Ω |u(τ, x)| ≤ u (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='7) 5 for some T < Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Then there is a quantity c(T, D0, ̺, ϑ, u), bounded for bounded arguments, such that sup τ∈[0,T) max � ∥(̺, ϑ, u)(τ, ·)∥W 3,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' sup x∈Ω 1 ̺(τ, x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' sup x∈Ω 1 ϑ(τ, x) � ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='8) In view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1, the conclusion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2 follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The rest of the paper is therefore devoted to the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' As observed in [8], the conditional regularity results established in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3 gives rise to stability with respect to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' More specifically, the maximal existence time Tmax is a lower semicontinuous function of the data with respect to the topologies in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Conditional regularity results in combination with the weak–strong uniqueness principle in the class of measure–valued solutions is an efficient tool for proving convergence of numerical schemes, see [6, Chapter 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The concept of measure–valued solutions to the Navier– Stokes–Fourier system with inhomogeneous Dirichlet boundary conditions has been introduced recently by Chaudhuri [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' 3 Energy estimates To begin, it is suitable to extend the boundary data into Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' For definiteness, we consider the (unique) solutions of the Dirichlet problem ∆x ˜ϑ = 0 in Ω, ˜ϑ|∂Ω = ϑB, divxS(Dx˜u) = 0 in Ω, ˜u|∂Ω = uB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) By abuse of notation, we use the same symbol ϑB, uB for both the boundary values and their C1 extensions ˜ϑ = ˜ϑ(x), ˜u = ˜u(x) inside Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' We start with the ballistic energy equality, see [2, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4], d dt � Ω �1 2̺|u − uB|2 + ̺e − ϑB̺s � dx + � Ω ϑB ϑ � S(Dxu) : Dxu + κ|∇xϑ|2 ϑ � dx = − � Ω � ̺u ⊗ u + pI − S(Dxu) � : DxuB dx + 1 2 � Ω ̺u · ∇x|uB|2 dx + � Ω ̺(u − uB) · f dx − � Ω ̺su · ∇xϑB dx + κ � Ω ∇xϑ ϑ ∇xϑB dx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) where we have introduced the entropy s = cv log(ϑ) − log(̺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' 6 Thus the choice (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) yields the following bounds sup t∈[0,T) � Ω ̺| log(ϑ)|(t, ·) dx ≤ c(T, D0, ̺, ϑ, u), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) � T 0 � Ω |∇xu|2 dx dt ≤ C(̺, ϑ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' data) ⇒ � T 0 ∥u∥2 W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) dt ≤ c(T, D0, ̺, ϑ, u), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4) � T 0 � Ω � |∇xϑ|2 + |∇x log(ϑ)|2� dx dt ≤ c(T, D0, ̺, ϑ, u), ⇒ � T 0 ∥ϑ∥2 W 1,2(Ω) dt + � T 0 ∥ log(ϑ)∥2 W 1,2(Ω) dt ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5) 4 Estimates of the velocity gradient This section is the heart of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' In principle, we follow the arguments similar to Fang, Zi, and Zhang [5, Section 3] but here adapted to the inhomogeneous boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1 Estimates of the velocity material derivative Let us introduce the material derivative of a function g, Dtg = ∂tg + u · ∇xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Accordingly, we may rewrite the momentum equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) as ̺Dtu + ∇xp = divxS + ̺f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) Now, consider the scalar product of the momentum equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) with Dt(u − uB), ̺|Dtu|2 + ∇xp · Dt(u − uB) = divxS(Dxu) · Dt(u − uB) + ̺f · Dt(u − uB) + ̺Dtu · DtuB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) The next step is integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Here and hereafter we use the hypothesis uB·n|∂Ω = 0 yielding Dt(u − uB)|∂Ω = (∂tu − u · ∇x(u − uB)) |∂Ω = −uB · ∇x(u − uB)|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) Writing divxS(Dxu) = µ∆xu + � η + µ 3 � ∇xdivxu, and making use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) we obtain � Ω divxS(Dxu) · Dt(u − uB) dx 7 = − � Ω S(Dxu) : ∇x∂tu dx − µ � Ω ∇xu : ∇x � u · ∇x(u − uB) � dx − � η + µ 3 � � Ω divxu divx � u · ∇x(u − uB) � dx = − 1 2 d dt � Ω S(Dxu) : Dxu dx − µ � Ω ∇xu : ∇x � u · ∇x(u − uB) � dx − � η + µ 3 � � Ω divxu divx � u · ∇x(u − uB) � dx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4) where, furthermore, � Ω ∇xu : ∇x(u · ∇xu) dx = � Ω ∇xu : (∇xu · ∇xu) dx + 1 2 � Ω u · ∇x|∇xu|2 dx = � Ω ∇xu : (∇xu · ∇xu) dx − 1 2 � Ω divxu|∇xu|2 dx (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5) Note carefully we have used u · n|∂Ω = 0 in the last integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Similarly, � Ω divxu divx(u · ∇xu) dx = � Ω divxu ∇xu : ∇t xu dx − 1 2 � Ω (divxu)3 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6) Thus summing up the previous observations, we get 1 2 d dt � Ω S(Dxu) : Dxu dx + 1 2 � Ω ̺|Dtu|2 dx + � Ω ∇xp · Dt(u − uB) dx ≤ c(T, D0, ̺, ϑ, u) � 1 + � Ω |∇xu|3 dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='7) Moreover, � Ω ∇xp · Dt(u − uB) dx = − � Ω p divx(Dt(u − uB)) dx = − � Ω p divxDtu dx + � Ω p divx(u · ∇xuB) dx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='8) where p divxDtu = ∂t(p divxu) − � ∂tp + divx(pu) � divxu + divx(pu)divxu + p divx(u · ∇xu) = ∂t(p divxu) − � ∂tp + divx(pu) � divxu + p∇xu : ∇t xu + divx � pu divxu � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' As u · n|∂Ω = 0, we have � Ω divx � pu divxu � dx = 0, 8 and the above estimates together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='7) give rise to 1 2 d dt � Ω S(Dxu) : Dxu dx − d dt � Ω pdivxu dx + 1 2 � Ω ̺|Dtu|2 dx ≤ c(T, D0, ̺, ϑ, u) � 1 + � Ω |∇xu|3 dx � − � Ω � ∂tp + divx(pu) � divxu dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Finally, we realize ∂tp + divx(pu) = ̺Dtϑ to conclude 1 2 d dt � Ω S(Dxu) : Dxu dx − d dt � Ω pdivxu dx + 1 2 � Ω ̺|Dtu|2 dx ≤ c(T, D0, ̺, ϑ, u) � 1 + � Ω ̺|Dtϑ||∇xu| dx + � Ω |∇xu|3 dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='9) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2 Higher order velocity material derivative estimates Following [5, Section 3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3], see also Hoff [12], we deduce ̺D2 t u + ∇x∂tp + divx(∇xp ⊗ u) = µ � ∆x∂tu + divx(∆xu ⊗ u) � + � η + µ 3 � � ∇xdivx∂tu + divx ((∇xdivxu) ⊗ u) � + ̺u · ∇xf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='10) Next, we compute DtuB = u · ∇xuB, D2 t uB = ∂tu · ∇xuB + u · ∇x(u · ∇xuB) = Dtu · ∇xuB − (u · ∇xu) · ∇xuB + u · ∇x(u · ∇xuB) = Dtu · ∇xuB + (u ⊗ u) : ∇2 xuB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='11) Consequently, we may rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='10) in the form ̺D2 t (u − uB) + ∇x∂tp + divx(∇xp ⊗ u) = µ � ∆x∂tu + divx(∆xu ⊗ u) � + � η + µ 3 � � ∇xdivx∂tu + divx ((∇xdivxu) ⊗ u) � + ̺u · ∇xf − ̺Dtu · ∇xuB − ̺(u ⊗ u) : ∇2 xuB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='12) The next step is considering the scalar product of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='12) with Dt(u − uB) and integrating over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The resulting integrals can be handled as follows: ̺D2 t (u − uB) · Dt(u − uB) = ̺1 2Dt|Dt(u − uB)|2 9 = 1 2̺ � ∂t|Dt(u − uB)|2 + u · ∇x|Dt(u − uB)|2� = 1 2∂t � ̺|Dt(u − uB)|2� + 1 2divx � ̺u|Dt(u − uB)|2� , where we have used the equation of continuity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Seeing that u · n|∂Ω = 0 we get � Ω ̺D2 t (u − uB) · Dt(u − uB) dx = d dt 1 2 � Ω ̺|Dt(u − uB)|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='13) Similarly, � Ω � ∇x∂tp + divx(∇xp ⊗ u) � Dt(u − uB) dx = − � Ω � ∂tp + divx(pu) � divxDt(u − uB) dx + � Ω � divx(pu)divxDt(u − uB) − ∇xp ⊗ u : ∇xDt(u − uB) � dx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='14) where � Ω ∇xp ⊗ u : ∇xDt(u − uB) dx = − � Ω p∇xu : ∇xDt(u − uB) dx + � Ω ∇x(pu) : ∇xDt(u − uB) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' In addition, as Dt(u−uB) vanishes on ∂Ω, we can perform by parts integration in the last integral obtaining � Ω ∇x(pu) : ∇xDt(u − uB) dx = � Ω divx(pu)divxDt(u − uB) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Thus, similarly to the preceding section, we conclude � Ω � ∇x∂tp + divx(∇xp ⊗ u) � Dt(u − uB) dx = − � Ω ̺DtϑdivxDt(u − uB) dx + � Ω p∇xu : ∇xDt(u − uB) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='15) Analogously, � Ω � ∆x∂tu + divx(∆xu ⊗ u) � Dt(u − uB) dx = − � Ω ∇x∂tu : ∇xDt(u − uB) dx − � Ω (∆xu ⊗ u) : ∇xDt(u − uB) dx = − � Ω ∇xDtu : ∇xDt(u − uB) dx − � Ω � ∆xu ⊗ u − ∇x(u · ∇xu) � : ∇xDt(u − uB) dx, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='16) 10 where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' using summation convention,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' � Ω � ∆xu ⊗ u � : ∇xDt(u − uB) dx = � Ω ∂xk � uj∂xkui � ∂xjDt(u − uB)i dx − � Ω ∂xkui∂xkuj∂xjDt(u − uB)i dx = � Ω ∂xj � uj∂xkui � ∂xkDt(u − uB)i dx − � Ω ∂xkui∂xkuj∂xjDt(u − uB)i dx = � Ω divxu ∇xu : ∇xDt(u − uB) dx + � Ω � uj∂xk∂xjui � ∂xkDt(u − uB)i dx − � Ω ∂xkui∂xkuj∂xjDt(u − uB)i dx = � Ω ∇x(u · ∇xu) : ∇xDt(u − uB) dx + � Ω divxu ∇xu : ∇xDt(u − uB) dx − � Ω ∂xjui∂xkuj∂xkDt(u − uB)i dx − � Ω ∂xkui∂xkuj∂xjDt(u − uB)i dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='17) Summing up (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='17) we conclude � Ω � ∆x∂tu + divx(∆xu ⊗ u) � Dt(u − uB) dx = − � Ω ∇xDtu : ∇xDt(u − uB) dx − � Ω divxu ∇xu : ∇xDt(u − uB) dx + � Ω ∂xjui∂xkuj∂xkDt(u − uB)i dx + � Ω ∂xkui∂xkuj∂xjDt(u − uB)i dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='18) Estimating the remaining integrals in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='12) in a similar manner we may infer 1 2 d dt � Ω ̺|Dt(u − uB)|2 dx + µ � Ω |∇xDt(u − uB)|2 dx + � η + µ 3 � � Ω |divxDt(u − uB)|2 dx ≤ c(T, D0, ̺, ϑ, u) � 1 + � Ω ̺|Dtϑ|2 dx + � Ω |∇xu|4 dx + � Ω ̺|Dtu|2 dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='19) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' [5, Section 3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3 Velocity decomposition Following the original idea of Sun, Wang, and Zhang [18], we decompose the velocity field in the form: u = v + w, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='20) divxS(Dxv) = ∇xp in (0, T) × Ω, v|∂Ω = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='21) 11 divxS(Dxw) = ̺Dtu − ̺f in (0, T) × Ω, w|∂Ω = uB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='22) Since divxS(Dx∂tv) = ∇x∂tp in (0, T) × Ω, v|∂Ω = 0, we get � Ω ∂tp divxv dx = − � Ω ∇x∂tp · v dx = 1 2 d dt � Ω S(Dxv) : Dxv dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='23) Moreover, the standard elliptic estimates for the Lam´e operator yield: ∥v∥W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) ≤ c(q, ̺, ϑ) for all 1 ≤ q < ∞, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='24) ∥v∥W 2,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) ≤ c(q, ̺, ϑ) � ∥∇x̺∥Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) + ∥∇xϑ∥Lq(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) � , 1 < q < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='25) Similarly, ∥w∥W 2,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) ≤ c(T, D0, ̺, ϑ, u) � 1 + ∥√̺∂tu∥L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) + ∥∇xu∥L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3×3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='26) The estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='24)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='26) are uniform in the time interval [0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4 Temperature estimates Similarly to Fang, Zi, Zhang [5, Section 3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4] we multiply the internal energy equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) on ∂tϑ and integrate over Ω obtaining cv � Ω ̺|Dtϑ|2 dx + κ 2 d dt � Ω |∇xϑ|2 dx = cv � Ω ̺Dtϑ u · ∇xϑ dx − � Ω ̺ϑ divxu Dtϑ dx + � Ω ̺ϑ divxu u · ∇xϑ dx + d dt � Ω ϑ S(Dxu) : ∇xu dx − µ � Ω ϑ � ∇xu + ∇t xu − 2 3divxuI � : � ∇x∂tu + ∇t x∂tu − 2 3divx∂tuI � dx − 2η � Ω ϑ divxu divx∂tu dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='27) Indeed the term involving the boundary integral is handled as −κ � Ω ∆xϑ ∂tϑ dx = −κ � ∂Ω ∂tϑB∇xϑ · n dSx + κ 2 d dt � Ω |∇xϑ|2 dx, where � ∂Ω ∂tϑB∇xϑ · n dSx = 0 12 as the boundary temperature is independent of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Similarly to Fang, Zi, Zhang [5, Section 3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4], we have to show that the intergrals � Ω ϑ ∇xu : ∇x∂tu dx, � Ω ϑ ∇xu : ∇t x∂tu dx, and � Ω ϑ divxu divx∂tu dx can be rewritten in the form compatible with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='19), meaning with the time derivatives replaced by material derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Fortunately, this step can be carried out in the present setting using only the boundary condition u · n|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Indeed we get � Ω ϑ ∇xu : ∇x∂tu dx = � Ω ϑ ∇xu : ∇x(Dtu) dx − � Ω ϑ ∇xu : ∇x(u · ∇xu) dx, where � Ω ϑ ∇xu : ∇x(u · ∇xu) dx = � Ω ϑ ∇xu : (∇xu · ∇xu) dx + 1 2 � Ω ϑ u · ∇x|∇xu|2 dx = � Ω ϑ ∇xu : (∇xu · ∇xu) dx − 1 2 � Ω |∇xu|2 ∇xϑ · u dx − 1 2 � Ω |∇xu|2 ϑdivxu dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Similarly, � Ω ϑ ∇xu : ∇t x∂tu dx = � Ω ϑ ∇xu : ∇t x(Dtu) dx − � Ω ϑ ∇xu : ∇t x(u · ∇xu) dx, where � Ω ϑ ∇xu : ∇t x(u · ∇xu) dx = � Ω ϑ ∇xu : (∇t xu · ∇t xu) dx + 1 2 � Ω ϑ u · ∇x(∇xu : ∇t xu) dx = � Ω ϑ ∇xu : (∇t xu · ∇t xu) dx − 1 2 � Ω (∇xu : ∇t xu) ∇xϑ · u dx − 1 2 � Ω (∇xu : ∇t xu) ϑdivxu dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Finally, � Ω ϑ divxu divx∂tu dx = � Ω ϑ divxu divxDtu dx − � Ω ϑ divxu divx(u · ∇xu) dx, where � Ω ϑ divxu divx(u · ∇xu) dx 13 = � Ω ϑ divxu (∇xu : ∇t xu) dx + 1 2 � Ω ϑu · ∇x|divxu|2 dx = � Ω ϑ divxu (∇xu : ∇t xu) dx − 1 2 � Ω |divxu|2 ∇xϑ · u dx − 1 2 � Ω |divxu|2 ϑdivxu dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' We conclude, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='7), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='19), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='27), � Ω |∇xϑ|2(τ, ·) dx + � τ 0 � Ω ̺|Dtϑ|2 dx dt ≤ c(T, D0, ̺, ϑ, u) � 1 + � τ 0 � Ω |∇xu|4 dx dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='28) Next, by virtue of the decomposition u = v + w and the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='24), � Ω |∇xu|4 dx <∼ � Ω |∇xv|4 dx + � Ω |∇xw|4 dx ≤ c(T, D0, ̺, ϑ, u) � 1 + � Ω |∇xw|4 dx � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='29) and, similarly, ∥w∥L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) ≤ ∥u∥L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) + ∥v∥L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='30) Recalling the Gagliardo–Nirenberg interpolation inequality in the form ∥∇xU∥2 L4(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) ≤ ∥U∥L∞(Ω)∥∆xU∥L2(Ω) whenever U|∂Ω = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='31) we may use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='29), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='30) to rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='28) in the form � Ω |∇xϑ|2(τ, ·) dx + � τ 0 � Ω ̺|Dtϑ|2 dx dt ≤ c(T, D0, ̺, ϑ, u) � 1 + � τ 0 � Ω |∇xϑ|2 dx dt + � τ 0 ∥w∥2 W 2,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='32) Finally, we use the elliptic estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='26) to conclude � Ω |∇xϑ|2(τ, ·) dx + � τ 0 � Ω ̺|Dtϑ|2 dx dt ≤ c(T, D0, ̺, ϑ, u) � 1 + � τ 0 � Ω � |∇xϑ|2 + |∇xu|2� dx dt + � τ 0 ∥√̺∂tu∥2 L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='33) Summing up (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='7), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='19), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='33) we may apply Gronwall’s lemma to obtain the following bounds: sup t∈[0,T) ∥u(t, ·)∥W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) ≤ c(T, D0, ̺, ϑ, u), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='34) sup t∈[0,T) ∥√̺Dtu(t, ·)∥L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) ≤ c(T, D0, ̺, ϑ, u), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='35) 14 sup t∈[0,T) ∥ϑ(t, ·)∥W 1,2(Ω) ≤ c(T, D0, ̺, ϑ, u), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='36) � T 0 � Ω |∇xDtu|2 dx dt ≤ c(T, D0, ̺, ϑ, u), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='37) � T 0 � Ω ̺|Dtϑ|2 dx dt ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='38) Moreover, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='24), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='31), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='35) sup t∈[0,T) ∥∇xu(t, ·)∥L4(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3×3) ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='39) In addition, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='38), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='39) and the standard parabolic estimates applied to the internal energy balance (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) yield � T 0 ∥ϑ∥2 W 2,2(Ω) dt ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='40) 5 Second energy bound It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='26), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='35) that sup t∈[0,T) ∥w(t, ·)∥W 2,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) ≤ c(T, D0, ̺, ϑ, u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) whence, by virtue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='24) and Sobolev embedding W 1,2(Ω) ֒→ L6(Ω), sup t∈[0,T) ∥∇xu(t, ·)∥2 L6(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3×3) ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) Moreover, as a consequence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='37), Dtu is bounded in L2(L6), which, combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2), gives rise to � T 0 ∥∂tu∥2 L6(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) dt ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) Finally, going back to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='22) we conclude � T 0 ∥w∥2 W 2,6(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) dt ≤ c(T, D0, ̺, ϑ, u), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4) and � T 0 ∥u∥2 W 1,q(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) dt ≤ c(T, D0, ̺, ϑ, u, q) for any 1 ≤ q < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5) 15 6 Estimates of the derivatives of the density Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5), we may proceed as in [19, Section 5] to deduce the bounds supt∈[0,T) � ∥∂t̺(t, ·)∥L6(Ω) + ∥̺(t, ·)∥W 1,6(Ω) � ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) Revisiting the momentum equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) we use (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) together with the other bounds established above to obtain � T 0 ∥u∥2 W 2,6(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) dt ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1 Positivity of the density and temperature It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) that divxu is bounded in L1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' L∞(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Thus the equation of continuity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) yields a positive lower bound on the density inf (t,x)∈[0,T)×Ω ̺(t, x) ≥ ̺ > 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) where the lower bound depends on the data as well as on the length T of the time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Similarly, rewriting the internal energy balance equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) in the form cv (∂tϑ + u · ∇xϑ) − κ ̺∆xϑ = 1 ̺S : Dxu − ϑdivxu (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4) we may apply the standard parabolic maximum/minimum principle to deduce inf (t,x)∈[0,T)×Ω ϑ(t, x) ≥ ϑ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5) 7 Parabolic regularity for the heat equation We rewrite the parabolic equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4) in terms of Θ = ϑ − ϑB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Recalling ∆xϑB = 0 we get cv (∂tΘ + u · ∇xϑ) − κ ̺∆xΘ = 1 ̺S : Dxu − ϑdivxu (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) with the homogeneous Dirichlet boundary conditions Θ|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) Now, we can apply all arguments of [10, Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='7] to Θ obtaining the bounds ∥ϑ∥Cα([0,T]×Ω) ≤ c(T, D0, ̺, ϑ, u) for some α > 0, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) ∥ϑ∥Lp(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='W 2,3(Ω)) + ∥∂tϑ∥Lp(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='L3(Ω)) ≤ c(T, D0, ̺, ϑ, u) for all 1 ≤ p < ∞, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4) together with ∥u∥Lp(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='W 2,6(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3)) + ∥∂tu∥Lp(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='L6(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3)) ≤ c(T, D0, ̺, ϑ, u) for any 1 ≤ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5) 16 8 Final estimates The bounds (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5) imply, in particular, sup (t,x)∈[0,T)×Ω |∇xu(t, x)| ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) Thus the desired higher order estimates can be obtained exactly as in [9, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Indeed the arguments of [9, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6] are based on differentiating the equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='1) with respect to time which gives rise to a parabolic problem for ∂tϑ with the homogeneous Dirichlet boundary conditions ∂tϑ|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Indeed we get cv∂2 ttϑ + cvu · ∇x∂tϑ − κ ̺∆x∂tϑ =−cv∂tu · ∇xϑ − 1 ̺2∂t̺ (κ∆xϑ + S(Dxu) : Dxu) +2 ̺ S(Dxu) : Dx∂tu − ∂tϑ divxu − ϑ divx∂tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The estimates obtained in the previous sections imply that the right–hand side of the above equation is bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' L2(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Thus multiplying the equation on ∆x∂tϑ and performing the standard by parts integration, we get the desired estimates as in [9, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' The remaining estimates are obtained exactly as in [9, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6] : sup t∈[0,T) ∥ϑ(t, ·)∥W 3,2(Ω) + sup t∈[0,T) ∥∂tϑ(t, ·)∥W 1,2(Ω) ≤ c(T, D0, ̺, ϑ, u), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='2) � T 0 � ∥∂tϑ∥2 W 2,2(Ω) + ∥ϑ∥2 W 4,2(Ω) � dt ≤ c(T, D0, ̺, ϑ, u), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3) sup t∈[0,T) ∥u(t, ·)∥W 3,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) + sup t∈[0,T) ∥∂tu(t, ·)∥W 1,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) ≤ c(T, D0, ̺, ϑ, u), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='4) � T 0 � ∥∂tu∥2 W 2,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) + ∥u∥2 W 4,2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='R3) � dt ≤ c(T, D0, ̺, ϑ, u), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='5) and sup t∈[0,T) ∥̺(t, ·)∥W 3,2(Ω) ≤ c(T, D0, ̺, ϑ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='6) We have completed the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Chaudhuri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' On weak(measure valued)–strong uniqueness for Navier–Stokes–Fourier system with Dirichlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' Archive Preprint Series, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFRT4oBgHgl3EQfJje8/content/2301.13496v1.pdf'} +page_content=' arxiv 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