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A Survey on Fourier Analysis Methods for Solving the Compressible Navier-Stokes Equations

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A Survey on Fourier Analysis Methods for Solving the Compressible Navier-Stokes Equations Editorial Board Supported by NSFC Honorary Editor General ZHOU GuangZhao (Zhou Guang Zhao) Editor General ZHU ZuoYan Institute of Hydrobiology, CAS Editor-in-Chief YANG Le (Yang Lo) Academy of Mathematics and Systems Science, CAS Executive Associate Editor-in-Chief LI WenLin Academy of Mathematics and Systems Science, CAS Members CAO DaoMin LI JianShu XI NanHua Academy of Mathematics and The Hong Kong University of Academy of Mathematics and Systems Science, CAS Science and Technology Systems Science, CAS CHEN ZhiMing LI KeZheng XIAO Jie Academy of Mathematics and Capital Normal University Tsinghua University Systems Science, CAS LIN FangHua XIN ZhouPing CHENG ChongQing New York University The Chinese University of Hong Kong Nanjing University LIU KeFeng XING ChaoPing University of California at Los Angeles DU DingZhu Nanyang Technological University University of Texas at Dallas LIU ZhangJu XU Fei Peking University DUAN HaiBao Academy of Mathematics and Academy of Mathematics and MA XiaoNan Systems Science, CAS Université Denis Diderot - Paris 7 Systems Science, CAS YAU Shing-Tung E WeiNan MA ZhiMing Harvard University Academy of Mathematics and Princeton University YE XiangDong Systems Science, CAS FAN JianQing University of Science and MOK NgaiMing Princeton University Technology of China The University of Hong Kong FENG KeQin YUAN YaXiang QIN HouRong Tsinghua University Academy of Mathematics and Nanjing University Systems Science, CAS FENG Qi SHI ZhongCi Academy of Mathematics and ZHANG JiPing Academy of Mathematics and Systems Science, CAS Peking University Systems Science, CAS ZHANG Ping GAO XiaoShan SHU Chi-Wang Academy of Mathematics and Academy of Mathematics and Brown University Systems Science, CAS Systems Science, CAS SIU Yum-Tong ZHANG ShouWu GE LiMing Harvard University Academy of Mathematics and Columbia University WANG FengYu Systems Science, CAS ZHANG WeiPing Beijing Normal University GRAHAM Fan Chung Nankai University WANG QiHua University of California at San Diego Academy of Mathematics and ZHOU XiangYu HONG JiaXing Systems Science, CAS Academy of Mathematics and Fudan Unviersity Systems Science, CAS WANG XuJia JI LiZhen Australian National University ZHU XiPing University of Michigan Sun Yat-Sen University WANG YueFei LAU Ka-Sing Academy of Mathematics and ZUO Kang The Chinese University of Hong Kong Systems Science, CAS Johannes Gutenberg-University Mainz Editorial Staff CHAI Zhao YANG ZhiHua ZHANG RuiYan [email protected] [email protected] [email protected] SCIENCE CHINA Mathematics . ARTICLES . February 2012 Vol. 55 No. 2: 245{275 doi: 10.1007/s11425-011-4357-8 A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations Dedicated to the NSFC-CNRS Chinese-French summer institute on fluid mechanics in 2010 DANCHIN Rapha¨el LAMA UMR 8050, Universit´eParis-Est, 61, avenue du G´en´eral de Gaulle, 94010 Cr´eteilCedex, France Email: [email protected] Received March 11, 2011; accepted May 17, 2011 Abstract Fourier analysis methods and in particular techniques based on Littlewood-Paley decomposition and paraproduct have known a growing interest recently for the study of nonlinear evolutionary equations. In this survey paper, we explain how these methods may be implemented so as to study the compresible Navier-Stokes equations in the whole space. We shall investigate both the initial value problem in critical Besov spaces and the low Mach number asymptotics. Keywords compressible fluids, critical regularity, Besov spaces, Fourier analysis, Mach number MSC(2010) 35B25, 35B40, 76N10 Citation: Danchin R. A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations. Sci China Math, 2012, 55(2): 245{275, doi: 10.1007/s11425-011-4357-8 1 Introduction This survey aims at presenting a few recent results for the compressible Navier-Stokes equations that have been proved by means of modern Fourier analysis methods. These techniques are relevant in any context where a good notion of Fourier transform is available. Here, for simplicity, we shall concentrate on the whole space case Rd (that is, the space variable of the solutions to the Navier-Stokes equations will be in Rd). However, our approach may be easily adapted to periodic boundary conditions x 2 Td and more generally to x 2 Td1 × Rd2 and so on (see, for example, [16]). Let us also stress that the methods that we introduce here are quite robust. As a matter of fact, we expect this survey to be a guidebook for solving more elaborate evolutionary nonlinear partial differential equations not necessarily related to fluid mechanics. In addition to the Fourier analysis techniques, adopting a critical functional framework will be another thread of these notes. Let us present the idea on an abstract model problem. Let X be a Banach space of functions v = v(t; x) with t 2 R+ and x 2 Rd; and B : X × X ! X; a continuous bilinear map with norm M: Then the standard fixed point theorem ensures that there exists a unique solution v in B(0; 2kv0kX ) to v = v0 + B(v; v); (E) whenever 4Mkv0kX < 1: (1.1) ⃝c Science China Press and Springer-Verlag Berlin Heidelberg 2012 math.scichina.com www.springerlink.com 246 Danchin R Sci China Math February 2012 Vol. 55 No. 2 Assume that (E) has the following scaling invariance property: there exists a triplet (α; β; γ) such that for all λ > 0; the transform α β γ v 7! vλ with vλ(t; x) = λ v(λ t; λ x) preserves Equation (E) if v0 has been changed into α γ v0,λ : x 7−! λ v0(λ x): Hence, either (E) may be solved for any data in X (which is not the most common case) or the condition (1.1) itself has to be invariant by rescaling, that is, kv0,λkX ≈ kv0kX ; 8 λ > 0: If so, we shall say that the space X is critical for (E). This simple argument has been used a number of times for solving evolutionary equations such as nonlinear Schr¨odinger,wave or heat equations. Let us explain how to implement it on the incompressible Navier-Stokes equations ( @tu + div(u ⊗ u) − µ∆u + rΠ = 0; (NSI) divu = 0: At the formal level, solving (NSI) amounts to solving (E) with Z t t∆ (t−τ)∆ v0 := e u0 and B(v; v)(t) = − e Pdiv(v ⊗ v) dτ; 0 t∆ where (e )t>0 stands for the heat semi-group and P the Leray projector over divergence-free vector-fields. Now, it is clear that the above equation is invariant for all λ > 0 by the transform 2 v 7! vλ with vλ(t; x) = λv(λ t; λx): Hence, any space X which is norm invariant for every λ > 0 by the above transform is critical for the incompressible Navier-Stokes equation1) . In the context of the incompressible Navier-Stokes equations, this idea has been first implemented by Fujita and Kato [20] (see also the work by Chemin [5]). Here { } + d −1 1 + d − 1 X = v 2 C(R ; H_ 2 ) such that t 4 v 2 C(R ; H_ 2 2 ) d −1 and the initial data is in the homogeneous Sobolev space H_ 2 . There are a number of critical functional spaces in which (NSI) may be globally solved for small data, for instance, •C(R+; Ld) (see the works by Giga [23], Kato [24] and Furioli-Lemari´e-Terraneo [21]); d −1 d +1 •C R+ _ p \ 1 R+ _ p ( ; Bp;1 ) L ( ; Bp;1 ) and more general Besov spaces (see the works by Cannone-Meyer- Planchon [4] and by Kozono and Yamazaki [29]). Presenting similar results in the context of compressible viscous fluids is the main goal of these notes. To simplify the presentation, we focus on the barotropic Navier-Stokes equations: ( @tρ + div(ρu) = 0; 0 @t(ρu) + div (ρu ⊗ u) − µ∆u − µ rdiv u + rP = 0; where • ρ = ρ(t; x) 2 R+ (with t 2 R+ and x 2 Rd) is the density, • u = u(t; x) 2 Rd is the velocity field, 1) Which, of course, does not mean that (NSI) may be solved in any such space! Danchin R Sci China Math February 2012 Vol. 55 No. 2 247 • the viscosity coefficients µ and µ0 satisfy µ > 0 and ν := µ + µ0 > 0: In order to close the system, we assume the pressure P to be a given (suitably smooth) function of ρ. This is the so-called barotropic assumption. In the viscous case that we shall consider, this assumption is somewhat irrelevant from a physical viewpoint. However, the above system already contains many features of the full model as far as mathematical results are concerned. We supplement the system with the following boundary conditions: • u decays to zero at infinity, • ρ tends to some positive constantρ ¯ at infinity. We shall takeρ ¯ = 1 for simplicity. Denoting ρ = 1+a and assuming that the density is positive everywhere, the barotropic system rewrites ( @ta + u · ra = −(1 + a)divu; (1.2) @tu − Au = −u · ru − J(a)Au − rG(a); where A := µ∆ − µ0rdiv;J(a) := a=(1 + a) and G is a smooth function with G(0) = 0: Strictly speaking, the above system does not own any scaling invariance property in the above meaning. However, we notice that up to a change of G into λ2G; System (1.2) is invariant by the rescaling a(t; x) ! a(λ2t; λx); u(t; x) ! λu(λ2t; λx): (1.3) To some extent, the term G is lower order. Therefore, we expect the critical functional spaces for the velocity to be the same ones as in the incompressible case, whereas one more derivative has to be taken for the density. This heuristics will be made more explicit later on in the paper. Our paper unfolds as follows.
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