流体动力学方程的Fourier局部化方法
|
摘要
Fourier局部化方法是研究流体动力学方程(组)的一个崭新的工具,它主要基于经典的Littlewood-Paley理论和Bony的仿积分解技术。Meyer,Chemin,Cannone,Planchon及其法国学派发展了一整套研究不可压缩流体动力学方程的技术和方法,得到了一系列深刻的结果。与此同时,Danchin又将这些方法应用到了可压缩流体动力学方程的研究中。最近,Danchin,Abidi,Hmidi,Keraani等数学家进一步发展了Fourier局部化方法,建立了一些新的局部化引理和交换子估计,并将其应用到了一些流体动力学方程的研究中。
本文致力于利用Fourier局部化方法来研究二维分数阶Navier-Stokes方程,临界耗散Burgers方程,旋转浅水方程和二维分数次耗散chemotaxis方程等几类流体动力学方程,主要研究解的局部适定性,整体适定性,以及粘性方程解的无粘极限等问题。为此,我们借助于Fourier局部化方法首次建立了分数次Laplace算子的交换子估计和分数阶输运扩散方程的一致时空估计,该一致估计对散度自由和非散度自由的系数都是有效的,从而为研究(分数次耗散)不可压和可压流体动力学方程提供了一个重要的工具。
本文具体内容如下:
第二章回忆Besov空间理论和Fourier局部化方法,列举Besov空间的一些基本性质和Bony仿积估计,同时介绍Fourier局部化方法中的一些新的局部化引理和交换子估计。
第三章研究二维分数阶Navier-Stokes方程的无粘极限,并估计其极限的收敛速度。此外,还证明初始正则性在临界Besov空间的一致保持性。为了证明这些结果,我们首先利用具有正的小正则性指标的齐次Besov空间的差商刻画来证明一个分数次Laplace算子的交换子估计。然后我们利用这个交换子估计结合Fourier局部化技巧来证明涡度方程的一个频段层次上的正则效应,进而来证明粘性速度的Lipschitz范数关于粘性v的一致有界性。
第四章研究临界耗散Burgers方程(?)_tu+u(?)_xu+(-Δu)~(1/2)=0的Cauchy问题。文中我们利用Fourier局部化方法结合Lagrangian坐标变换以及分数次Laplace算子的交换子估计首次建立了分数阶输运扩散方程在Besov框架下的一致时空估计,这是证明局部适定性的关键,同时也为其他相关方程的研究提供了一个很好的工具。然后我们利用连续模方法以及Fourier局部化技巧来证明临界耗散Burgers方程在临界Besov空间(?)_(p,1)~(1/p)(R)(p∈[1,∞))中的整体适定性。
第五章研究带有毛细管作用项的粘性旋转浅水方程。我们证明Cauchy问题在关于初值的低正则性假设下的局部存在性和唯一性,关于初始高度要求远离零点有界。为此,我们首先进行变量替换并利用Hodge分解将向量场分解成一个可压部分和一个不可压部分,由于Coriolis力的旋转效应,从而得到一个耦合系统。然后,借助于Fourier局部化方法,我们得到相应线性系统的先验估计。因为Coriolis frequency的出现,我们必须对高频部分和低频部分进行不同的估计。最后,我们使用一个经典的迭代方法来构造逼近解并证明局部存在性和唯一性。
第六章研究chemotaxis的Keller-Segel模型的推广一分数次耗散方程u_t+(-Δ)~(α/2)u=▽·(u▽(Δ~(-1)u))的Cauchy问题,这里初值u_0属于临界的Besov空间(?)_(2,r)~(1-α)(R~2),其中r∈[1,∞],1＜α＜2。利用mixed时空空间框架下的线性耗散方程的估计,Chemin的“单模”方法,Fourier局部化技巧和Littlewood-Paley理论,我们得到一个局部适定性结果。此外,我们还考虑类似的“双抛物”模型。
Fourier localization method based on the classical Littlewood-Paley theory and Bony's paraproduct decomposition is a very new tool in the study of fluid dynamics equations.Meyer,Chemin,Cannone,Planchon and their French school developed an entire set of techniques and methods in the study of incompressible fluid dynamics equations.Moreover,Danchin applied these methods to the study of compressible fluid dynamics equations.Recently,some mathematicians including Danchin,Abidi,Hmidi and Keraani took the development of Fourier localization method to a new higher level,established some new localization lemmas and commutator estimates and used those to study some fluid dynamics equations.
This thesis is devoted to the study of several types of fluid dynamics equations: two-dimensional Navier-Stokes equations with fractional diffusion,critical Burgers equation,rotating shallow water equations and two-dimensional chemotaxis models with fractional diffusion.We study their basic properties such as the local well-posedness,global well-posedness,inviscid limit,etc.With help of Fourier localization method,we for the first time establish the commutator estimate for the fractional Laplacian operator and the uniform space-time estimate for the transport diffusion equations with fractional diffusion.This estimate is valid for both divergence-free and non-divergence-free coefficients,thus provide an important tool for the study of incompressible and compressible fluid dynamics equations(with fractional diffusion).
The detail of this thesis is arranged as follows.
In the second chapter,we recall the theory of Besov spaces and Fourier localization method.We list some basic properties of Besov spaces and Bony's paraproduct,and introduce some new localization lemmas and commutator estimates in Fourier localization method.
In the third chapter,we study the inviscid limit of the 2-D Navier-Stokes equations with fractional diffusion and establish the convergence rate of the inviscid limit for vanishing viscosity.We prove also the uniform persistence of the initial regularity in some critical Besov spaces.In order to prove these results,we first prove a commutator estimate for the fractional Laplacian operator by making use of the "difference quotient" description for homogeneous Besov spaces with small positive regularity index.Then we use this commutator estimate together with the Fourier localization technique to prove a regularization effect of the vorticity equation which allows us to bound the Lipschitz norm of the viscous velocity uniformly on the viscosity v.
In the fourth chapter,we study the Cauchy problem for the critical Burgers equation(?)_tu+u(?)_xu+(-△u)~(1/2)=0.We make use of the Fourier localization method together with Lagrangian coordinates transformation and the commutator estimate for the fractional Laplacian operator to establish the uniform spacetime estimate for the transport diffusion equations with fractional diffusion in the frame of Besov spaces for the first time.This estimate is the key to the proof of the local well-posedness,and provide a very useful tool for the study of the other relevant equations.Then we make use of the method of modulus of continuity and Fourier localization technique to prove the global well-posedness of the critical Burgers equation in critical Besov spaces(?)_(p,1)~(1/p)(R) with p∈[1,∞).
In the fifth chapter,we study the viscous rotating shallow water equations with a term of capillarity.We prove the local in time existence and uniqueness for the Cauchy problem under low regularity assumptions on the initial data as well as the initial height bounded away from zero.To prove this result,we firstly perform variable substitution and make use of the Hodge's decomposition to separate the vector field into a compressible part and an incompressible part, and obtain a coupled system due to the rotating effect of the Coriolis force. Secondly,with the help of the Fourier localization method,we obtain the a priori estimates of the corresponding linear system.Because of the appearance of the Coriolis frequency,we must perform different estimates for the high frequencies and the low frequencies respectively.Finally,we use a classical iterate method to construct approximate solutions and prove the local in time existence and uniqueness.
In the last chapter,we study the Cauchy problem for the fractional diffusion equation u_t+(-△)~(α/2)u=▽·(u▽(△~(-1)u)),generalizing the Keller-Segel model of chemotaxis,for the initial data uo in critical Besov spaces(?)_(2,r)~(1-α)(R~2) with r∈[1,∞],where 1＜α＜2.Making use of some estimates of the linear dissipative equation in the frame of mixed time-space spaces,the Chemin "mononorm method",Fourier localization technique and the Littlewood-Paley theory, we get a local well-posedness result.Furthermore,we also consider analogous "doubly parabolic" models.
本文致力于利用Fourier局部化方法来研究二维分数阶Navier-Stokes方程,临界耗散Burgers方程,旋转浅水方程和二维分数次耗散chemotaxis方程等几类流体动力学方程,主要研究解的局部适定性,整体适定性,以及粘性方程解的无粘极限等问题。为此,我们借助于Fourier局部化方法首次建立了分数次Laplace算子的交换子估计和分数阶输运扩散方程的一致时空估计,该一致估计对散度自由和非散度自由的系数都是有效的,从而为研究(分数次耗散)不可压和可压流体动力学方程提供了一个重要的工具。
本文具体内容如下:
第二章回忆Besov空间理论和Fourier局部化方法,列举Besov空间的一些基本性质和Bony仿积估计,同时介绍Fourier局部化方法中的一些新的局部化引理和交换子估计。
第三章研究二维分数阶Navier-Stokes方程的无粘极限,并估计其极限的收敛速度。此外,还证明初始正则性在临界Besov空间的一致保持性。为了证明这些结果,我们首先利用具有正的小正则性指标的齐次Besov空间的差商刻画来证明一个分数次Laplace算子的交换子估计。然后我们利用这个交换子估计结合Fourier局部化技巧来证明涡度方程的一个频段层次上的正则效应,进而来证明粘性速度的Lipschitz范数关于粘性v的一致有界性。
第四章研究临界耗散Burgers方程(?)_tu+u(?)_xu+(-Δu)~(1/2)=0的Cauchy问题。文中我们利用Fourier局部化方法结合Lagrangian坐标变换以及分数次Laplace算子的交换子估计首次建立了分数阶输运扩散方程在Besov框架下的一致时空估计,这是证明局部适定性的关键,同时也为其他相关方程的研究提供了一个很好的工具。然后我们利用连续模方法以及Fourier局部化技巧来证明临界耗散Burgers方程在临界Besov空间(?)_(p,1)~(1/p)(R)(p∈[1,∞))中的整体适定性。
第五章研究带有毛细管作用项的粘性旋转浅水方程。我们证明Cauchy问题在关于初值的低正则性假设下的局部存在性和唯一性,关于初始高度要求远离零点有界。为此,我们首先进行变量替换并利用Hodge分解将向量场分解成一个可压部分和一个不可压部分,由于Coriolis力的旋转效应,从而得到一个耦合系统。然后,借助于Fourier局部化方法,我们得到相应线性系统的先验估计。因为Coriolis frequency的出现,我们必须对高频部分和低频部分进行不同的估计。最后,我们使用一个经典的迭代方法来构造逼近解并证明局部存在性和唯一性。
第六章研究chemotaxis的Keller-Segel模型的推广一分数次耗散方程u_t+(-Δ)~(α/2)u=▽·(u▽(Δ~(-1)u))的Cauchy问题,这里初值u_0属于临界的Besov空间(?)_(2,r)~(1-α)(R~2),其中r∈[1,∞],1＜α＜2。利用mixed时空空间框架下的线性耗散方程的估计,Chemin的“单模”方法,Fourier局部化技巧和Littlewood-Paley理论,我们得到一个局部适定性结果。此外,我们还考虑类似的“双抛物”模型。
Fourier localization method based on the classical Littlewood-Paley theory and Bony's paraproduct decomposition is a very new tool in the study of fluid dynamics equations.Meyer,Chemin,Cannone,Planchon and their French school developed an entire set of techniques and methods in the study of incompressible fluid dynamics equations.Moreover,Danchin applied these methods to the study of compressible fluid dynamics equations.Recently,some mathematicians including Danchin,Abidi,Hmidi and Keraani took the development of Fourier localization method to a new higher level,established some new localization lemmas and commutator estimates and used those to study some fluid dynamics equations.
This thesis is devoted to the study of several types of fluid dynamics equations: two-dimensional Navier-Stokes equations with fractional diffusion,critical Burgers equation,rotating shallow water equations and two-dimensional chemotaxis models with fractional diffusion.We study their basic properties such as the local well-posedness,global well-posedness,inviscid limit,etc.With help of Fourier localization method,we for the first time establish the commutator estimate for the fractional Laplacian operator and the uniform space-time estimate for the transport diffusion equations with fractional diffusion.This estimate is valid for both divergence-free and non-divergence-free coefficients,thus provide an important tool for the study of incompressible and compressible fluid dynamics equations(with fractional diffusion).
The detail of this thesis is arranged as follows.
In the second chapter,we recall the theory of Besov spaces and Fourier localization method.We list some basic properties of Besov spaces and Bony's paraproduct,and introduce some new localization lemmas and commutator estimates in Fourier localization method.
In the third chapter,we study the inviscid limit of the 2-D Navier-Stokes equations with fractional diffusion and establish the convergence rate of the inviscid limit for vanishing viscosity.We prove also the uniform persistence of the initial regularity in some critical Besov spaces.In order to prove these results,we first prove a commutator estimate for the fractional Laplacian operator by making use of the "difference quotient" description for homogeneous Besov spaces with small positive regularity index.Then we use this commutator estimate together with the Fourier localization technique to prove a regularization effect of the vorticity equation which allows us to bound the Lipschitz norm of the viscous velocity uniformly on the viscosity v.
In the fourth chapter,we study the Cauchy problem for the critical Burgers equation(?)_tu+u(?)_xu+(-△u)~(1/2)=0.We make use of the Fourier localization method together with Lagrangian coordinates transformation and the commutator estimate for the fractional Laplacian operator to establish the uniform spacetime estimate for the transport diffusion equations with fractional diffusion in the frame of Besov spaces for the first time.This estimate is the key to the proof of the local well-posedness,and provide a very useful tool for the study of the other relevant equations.Then we make use of the method of modulus of continuity and Fourier localization technique to prove the global well-posedness of the critical Burgers equation in critical Besov spaces(?)_(p,1)~(1/p)(R) with p∈[1,∞).
In the fifth chapter,we study the viscous rotating shallow water equations with a term of capillarity.We prove the local in time existence and uniqueness for the Cauchy problem under low regularity assumptions on the initial data as well as the initial height bounded away from zero.To prove this result,we firstly perform variable substitution and make use of the Hodge's decomposition to separate the vector field into a compressible part and an incompressible part, and obtain a coupled system due to the rotating effect of the Coriolis force. Secondly,with the help of the Fourier localization method,we obtain the a priori estimates of the corresponding linear system.Because of the appearance of the Coriolis frequency,we must perform different estimates for the high frequencies and the low frequencies respectively.Finally,we use a classical iterate method to construct approximate solutions and prove the local in time existence and uniqueness.
In the last chapter,we study the Cauchy problem for the fractional diffusion equation u_t+(-△)~(α/2)u=▽·(u▽(△~(-1)u)),generalizing the Keller-Segel model of chemotaxis,for the initial data uo in critical Besov spaces(?)_(2,r)~(1-α)(R~2) with r∈[1,∞],where 1＜α＜2.Making use of some estimates of the linear dissipative equation in the frame of mixed time-space spaces,the Chemin "mononorm method",Fourier localization technique and the Littlewood-Paley theory, we get a local well-posedness result.Furthermore,we also consider analogous "doubly parabolic" models.
引文
[1]H.Abidi,T.Hmidi,On the global well-posedness of the critical quasi-geostrophic equation,SIAM J.Math.Anal.40(2008),no.1,167-185.
[2]N.Alibaud,Entropy formulation for fractal conservation laws,J.evol.equ.7(2007),145-175.
[3]N.Alibaud,J.Droniou,Occurence and non-apperance of schockes in fractal Burgers equations,Journal of Hyperbolic Differential Equations,vol.4,No.3(2007)479-499.
[4]J.T.Beale,T.Kato,A.Majda,Remarks on the breakdown of smooth solutions for 3-D Euler equations,Comm.Math.Phys 94(1984),61-66.
[5]J.Bergh,J.L(o|¨)fstr(o|¨)m,Interpolation Spaces,an Introduction,Springer-Verlag,New York,1976.
[6]P.Biler,Existence and nonexistence of solutions for a model of gravitational interaction of particles Ⅲ,Colloquium Math.68(1995),229-239.
[7]P.Biler,Local and global solvability of parabolic systems modelling chemotaxis,Adv.Math.Sci.Appl.8(1998),715-743.
[8]P.Biler,A note on the paper of Y.Naito,33-40,in:Self-Similar Solutions in Nonlinear PDE,Banach Center Publications 74,Polish Acad.Sci.,Warsaw,2006.
[9]P.Biler,M.Cannone,I.Guerra and G.Karch,Global regular and singular solutions for a model of gravitating particles,Math.Ann.330(2004),693-708.
[10]P.Biler,L.Corrias,J.Dolbeault,in preparation.
[11]P.Biler,G.Karch,Blow up of solutions of nonlinear evolution equations with Levy diffusion operators,in preparation.
[12]P.Biler,W.A.Woyczynski,Global and exploding solutions for nonlocal quadratic evolution problems,SIAM J.Appl.Math.59(1998),845-869.
[13]P.Biler,W.A.Woyczynski,General nonlocal diffusion-convection mean.field models:nonexistence of global solutions,Physica A 379(2007),523-533.
[14]D.Bresch,B.Desjardins,Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasigeostrophic model,Comm.Math.Phys.238,211-223(2003).
[15]D.Bresch,B.Desjardins,Some diffusive capillary models for Korteweg type,C.R.Mecanique 331,2003.
[16]A.T.Bui,Existence and uniqueness of a classical solution of an initial boundary value problem of the theory of shallow waters,SIAM J.Math.Anal.12,229-241(1981).
[17]V.Calvez,L.Corrias,The parabolic-parabolic Keller-Segel model in R~2,1-26,arXiv:math.AP/0712.3169v1.
[18]M.Cannone,Harmonic analysis tools for solving the incompressible Navier-Stokes equations,161-244,in:Handbook of Mathematical Fluid Dynamics,vol.Ⅲ,S.J.Friedlander,D.Serre,eds.,Elsevier,2004.
[19]D.Chae,Local existence and blow-up criterion for the Euler equations in the Besov spaces,Asymptot.Anal.38(2004),no.3-4,339-358.
[20]D.Chae,J.Lee,On the global well-posedness and stability of the Navier-Stokes and the related equations,Contributions to current challenges in mathematical fluid mechanics,31-51,Adv.Math.Fluid Mech.,Birkh(a|¨)user,Basel,2004.
[21]J.Y.Chemin,Perfect Incompressible Fluids,Oxford Lecture Ser.Math.Appl.,vol.14,The Clarendon Press/Oxford University Press,New York,1998.
[22]J.Y.Chemin,Theoremes d'unicite pour le systeme de Navier-Stokes tridimensionnel,J.d'Analyse Math.,77(1999),27-50.
[231 Q.Chen,C.Miao and Z.Zhang,A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation,Comm.Math.Phys.271,821-838(2007).
[24]Q.Chen,C.Miao,Z.Zhang,Well-posedness for the viscous shallow water equations in critical spaces,SIAM.J.Math.Anal.Vol.40(2008) 443-474.
[25]Q.Chen,C.Miao,Z.Zhang,Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities,ArXiv:math.AP/0811.4215v1,2008.
[26]B.Cheng,E.Tadmor,Long time existence of smooth solutions.for the rapidly rotating shallow-water and Euler equations,ArXiv:math.AP/OTO6.0758v1,2007.
[27]P.Constantin,D.Cordoba and J.Wu,On the critical dissipative quasi-geostrophic equation,Dedicated to Professors Ciprian Foias and Roger Temam(Bloomington,IN,2000).Indiana Univ.Math.J.50(2001),97-107.
[28]A.Cordoba and D.Cordoba,A maximum principle applied to quasi-geostrophic equations,Comm.Math.Phys.,249(2004),511-528.
[29]R.Danchin,Global existence in critical spaces for compressible Navier-Stokes equations,Invent.Math.141,579-614(2000).
[30]R.Danchin,Global existence in critical spaces for flows of compressible viscous and heat-conductive gases,Arch.Rational Mech.Anal.,160(2001),1-39.
[31]R.Danchin,Density-dependent incompressible viscous fluids in critical spaces,Proc.Roy.Soc.Edinburgh Sect.A,133(2003),1311-1334.
[32]R.Danchin,On the uniqueness in critical spaces for compressible Navier-Stokes equations,Nonlinear Differential Equations Appl.,12(2005),111-128.
[33]R.Danchin,Fourier analysis methods for POE's,preprint(2005),Lab.d'Analyse et de Math.Appliquees,UMR 8050,http://perso-math.univmlv.fr/users/danchin.raphael.
[34]R.Danchin,Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients,Rev.Mat.Iberoamericana 21(3)(2005)863-888.
[35]R.Danchin,Uniform Estimates for Transport-Diffusion Equations,Journal of Hyperbolic Differential Equations,vol.4,No.1(2007)1-17.
[36]R.Danchin,Well-Posedness in Critical Spaces for Barotropic Viscous Fluids with Truly Not Constant Density,Communications in Partial Differential Equations,32:9,1373-1397(2007).
[37]R.Danchin,B.Desjardins,Existence of solutions for compressible fluid models of Korteweg type,Annales de I'IHP,Analyse non lineaire 18,97-133(2001).
[38]H.Dong,D.Du,Global well-posedness and a decay estimate.for the critical dissipative quasi-geostrophic equation in the whole space,Discrete Contin.Dyn.Syst.21(2008),no.4,1095-1101.
[39]H.Dong,D.Du,D.Li,Finite time sigularities and global well-posedness for fractal Burgers equation,Indiana Univ.Math.J.58(2009).
[40]C.Escudero,Chemotactic collapse and mesenchymal morphogenesis,Phys.Rev.E 72(2005),022903.
[41]C.Escudero,The fractional Keller-Segel model,Nonlinearity 19(2006),2909-2918.
[42]H.Fujita,T.Kato,On the nonstationnary Navier-Stokes system,rend.Sem.Mat.Univ.Padova,32(1962),243-260.
[43]Z.Guo,Q.Jiu,Z.Xin,Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients,SIAM J.Math.Anal.39,1402-1427(2008).
[44]C.Hao,L.Hsiao,H.Li,Cauchy problem for viscous rotating shallow water equations,arXiv:0806.4504v3.
[45]B.Haspot,Cauchy problem for viscous shallow water equations with a term of capillarity,ArXiv:math.AP/0803.1939v1,2008.
[46]T.Hmidi,Regularite h(o|¨)lderienne des poches de tourbillon visqueuses,J.Math.pures Appl.(9) 84(11)(2005) 1455-1495.
[47]T.Hmidi,S.Keraani,Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces,Advances in Mathematics,Volume 214,Issue 2,1 October 2007,Pages 618-638.
[48]T.Hmidi,S.Keraani,Inviscid limit for the two-dimensional Navier-Stokes system in a critical Besov space,Asymptotic Analysis,53(3) 2007,125-138.
[49]T.Hmidi,S.Keraani,Incompressible viscous flows in borderline Besov spaces,Arch.Ration.Mech.Anal.189(2008),no.2,283-300.
[50]T.Hmidi,S.Keraani,Existence globale pour le systeme d'Euler incompressible 2-D dans B_(∞,1)~1,C.R.Math.Acad.Sci.Paris 341(2005),no.11,655-658.
[51]G.Karch,Scaling in nonlinear parabolic equations,J.Math.Anal.Appl.234(1999),534-558.
[52]G.Karch,Scaling in nonlinear parabolic equations:Applications to Debye system,in:Proceedings of Conference on Disordered and Complex Systems,London,10-14 July,2000,American Institute of Physics Publishing 553,2001.[53]G.Karch,C.Miao,X.Xu,On convergence of solutions of fractal Burgers equation toward rarefaction waves,SIAM J.Math.Anal.39(2007)1536-1549.
[54]T.Kato,G.Ponce,Commutator estimates and the Euler and Navier-Stokes equations,Communications on Pure and Applied Mathematics,41(1988),891-907.
[55]A.Kiselev,F.Nazarov,R.Shterenberg,Blow up and regularity for fractal Burgers equation,Dyn.Partial Differ.Equ.5(2008),no.3,211-240.
[56]A.Kiselev,F.Nazarov and A.Volberg,Global well-posedness for the critical 2D dissipative quasi-geostrophic equation,Inventiones Math.167(2007) 445-453.
[57]P.E.Kloeden,Global existence of classical solutions in the dissipative shallow water equations,SIAM J.Math.Anal.16,301-315(1985).
[58]P.G.Lemarie-Rieusset,Recent development in the Navier-Stokes problem,Chapman Hall/CRC,2002.
[59]J.Leray,Sur le mouvement d'un liquide visqueux remplissant l'espace,Acta mathematica,63(1934),193-248.
[60]H.Li,J.Li,Z.Xin,Vanishing of vacuum states and blow-up phenomena for the compressible Navier-Stokes equations,Comm.Math.Phys.281,401-444(2008).
[61]A.Majda,Vorticity and the mathematical theory of an incompressible fluid flow,Communications on Pure and Applied Mathematics,38(1986),187-220.
[62]F.Marche,Derivation of a new two-dimensional viscous shallow water model with varying topography,bottom friction and capillary effects,European J.Mech.B/Fluids,26,49-63(2007).
[63]A.Mellet,A.Vasseur,On the barotropic compressible Navier-Stokes equations,Comm.Partial Diff.Eqns.32(1-3),431-452(2007).
[64]Y.Meyer,Wavelets,paraproducts and Navier-Stokes equations,Current developments in Math.,International Press,(1996),105-212.
[65]C.Miao,Time-space estimates of solutions to general semilinear parabolic equations,Tokyo J.Math.24(2001),245-276.
[66]C.Miao,Harmonic Analysis and Applications to Partial Differential Equations (Second Edition),Monographs on Modern pure mathematics,No.89.Science Press,Beijing,2004.
[67]C.Miao,Harmonic Analysis Method of Partial Differential Equations,Monographs on Modern pure mathematics,No.117.Science Press,Beijing,2008.
[68]C.Miao,Y.Gu,Space-time estimates for parabolic type operator and application to nonlinear parabolic equations,J.Partial Differential Equations 11(1998),301-312.
[69]C.Miao,B.Yuan,Solutions to some nonlinear parabolic equations in pseudomeasure spaces,Math.Nachr.280(2007),171-186.
[70]C.Miao,B.Yuan and B.Zhang,Well-posedness of the Cauchy problem for the fractional power dissipative equations,Nonlinear Analysis 68(2008) 461-484.
[71]C.Miao,B.Zhang,The Cauchy problem for semilinear parabolic equations in Besov spaces,Houston J.Math.30(2004),829-878.
[72]Y.Naito,Asymptotically self-similar solutions for the parabolic systems modelling chemotaxis,149-160,in:Self-Similar Solutions in Nonlinear PDE,Banach Center Publications 74,Polish Acad.Sci.,Warsaw,2006.
[73]Y.Naito,T.Suzuki,K.Yoshida,Self-similar solutions to a parabolic system modeling chemotaxis,J.Differential Equations 184(2002),386-421.
[74]L.Nirenberg,On elliptic partial differential equations,Ann.Scuola Norm.Sup.Pisa,13(1959),115-162.
[75]A.Raczynski,Stability property of the two-dimensional Keller-Segel model,Asymptot.Anal.61(2009),no.1,35-59.
[76]T.Runst,W.Sickel,Sobolev Spaces of Fractional Order,Nemytskij Operators,and Nonlinear Partial Differential Equations,de Gruyter Series in Nonlinear Analysis and Applications,vol.3,Walter de Gruyter & Co.,Berlin,1996.
[77]E.M.Stein,Singular Integrals and Differentiability Properties of Functions,Princeton University Press,Princeton,New Jersey,1970.
[78]L.Sundbye,Global existence for Dirichlet problem for the viscous shallow water equations,J.Math.Anal.Appl.202,236-258(1996).
[79]L.Sundbye,Global existence for the Cauchy problem for the viscous shallow water equations,Rocky Mountain J.Math.28,1135-1152(1998).
[80]H.Triebel,Theory of Function Spaces,Monographs in Mathematics,Vol.78.Basel-Boston -Stuttgart:Birkhuser Verlag,DM 90.00,1983.
[81]M.Vishik,Hydrodynamics in Besove Spaces,Arch.Rational Mech.Anal 145(1998),197-214.
[82]W.Wang,C.Xu,The Cauchy problem for viscous shallow water equations,Rev.Mat.Iberoamericana 21,1-24(2005).
[83]G.Wu,J.Yuan,Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces,J.Math.Anal.Appl.340(2008)1326-1335.
[84]J.Wu,The generalized incompressible Navier-Stokes equations in Besov spaces,Dyn.Partial Differ.Equ.1(2004),no.4,381-400.
[85]J.Wu,The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation,Nonlinearity 18,139-154(2005).
[86]J.Wu,Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces,Comm.Math.Phys.263(2006),no.3,803-831.
[2]N.Alibaud,Entropy formulation for fractal conservation laws,J.evol.equ.7(2007),145-175.
[3]N.Alibaud,J.Droniou,Occurence and non-apperance of schockes in fractal Burgers equations,Journal of Hyperbolic Differential Equations,vol.4,No.3(2007)479-499.
[4]J.T.Beale,T.Kato,A.Majda,Remarks on the breakdown of smooth solutions for 3-D Euler equations,Comm.Math.Phys 94(1984),61-66.
[5]J.Bergh,J.L(o|¨)fstr(o|¨)m,Interpolation Spaces,an Introduction,Springer-Verlag,New York,1976.
[6]P.Biler,Existence and nonexistence of solutions for a model of gravitational interaction of particles Ⅲ,Colloquium Math.68(1995),229-239.
[7]P.Biler,Local and global solvability of parabolic systems modelling chemotaxis,Adv.Math.Sci.Appl.8(1998),715-743.
[8]P.Biler,A note on the paper of Y.Naito,33-40,in:Self-Similar Solutions in Nonlinear PDE,Banach Center Publications 74,Polish Acad.Sci.,Warsaw,2006.
[9]P.Biler,M.Cannone,I.Guerra and G.Karch,Global regular and singular solutions for a model of gravitating particles,Math.Ann.330(2004),693-708.
[10]P.Biler,L.Corrias,J.Dolbeault,in preparation.
[11]P.Biler,G.Karch,Blow up of solutions of nonlinear evolution equations with Levy diffusion operators,in preparation.
[12]P.Biler,W.A.Woyczynski,Global and exploding solutions for nonlocal quadratic evolution problems,SIAM J.Appl.Math.59(1998),845-869.
[13]P.Biler,W.A.Woyczynski,General nonlocal diffusion-convection mean.field models:nonexistence of global solutions,Physica A 379(2007),523-533.
[14]D.Bresch,B.Desjardins,Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasigeostrophic model,Comm.Math.Phys.238,211-223(2003).
[15]D.Bresch,B.Desjardins,Some diffusive capillary models for Korteweg type,C.R.Mecanique 331,2003.
[16]A.T.Bui,Existence and uniqueness of a classical solution of an initial boundary value problem of the theory of shallow waters,SIAM J.Math.Anal.12,229-241(1981).
[17]V.Calvez,L.Corrias,The parabolic-parabolic Keller-Segel model in R~2,1-26,arXiv:math.AP/0712.3169v1.
[18]M.Cannone,Harmonic analysis tools for solving the incompressible Navier-Stokes equations,161-244,in:Handbook of Mathematical Fluid Dynamics,vol.Ⅲ,S.J.Friedlander,D.Serre,eds.,Elsevier,2004.
[19]D.Chae,Local existence and blow-up criterion for the Euler equations in the Besov spaces,Asymptot.Anal.38(2004),no.3-4,339-358.
[20]D.Chae,J.Lee,On the global well-posedness and stability of the Navier-Stokes and the related equations,Contributions to current challenges in mathematical fluid mechanics,31-51,Adv.Math.Fluid Mech.,Birkh(a|¨)user,Basel,2004.
[21]J.Y.Chemin,Perfect Incompressible Fluids,Oxford Lecture Ser.Math.Appl.,vol.14,The Clarendon Press/Oxford University Press,New York,1998.
[22]J.Y.Chemin,Theoremes d'unicite pour le systeme de Navier-Stokes tridimensionnel,J.d'Analyse Math.,77(1999),27-50.
[231 Q.Chen,C.Miao and Z.Zhang,A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation,Comm.Math.Phys.271,821-838(2007).
[24]Q.Chen,C.Miao,Z.Zhang,Well-posedness for the viscous shallow water equations in critical spaces,SIAM.J.Math.Anal.Vol.40(2008) 443-474.
[25]Q.Chen,C.Miao,Z.Zhang,Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities,ArXiv:math.AP/0811.4215v1,2008.
[26]B.Cheng,E.Tadmor,Long time existence of smooth solutions.for the rapidly rotating shallow-water and Euler equations,ArXiv:math.AP/OTO6.0758v1,2007.
[27]P.Constantin,D.Cordoba and J.Wu,On the critical dissipative quasi-geostrophic equation,Dedicated to Professors Ciprian Foias and Roger Temam(Bloomington,IN,2000).Indiana Univ.Math.J.50(2001),97-107.
[28]A.Cordoba and D.Cordoba,A maximum principle applied to quasi-geostrophic equations,Comm.Math.Phys.,249(2004),511-528.
[29]R.Danchin,Global existence in critical spaces for compressible Navier-Stokes equations,Invent.Math.141,579-614(2000).
[30]R.Danchin,Global existence in critical spaces for flows of compressible viscous and heat-conductive gases,Arch.Rational Mech.Anal.,160(2001),1-39.
[31]R.Danchin,Density-dependent incompressible viscous fluids in critical spaces,Proc.Roy.Soc.Edinburgh Sect.A,133(2003),1311-1334.
[32]R.Danchin,On the uniqueness in critical spaces for compressible Navier-Stokes equations,Nonlinear Differential Equations Appl.,12(2005),111-128.
[33]R.Danchin,Fourier analysis methods for POE's,preprint(2005),Lab.d'Analyse et de Math.Appliquees,UMR 8050,http://perso-math.univmlv.fr/users/danchin.raphael.
[34]R.Danchin,Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients,Rev.Mat.Iberoamericana 21(3)(2005)863-888.
[35]R.Danchin,Uniform Estimates for Transport-Diffusion Equations,Journal of Hyperbolic Differential Equations,vol.4,No.1(2007)1-17.
[36]R.Danchin,Well-Posedness in Critical Spaces for Barotropic Viscous Fluids with Truly Not Constant Density,Communications in Partial Differential Equations,32:9,1373-1397(2007).
[37]R.Danchin,B.Desjardins,Existence of solutions for compressible fluid models of Korteweg type,Annales de I'IHP,Analyse non lineaire 18,97-133(2001).
[38]H.Dong,D.Du,Global well-posedness and a decay estimate.for the critical dissipative quasi-geostrophic equation in the whole space,Discrete Contin.Dyn.Syst.21(2008),no.4,1095-1101.
[39]H.Dong,D.Du,D.Li,Finite time sigularities and global well-posedness for fractal Burgers equation,Indiana Univ.Math.J.58(2009).
[40]C.Escudero,Chemotactic collapse and mesenchymal morphogenesis,Phys.Rev.E 72(2005),022903.
[41]C.Escudero,The fractional Keller-Segel model,Nonlinearity 19(2006),2909-2918.
[42]H.Fujita,T.Kato,On the nonstationnary Navier-Stokes system,rend.Sem.Mat.Univ.Padova,32(1962),243-260.
[43]Z.Guo,Q.Jiu,Z.Xin,Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients,SIAM J.Math.Anal.39,1402-1427(2008).
[44]C.Hao,L.Hsiao,H.Li,Cauchy problem for viscous rotating shallow water equations,arXiv:0806.4504v3.
[45]B.Haspot,Cauchy problem for viscous shallow water equations with a term of capillarity,ArXiv:math.AP/0803.1939v1,2008.
[46]T.Hmidi,Regularite h(o|¨)lderienne des poches de tourbillon visqueuses,J.Math.pures Appl.(9) 84(11)(2005) 1455-1495.
[47]T.Hmidi,S.Keraani,Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces,Advances in Mathematics,Volume 214,Issue 2,1 October 2007,Pages 618-638.
[48]T.Hmidi,S.Keraani,Inviscid limit for the two-dimensional Navier-Stokes system in a critical Besov space,Asymptotic Analysis,53(3) 2007,125-138.
[49]T.Hmidi,S.Keraani,Incompressible viscous flows in borderline Besov spaces,Arch.Ration.Mech.Anal.189(2008),no.2,283-300.
[50]T.Hmidi,S.Keraani,Existence globale pour le systeme d'Euler incompressible 2-D dans B_(∞,1)~1,C.R.Math.Acad.Sci.Paris 341(2005),no.11,655-658.
[51]G.Karch,Scaling in nonlinear parabolic equations,J.Math.Anal.Appl.234(1999),534-558.
[52]G.Karch,Scaling in nonlinear parabolic equations:Applications to Debye system,in:Proceedings of Conference on Disordered and Complex Systems,London,10-14 July,2000,American Institute of Physics Publishing 553,2001.[53]G.Karch,C.Miao,X.Xu,On convergence of solutions of fractal Burgers equation toward rarefaction waves,SIAM J.Math.Anal.39(2007)1536-1549.
[54]T.Kato,G.Ponce,Commutator estimates and the Euler and Navier-Stokes equations,Communications on Pure and Applied Mathematics,41(1988),891-907.
[55]A.Kiselev,F.Nazarov,R.Shterenberg,Blow up and regularity for fractal Burgers equation,Dyn.Partial Differ.Equ.5(2008),no.3,211-240.
[56]A.Kiselev,F.Nazarov and A.Volberg,Global well-posedness for the critical 2D dissipative quasi-geostrophic equation,Inventiones Math.167(2007) 445-453.
[57]P.E.Kloeden,Global existence of classical solutions in the dissipative shallow water equations,SIAM J.Math.Anal.16,301-315(1985).
[58]P.G.Lemarie-Rieusset,Recent development in the Navier-Stokes problem,Chapman Hall/CRC,2002.
[59]J.Leray,Sur le mouvement d'un liquide visqueux remplissant l'espace,Acta mathematica,63(1934),193-248.
[60]H.Li,J.Li,Z.Xin,Vanishing of vacuum states and blow-up phenomena for the compressible Navier-Stokes equations,Comm.Math.Phys.281,401-444(2008).
[61]A.Majda,Vorticity and the mathematical theory of an incompressible fluid flow,Communications on Pure and Applied Mathematics,38(1986),187-220.
[62]F.Marche,Derivation of a new two-dimensional viscous shallow water model with varying topography,bottom friction and capillary effects,European J.Mech.B/Fluids,26,49-63(2007).
[63]A.Mellet,A.Vasseur,On the barotropic compressible Navier-Stokes equations,Comm.Partial Diff.Eqns.32(1-3),431-452(2007).
[64]Y.Meyer,Wavelets,paraproducts and Navier-Stokes equations,Current developments in Math.,International Press,(1996),105-212.
[65]C.Miao,Time-space estimates of solutions to general semilinear parabolic equations,Tokyo J.Math.24(2001),245-276.
[66]C.Miao,Harmonic Analysis and Applications to Partial Differential Equations (Second Edition),Monographs on Modern pure mathematics,No.89.Science Press,Beijing,2004.
[67]C.Miao,Harmonic Analysis Method of Partial Differential Equations,Monographs on Modern pure mathematics,No.117.Science Press,Beijing,2008.
[68]C.Miao,Y.Gu,Space-time estimates for parabolic type operator and application to nonlinear parabolic equations,J.Partial Differential Equations 11(1998),301-312.
[69]C.Miao,B.Yuan,Solutions to some nonlinear parabolic equations in pseudomeasure spaces,Math.Nachr.280(2007),171-186.
[70]C.Miao,B.Yuan and B.Zhang,Well-posedness of the Cauchy problem for the fractional power dissipative equations,Nonlinear Analysis 68(2008) 461-484.
[71]C.Miao,B.Zhang,The Cauchy problem for semilinear parabolic equations in Besov spaces,Houston J.Math.30(2004),829-878.
[72]Y.Naito,Asymptotically self-similar solutions for the parabolic systems modelling chemotaxis,149-160,in:Self-Similar Solutions in Nonlinear PDE,Banach Center Publications 74,Polish Acad.Sci.,Warsaw,2006.
[73]Y.Naito,T.Suzuki,K.Yoshida,Self-similar solutions to a parabolic system modeling chemotaxis,J.Differential Equations 184(2002),386-421.
[74]L.Nirenberg,On elliptic partial differential equations,Ann.Scuola Norm.Sup.Pisa,13(1959),115-162.
[75]A.Raczynski,Stability property of the two-dimensional Keller-Segel model,Asymptot.Anal.61(2009),no.1,35-59.
[76]T.Runst,W.Sickel,Sobolev Spaces of Fractional Order,Nemytskij Operators,and Nonlinear Partial Differential Equations,de Gruyter Series in Nonlinear Analysis and Applications,vol.3,Walter de Gruyter & Co.,Berlin,1996.
[77]E.M.Stein,Singular Integrals and Differentiability Properties of Functions,Princeton University Press,Princeton,New Jersey,1970.
[78]L.Sundbye,Global existence for Dirichlet problem for the viscous shallow water equations,J.Math.Anal.Appl.202,236-258(1996).
[79]L.Sundbye,Global existence for the Cauchy problem for the viscous shallow water equations,Rocky Mountain J.Math.28,1135-1152(1998).
[80]H.Triebel,Theory of Function Spaces,Monographs in Mathematics,Vol.78.Basel-Boston -Stuttgart:Birkhuser Verlag,DM 90.00,1983.
[81]M.Vishik,Hydrodynamics in Besove Spaces,Arch.Rational Mech.Anal 145(1998),197-214.
[82]W.Wang,C.Xu,The Cauchy problem for viscous shallow water equations,Rev.Mat.Iberoamericana 21,1-24(2005).
[83]G.Wu,J.Yuan,Well-posedness of the Cauchy problem for the fractional power dissipative equation in critical Besov spaces,J.Math.Anal.Appl.340(2008)1326-1335.
[84]J.Wu,The generalized incompressible Navier-Stokes equations in Besov spaces,Dyn.Partial Differ.Equ.1(2004),no.4,381-400.
[85]J.Wu,The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation,Nonlinearity 18,139-154(2005).
[86]J.Wu,Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces,Comm.Math.Phys.263(2006),no.3,803-831.