Fourier Localization Method For Fluid Dynamics Equations

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+(-â–³u)^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+(-â–³)^α/2u=▽·(uâ–½(â–³^-1u)),generalizing the Keller-Segel model of chemotaxis,for the initial data uo in critical Besov spaces(?)_2,r^1-α(R²) 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.