On The Analysis Of The Gevrey Class Regularity For The Solutions Of Fluid Dynamic Equations Of Incompressible Flow

It is well known that Sobolev space is the most suitable function space for study-ing fluid dynamic equations,because the definition of energy in Sobolev space is very simple.However,there are not many satisfactory results for many basic problems of fluid dynamic equations in the framework of Sobolev space,for example the Prandtl boundary layer is not well-posed in Sobolev space in many cases.On the other hand,these equations are locally solvable in analytic function space by use of the Cauchy-Kovalevskaya theorem.The analytic function space does not contain the functions with compact support,thus it is not an appropriate function space to study the fluid dynamic equations.It is then natural to consider the Gevrey space as an intermediate function space of the Sobolev space and analytic function space.This thesis is a study on the Gevrey regularity of solutions to several homogeneous incompressible fluid dynamic equations,precisely,the incompressible Navier-Stokes equation,the incompressible Eu-ler equation,the ideal incompressible Magnetohydrodynamic(MHD)equations,and the incompressible Boussinesq equations.Among these physical models,the incompressible Navier-Stokes equation is the most basic model and the difference between the incom-pressible Navier-Stokes equations and the incompressible Euler equations is the viscous term and the boundary condition.The main difference between the ideal incompress-ible MHD equations and the incompressible Euler equations is the coupled magnetic equations,and this will enhance the nonlinearity of the ideal incompressible MHD equa-tions.The incompressible Boussinesq equations can be viewed as a symplified model to understand some key features of the incompressible Navier-Stokes equations,and it's difference from the incompressible Navier-Stokes equations is that the outer force term is replaced by the unknowns.Because of the inner connections of these models,we put them together to start our research.Since C.Foias and R.Temam applied the Fourier space method to study the Gevrey class regularity of the incompressible Navier-Stokes equations in their pioneer-ing work[47],this Gevrey class technique has become a standard tool to studying the analyticity regularity and estimating the analytical radius of solutions to various dissipative evolution equations,for example[17,37,46,57,97].By choosing an appro-priate raidus function,C.D.Levermore and M.Oliver extended the method to the non-dissipative generalized incompressible Euler equations in[81],and they obtained the decay estimate for the radius of analyticity for the solutions to the incompressible Euler equations.Later,I.Kukavica and V.Vicol improved the work of C.D.Levermore and M.Oliver in[78],and they obtained the lower bound on the radius of space an-alyticity which depend exponentially on the infinity norm of the gradient.Especially,I.Kukavica and V.Vicol also studied the analyticity and Gevrey class regularity of solutions to the incompressible Euler equation on the half space in[79],they introduced a new method to study the Gevrey class regularity for solutions to the incompressible Euler equations when considered in domain with boundary.This thesis is motivated by the above mentioned works.This thesis is made up of six chapters.In the first chapter,as an introduction,we will introduce the backgroud and the recent progress on this topic.In the second chapter,we will present in detail the definition and properties of Gevrey class.Meanwhile we will introduce some main known results on the topic and the main innovation of our work.In the third chapter,we study the vanishing vicosity limit of solutions to the in-compressible Navier-Stokes equation in Gevrey class space.We prove that the solutions to the Navier-Stokes equations converge strongly to the solutions of the incompressible Euler equation in Gevrey class space on the torus,which is a preliminary step for the study of the boundary layer theory in Gevrey space.In the fourth chapter,we will study the propagation of weighted Gevrey class regularity for the incompressible Euler equations.This is motivated by the no-slip Prandtl boundary layer problem.Since we work on the half plane,the Fourier space method can not apply to this case and we will use the Sobolev-Gevrey class approach which is introduce by I.Kukavica and V.Vicol[79].Due to the appearence of weight function,the estimate of the nonlinear pressure term is much more difficult and this is the mian innovation of our work.In the fifth chapter,we study the propagation of Gevrey class regularity of solutions to the three dimensional ideal incompressible MHD equations and give the lower bound for the radius of Gevrey class for the solutions.Although this work is similar to the classical work of incompressible Euler equation,the structure of the equation is much more complicate and the computation is much more complicate.Since the Gevrey space theory is suitable for the study of the boundary layer of MHD equations,this work is a preliminary step for the study of the boundary layer of MHD equations.In the last chapter,we study the analytic smoothing effect of the solutions to the incompressible Boussinesq equations.Although this result is similar to the classical results on the incompressible Navier-Stokes equations,the stucture of the equations is different and the construction of the approximate solutions is more complcate.This may provide fundamental theoretical basis for the research of the corresponding boundary layer theory in the framework of Gevrey space.