The Well-posed Ness Of Some Fluid Dynamics Equations And The Density Patch Problem

In this dissertation,we study the global regularity of the density patch for the inhomogeneous incompressible Navier-Stokes equations and the well-posed problem of two kinds of fluid equations.The full text is divided into five chapters,as follows:The first chapter is the introduction.The physical background,the defini-tion of correlation function space,the main results and innovations of this paper are reviewed.In Chapter 2,the Cauchy problem of the incompressible Navier-Stokes-Fokker-Planck equation is studied.By using the Garlakin approximation method to construct the approximate solution of the system,and using the Littlewood-Paley theory,the local well-posedness of the system in Hilbert space is proved by using the continuity method.Secondly,we prove the existence of solutions on some subcritical LP-type spaces.The method of maximum regularity estimation for heat operators and the continuity argument are mainly used.In Chapter 3,the local existence and uniqueness of the solutions of the non-dissipative homogeneous incompressible MHD equation under the energy framework Hs-1(Rd)× Hs(Rd),s>d/2 is proved by a useful commutator estimate given by us.In Chapter 4,we study the global regularity of density patches for the three-dimensional inhomogeneous incompressible Navier-Stokes equations on Euclidean space.Under the assumptions that the initial density is taken as the density patch,and the patch boundary is Ck,γ(k＝1,2),and the velocity field satis-fies certain regularity and smallness,we show that the boundary of the density patch remains Ck,γ(k=1,2)regularity with the evolution of the time.The main methods used are the time-weighted energy estimation,Stokes estimation,the properties of singular integrals,the disappearance of even kernel of singular integrals on the hemisphere,and the Chemin’s striated regularity technique.In Chapter 5,the Cauchy problem of two-dimensional incompressible MHD equations with discontinuous initial density is investigated.On the one hand,when(ρ0,u0,b0)∈ L∞(R2)×Hs(R2)× Hs(R2)and p0 is away from vacuum,the global well-posedness is established.In particular,the uniqueness of the solution is proved by using Lagrangian method.On the other hand,we consider the global regularity of density patch.The proof mainly depends on careful time-weighted energy estimation,Stokes estimation,singular integral operator,linear interpolation and Lagrangian method.