Regularity Of Solution For Incompressible Navier-Stokes Equations And MGD Equations

As a macroscopic model describing the movement of matter,the hydrody-namic equation model is a very important group of nonlinear partial differential e-quations that we know and understand natural phenomena.It has been occupying the core research field of mathematical physics.Among them,the Navier-Stokes equations are named after Claude-Lions-Navier and George-Gabriel-Stokes and are the basic equations for describing the viscous fluid.In addition,the magneto-hydrodynamics equations?MHD equations?describe the movement of conductive fluids in the electromagnetic field and have important physical applications in the fields of astrophysics,geophysics,aerodynamics and cosmic plasma physics back-ground.In this paper,we discuss the mathematical problems of these two kinds of equations.By using the classical energy method,the fixed point theorem of con-tractive mapping,Plancherel theorem,Fourier transformation,Littlewood-Paley paraproduct decomposition technique and Sobolev imbedding theorem,and some important inequalities,For example,arithmetic geometric mean value inequali-ty,Cauchy-Schwarz inequality,H¨older inequality,Gagliardo-Nirenberg inequality,Sobolev interpolation inequality,as well as Gronwall inequality,we study the glob-al regularity of the incompressible Navier-Stokes equations and the existence and uniqueness of local C^1?solution for the ideal incompressible MHD equations.This paper is divided into six parts,as follows:In Chapter 1,we introduce the research background and research progress of Navier-Stokes equations and MHD equations.Moreover,the research model,preliminary knowledge,research contents and main results are given in this paper.In Chapter 2,we study the global well-posedness of a three dimensional in-compressible Navier-Stokes equations in the case of large initial values.The new model is different from the original Navier-Stokes equation,we adjust the convec-tion term u·?u in the original Navier-Stokes equation to be about(D^-12 u)·?u,where D is a Fourier multiplier whose symbol is m???=|?|.Firstly,some related research results are reviewed,and the definitions,properties and important lem-mas are given.Secondly,the local well-posedness of the model is proved.Finally,by means of the energy estimation method and the related theory of Sobolev s-pace,we proved that the Navier-Stokes model is well-posed when any initial value u₀belongs to Sobolev space L²?R³?.In Chapter 3,we study the global regularity of a three-dimensional logarithmic sub-dissipative Navier-Stokes model.This system takes the form of?_tu+(D^-1/2u)·?u+?p=-A²u,where D and A are Fourier multipliers defined by D=|?|and A=|?|ln^-1/4?e+?ln?e+|?|??with??0.The symbols of the D and A are m???=|?|and h???=|?|/g???respectively,where g???=ln^1/4?e+?ln?e+|?|??,??0.It is clear that for the Navier-Stokes equations,global regularity is true under the assumption that h???=|?|^?for??5/4.Here by adjusting the advection term we greatly weaken the dissipation term to h???=|?|/g???.We prove the global well-posedness for any smooth initial data in H^s?R³?,s?3 by using the energy method.In Chapter 4,we consider the global existence of the two-dimensional Navier-Stokes flow in the exterior of a moving or rotating obstacle.Bogovski?i operator on a subset of R²is used in this paper.One important thing is to show that the solution of the equations do not blow up in finite time in the sense of some L²norm.We also obtain the global existence for the 2D Navier-Stokes equations with linearly growing initial velocity.In Chapter 5,we consider a model of inhomogeneous three-dimensional Navier-Stokes equations.By using the energy method,Littlewood-Paley para-product decomposition techniques and Sobolev embedding theorem,we study the global regularity of solutions.The dissipative term?u in the classical inhomo-geneous Navier-Stokes equations is replaced by-D²u and a new inhomogeneous Navier-Stokes equations model is obtained,where D is a Fourier multiplier whose symbol is m???=|?|^5/4.The blow-up criterion and the global regularity of the solution for the model are obtained about the initial data??₀,u₀??H^3/2+?×H^?,where?and?are arbitrary small positive constants.In Chapter 6,we mainly study the ideal incompressible MHD equations mod-el.For two-dimensional and three-dimensional ideal incompressible magnetohy-drodynamic models,we prove that the local existence and uniqueness of classical solutions for the MHD system in H¨older space when the general initial data belongs to C^1,??Rⁿ?.