Study On The Properties Of Several Kinds Of Fluid Equations Coupled With Navier-Stokes Equations

There are three parts in this paper.Chapter 2 is the first part of this paper,in which some basic inequalities and lemmas are introduced.The second part includes Chapter 3,Chapter 4 and Chapter 5.The well-posedness problems of finite energy weak solutions to several kinds of fluid equations are considered in the second part.Chapter 6 is the last part of this paper.In Chapter 6,we consider the uniqueness of the inverse boundary value problem for the steady incompressible magnetohydrodynamic equations.In Chapter 3,we discuss the weak-strong uniqueness of the finite energy weak solutions for Navier-Stokes-Poisson equations.By using the relative entropy inequality and Gronwall inequality,we prove the weak-strong uniqueness for NSP equations.In Chapter 4,we consider the existence and weak-strong uniqueness of finite energy weak solutions to three-dimensional compressible magnetohydrodynamic(MHD)equations with nonmonotonic pressure.Firstly,the existence of finite energy weak solutions for compressible MHD equations is proved by using the proof ideas of Lions and Feileisl,and the triple approximation method.We consider the approximation equations of MHD equations:with β>0 suitable large and ε>0,δ>0 properly small.By Faedo-Galerkin approximation method,it is proved existence of the approximate solution to this approximation equations under appropriate initial boundary conditions.Due to the addition of artificial diffusion term and artificial pressure term in the approximation equations,the density function of the approximation equations has higher integrability and batter smoothness.Then,we deduce the(uniform)prior estimates,and use these prior estimates to derive the limit of artificial diffusion and the limit of artificial pressure term.It is proved that the limit function is the finite energy weak solution of MHD equations.Finally,the weak-strong uniqueness of finite energy weak solution to MHD equations is proved by the relative entropy method.In Chapter 5,we use relative entropy method to prove weak-strong uniqueness of the finite energy weak solution to two-dimensional compressible magnetohydrodynamic equations with Coulomb force.In Chapter 6,we consider the inverse boundary value problem for the steady incompressible magnetohydrodynamic(MHD)equations in a bounded domain.It is firstly proved by operator theory that MHD equation has a special asymptotic solution.Then,by using the linearization method,the inverse boundary value problem of MHD equations is transformed into the inverse boundary value problem of two different Stokes equations.Finally,it is proved that the viscosity coefficient,and magnetic diffusion coefficient in the steady incompressible MHD equations can be uniquely determined by its Cauchy data.