Regularity Of Solutions For 2D Compressible Navier-Stokes Equations And 1D MHD Equations

The isentropic compressible Navier-Stokes equations describes the motion law of viscous compressible fluids,while the magnetohydrodynamics(MHD)equations gives the interaction law between magnetic field and conductive fluid(plasma,liquid metal,etc.).The mathematical research of these two physical models has important theoretical and practical significance.We mainly study the global existence and regularity of the solutions of these two kinds of hydrodynamics equations.The main contents of this paper are as follows:In Chapter 1,we introduce the research background,research models,research situation at home and abroad and the main results of this paper.In Chapter 2,we study the isentropic compressible Navier-Stokes equations in a 2D bounded domain.Under the assumptions that the initial energy is suitably small,the initial density allows large oscillation and can contain vacuum,we prove the existence and uniqueness of the global classical solutions of the Navier-Stokes equations with slip boundary condition in a bounded domain.We arrange the contents of this chapter as follows:in the first section,we introduce some general notations,known conclusions and lemmas to be used;in the second section,we give our main results;in the third section,we give some a priori estimates of smooth solutions,including lower-order estimates and higher-order estimates.The purpose of these estimates is to extend the local classical solution to be a global one when the initial energy is suitably small.In particular,in order to obtain the necessary estimates of boundary integral terms,we make full use of the slip boundary condition and the fact that u is a tangent vector on the boundary.The estimation of boundary integral terms is one of the main difficulties in our paper.Finally,in the fourth section,we prove the main results of the second section by using the estimates we have proved.In Chapter 3,we study the MHD equations and obtain the global strong solutions to the MHD equations with density-dependent viscosity and degenerate heat-conductivity in unbounded domains.The arrangement of this chapter is as follows:in the first section,we give the main results in this chapter;in the second section,we get some a priori estimates of the solutions.Compared with the previous work on bounded domains,the research method on unbounded domains is very different.One of the main difficulties is to give the lower bound of specific volume.In the third section,we give the proof of the main theorem in this chapter.In Chapter 4,we give a summary of our work and some topics which we plan to study in the future.