Study On Some Free Boundary Problems With Vacuum

Fluid dynamic equations are one of main research fields of partial differential equations.The Navier-Stokes equation is the main model in the field of fluid dynamic equations.Meanwhile,other models related to Navier-Stokes equations are also the focous of research.Thus,the object of this paper is devoted to the study on local and global well posedness of some fluid equations with the vacuum free boundary.The main contents are as follows:In chapter 1,we mainly introduce the background,significance,the research status all around the world and related research problems of the research on the vacuum free boundary problems.In chapter 2,the preliminary knowledge is introduced.The main contents include related basic knowledge,functional spaces,several common inequalities,the related knowledge of Lagrangian transformation and description of related symbol.In chapter 3,we study the global existence and uniqueness of smooth solution for a onedimensional compressible viscous two-phase flow model with a vacuum free boundary,including the physical vacuum condition((27)(27)31 ?).The method is that we need to obtain the uniform bounds of the solution energy by using energy estimates,Hardy inequality and Ellipse estimation after the Lagrangian coordinate transformation of the equation.In chapter 4,we study the local existence and uniqueness of strong solution for threedimensional non-isentropic compressible Navier-Stokes-Poisson equations with a vacuum free boundary.The method is that we need to introduce the energy functional in Lagrangian coordinates,and obtain the uniform estimates of velocity and temperature by energy method,Hardy inequality and Ellipse estimation.Finally,the existence and uniqueness of solution are obtained by constructing approximation equations,and using Galerkin method and fixed point argument.In chapter 5,the main results of the full paper are summarized and we present the relevant questions and future research direction.