The Existence Of Solutions For The Equations Of Compressible Viscous Fluids

This dissertation is mainly concerned with the global existence and decay of solutions for compressible viscous and heat conduction equations in the whole space.It is divided into four chapters.In the first chapter,the background and significance on the problem of viscous heat-conductive flows are introduced.The current situation and development trend of the system of compressible viscous heat-conductive flows studies in this paper are also introduced and main results of the existence and decay for the model are then presented in this chapter.The second chapter is mainly about basic knowledge.The important inequalities for calculating the decay rate and homogenization principle for the formal solutions of the model that need to be used are given.In the third chapter,a priori estimates and local existence are given,the local solutions are found under general condition and the global solution can be found by a combination of a local existence theorem and some a priori estimates.In the fourth chapter,through the Fourier transform on the linear part of the system,the problem is translated into a problem similar to the initial value problem of the heat conduction equations,and get the formal solution to the problem.The classical energy method,the linear spectral theory,Young inequalities,H¨older inequalities,Cauchy-Schwartz inequalities and the integration by parts are then applied to get the global solution of the equations of motion of compressible viscous and heat conductive fluids and decay rate for the model is obtained by estimating the solution.There have been a lot of important works on the models of compressible viscous and heatconductive flows.In this paper,the Cauchy problem for the equations of motion of compressible viscous and heat conduction fluid is studied under some restricted conditions.To a certain extent,the research results in the relevant literature are studied in the general situation and the existence and decay rate of solutions to the corresponding problems are obtained.