Global Solutions To Certain Compressible Navier-Stokes Type Equations With Large Initial Data

This thesis is concerned with the global wellposedness of the Cauchy problem of certain com-pressible Navier-Stokes type equations.The models under our consideration include the compressible Navier-Stokes-Cucker-Smale-Fokker-Planck equation and one-dimensional compressible Navier-Stokes equations with temperature dependent viscosity.For the Cauchy problems of these two types of equa-tions,the corresponding global solvability results with small initial data are well established,while for the corresponding results with large initial data,fewer results are available up to now and the main purpose of this thesis is devoted to such a problem.Our main results contain the following two parts:Firstly,for the Cauchy problem of one-dimansional and two-dimensional compressible Navier-Stokes-Cucker-Smale-Fokker-Planck equations,two global solvability results are obtained for large data which may contains vacuum;Secondly,we obtain the nonlinear stability of the superpositions of rarefaction wave and viscous contact wave for the Cauchy problem of one-dimensional compressible Navier-Stokes equation with temperature dependent viscosity and large initial perturbation.This thesis is divided into the following four chapters:In the first chapter,we first introduce some former results related,state the problems we want to study and then give the results we obtained.In the second chapter,we study the Cauchy problem of the particle-fluid coupled compressible Navier-Stokes-Cucker-Smale-Fokker-Planck equations.For weak solutions to the related compressible and incompressible models,[15]deals with the global existence of weak solutions to incompressible Navier-Stokes-Vlasov-Fokker-Planck equations in two and three dimensional setting,while the global existence of weak solutions to compressible Navier-Stokes-Vlasov equations in three dimensional setting is studied in[114,117].For classical solutions with small initial data,the existence of global classical solutions to the three dimensional incompressible Euler equations coupled with Fokker-Planck equation has been obtained in[13].In[16],for compressible Navier-Stokes-Vlasov-Fokker-Planck equations,the global classical solutions near equilibrium state is constructed and the corresponding convergence rates have also been obtained.All the above results are concentrated either on the global existence of weak solutions or on the existence of classical solutions under small initial data.The problem we want to consider is:is it possible to obtain the global existence of classical solution to the Cauchy problem of the compressible Navier-Stokes-Cucker-Smale-Fokker-Planck equations with large initial data which may contain vacuum?This is the main concern of the second chapter.In the case of one and two dimensional setting,we give an affirmative answer to this question(see Theorem 2.1 and Theorem 2.2)In the third chapter,we study the nonlinear stability of basic wave patterns to the Cauchy problem of one-dimensional compressible Navier-Stokes equations with temperature dependent viscosity.For the corresponding results with constant transport coefficients and small initial perturbation,the nonlinear stability of viscous shock profiles is obtained in[69],[88]and[81],the corresponding result for viscous contact waves is studied in[54],while the time asymptotic nonlinear stability of rarefaction waves is es-tablished in[70].The arguments employed in the above mentioned manuscripts can be adopted to yield the corresponding nonlinear stability results for the Cauchy problem of one-dimensional compressible Navier-Stokes equations with temperature dependent viscosity and small initial perturbation.But for the corresponding nonlinear stability result with large initial perturbation,the only result available up to now is[55],where the nonlinear stability of the superposition of rarefaction wave and viscous contact wave is established for the case of constant transport coefficient.The problem we want to study in this chapter is:For the Cauchy problem of the one-dimensional compressible Navier-Stokes equations with temperature dependent viscosity,can we obtain the nonlinear stability of the superposition of rarefac-tion wave and viscous contact wave with large initial perturbation?In chapter 3,we give an affirmative answer to such a problem(see Theorem 3.1)In the fourth chapter,we will list some problems which are now under our current study.The first one is for the Cauchy problem of the three dimensional compressible Navier-Stokes-Cucker-Smale-Fokker-Planck equations,can we get a global solvability result similar to that of[49]for certain class of initial value which is allowed to have large oscillations and may contain vacuum?Moreover we want to study the global existence of weak solutions of the Cauchy problem to this coupled system with large initial data.The second problem is to study the nonlinear stability of certain basic wave patterns of the Cauchy problem of one-dimensional viscous radiative and reactive gas(cf.[76])and the main purpose is to obtain the result on the nonlinear stability of the superposition of rarefaction wave and viscous contact wave with large initial perturbation similar to that of[46]for one-dimensional viscous heat-conducting ideal polytropic gas with temperature dependent viscosity.