Researches On The Global Well-posedness For Some Classes Of Nonlinear Hyperbolic-Parabolic Coupled Evolutionary Equations

In this dissertation, we mainly study the global well-posedness of solutions to some classes of nonlinear hyperbolic-parabolic coupled evolutionary equations. We have obtained some results with the theoretical value. This dissertation is divided into the following six chapters.Chapter1is the preface, in which we mainly introduce the relative research back-ground, present research situation, the fundamental sense of the global well-posedness and some useful lemmas.In Chapter2, we mainly consider the regularity of solutions to nonlinear com-pressible ideal magnetohydrodynamics equations with large initial data. By the em-bedding theorem and a series of delicate interpolation inequalities, we establish the global existence of solutions in Hi(i=2,4).In Chapter3, we mainly study the global well-posedness of solutions to nonlinear compressible radiative magnetohydrodynamics equations. By the energy method and some priori estimates, we overcome the difficulties caused by thermoradiation and es-tablish the global existence and exponential stability of solutions for this system. The difficulties encountered are:(1) how to get the uniform boundedness of specific vol-ume;(2) how to overcome the difficulties caused by thermoradiation. The novelty in this chapter is:we introduce the appropriate parameters to give a proper expression of the specific volume. By a series of delicate priori estimates, we overcome the difficul-ties caused by thermoradiation of high temperature and establish the positively lower bound and upper bound of the specific volume.In Chapter4, we establish the global existence and exponential stability of solu-tions to a class of nonlinear non-Newtonian fluid equations. The difficulties encoun-tered include:(1) how to overcome the difficulty that the heat conductivity coefficients depend nonlinearly on the gradient of temperature when proving the uniform esti-mates of the specific volume of the system;(2) how to derive the appropriate priori estimates. The novelty in this chapter is:by the p'th binomial inequality, we divide the thermal conductivity into two parts so that to overcome the difficulty caused by that the thermal conductivity depends nonlinearly on the gradient of temperature, and fi-nally obtain the global well-posedness of solutions.In Chapter5, we consider a nonlinear thermoelastic system with second sound. By the multiplier techniques and energy method, we obtain the delicate energy functional and Lyapunov functional, and prove the global existence and exponential stability of solutions to this system.In Chapter6, we summarize our work in this dissertation and prospect the future research.