Study On The Behavior Of Solutions Of Turbulent And Radiative Flow Equations

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 thesis is devoted to the study on global existence and time-decay rates of turbulence flow equations and radiation hydrodynamics model.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 radiation hydrodynamics model and turbulence flow equations.In chapter 2,the preliminary knowledge is introduced.The main contents include related basic knowledge,functional spaces,several common inequalities and the theory of Littlewood-Paley and important lemmas.In chapter 3,we investigate the time-decay rates in critical Besov space for a P1-approximation radiation hydrodynamics model.Under a suitable additional condition involving only the low frequencies of the data,we exhibit the time decay rates for the global solutions.The proof relies on an application of Fourier analysis to a mixed parabolic-hyperbolic system,and on a refined time-weighted energy functional.In chapter 4,we study the global existence and the optimal decay rates in the H~2-framework for a turbulence flow equations.We obtain the global existence of smooth solutions to the system under the condition that the initial data in lower regular spaces are close to the constant equilibrium state.Moreover,the optimal decay rates is based on the spectral analysis and the smooth effect of dissipative operator.In Chapter 5,the main results of the full thesis are summarized and we present the relevant questions and future research direction.