The Well-Posedness And Time Decay Estimates For Several Types Of Fluid Models

The compressible non-isentropic Navier–Stokes–Poisson(NSP)system is one of main model in the research fields of the fluid dynamics equations.The object of this paper is devoted to using the Littlewood-Paley theory and energy estimation in harmonic analysis to study the well-posedness and the decay rates of the Cauchy problem for the multi-dimensional compressible non-isentropic NSP equations.The main contents are as follows:In chapter 1,we mainly introduces the background of the research and significance of the compressible non-isentropic NSP equations,the current research status at home and abroad,and the main content of this paper.In chapter 2,the preliminary knowledge is introduced.The main contents include related basic knowledge,functional spaces,several common inequalities,the localization theory of functional spaces and important lemmas.In chapter 3,we study the global well-posedness of strong solutions in the sense of small initial values of the multidimensional compressible non-isentropic NSP equations in the critical Besov space under~2L framework.In mixed hyperbolic-parabolic linear system,we exploited the smoothing effects of velocity in all spaces,the smoothing effect of the density in the low frequency regime and damping effect in high frequency regime,respectively.By employing the method of continuous,we finally obtain the global existence of strong solutions.In chapter 4,the global well-posedness and the time-decay rates of the multidimensional compressible non-isentropic NSP equations in the critical Besov spaces in frameworkL~p.Our main method is based on Fourier frequency localization and Bony's paraproduct decomposition.In particular,our results allow the model to tolerate high oscillation initial velocity fields in the L~p framework.In chapter 5,the main results of the full paper are summarized and we present the relevant questions and research direction in the future.