Large-time Behavior Of The 3D Compressible MHD Equations And The Nematic Liquid Crystal Equations

The fluid dynamics equations describe the motion of the fluid and reflect the laws and properties of fluid motion.As two important types of nonlinear partial differential equations,the magnetohydrodynamic（MHD）equations and the liquid crystal equations have extensive application values in production,life and scientific research.The study of their definite solutions problem has always been one of the hot issues of scientific research for the scholars at home and abroad.In this paper,we mainly focus on the large time behavior of the solutions to the Cauchy problem for the following two classes equations:First of all,we investigate the optimal large–time behavior for higher–order spa-tial derivatives of strong solutions to the 3D full compressible MHD equations.More precisely,we employ delicate energy estimates and low–frequency and high–frequency decomposition to show that the third–order spatial derivatives of density,velocity and absolute temperature converge to their corresponding equilibrium states at the L²–rate （1+t）^{-3/4（2/p-1）-3/2}（1≤p<6/5）.It is worth mentioning that our innovation is to obtain the optimal decay rate of the third-order spatial derivative of the solutions,which is the same as that of the heat equation,and particularly improves the L²–rate （1+t）^{-3/4（2/p-1）-1/2}and （1+t）^{-3/4（2/p-1）-1} in the previous related work.Secondly,we establish the optimal time–decay rates of the 3D compressible nemat-ic liquid crystal equations.The global existence of low–energy weak solutions for the3D compressible nematic liquid crystal flows with discontinuous initial data and large oscillations has been proved by Wu–Tan under the assumptions that the initial energy is small and the initial density has positive lower and upper bounds.However,up to now,the time–decay rates of these solutions have remained an open problem.In order to overcome the low regularity of these solutions,especially the difficulty caused by the lack of regularity of density,we prove the time–decay rates of the solutions in L^r–norm with 2≤r≤∞ via the energy estimates and low–frequency and high–frequency decomposi-tion.Moreover,if additionally the initial data satisfies some low–frequency assumptions,the optimal lower bound decay rates of solutions are also obtained.Therefore,our decay rates are optimal in this sense.