On The Wellposedness And Periodic Solution Of Some Kind Of Hydrodynamic Equations

In this paper,we mainly investigate the wellposedness and periodic solution problem of some kind of hydrodynamic equations.In the first chapter,we outline the research status with respect to the wellposedness and periodic solution problem of the hydrodynamic equations and summarize the contents and sig-nificance of this paper.In the second chapter,we collect some preliminaries related to our research contents.In the chapter 3,based on the high-low frequency decomposition method,after applying the Fourier transformation to the low frequency part of and using weighted energy method to the high frequency part of the unknown of the liquid crystal equation,we establish the existence of the periodic solution to this equation provided the external forces are sufficiently small in weighted Sobolev spaces.In chapter 4,we discuss the wellposedness of the incompressible liquid crystal equation and establish the uniquely existence of its global mild solution when the initial data are suffi-ciently small in some Besov-Morrey spaces.The large time behavior of the global mild solution and the self-similar solution are also established.In chapter 5,we first establish the maximal Lorentz regularity of the Stokes equation by the real interpolation method,then prove the uniquely existence of the local strong solution to the Navier-Stokes equation with external forces satisfying some kind of conditions in certain Besov-Morrey spaces.The local strong solution can be extended to be global when the initial data and external forces have sufficiently small Besov-Morrey norms.In chapter 6,we consider some special regularity problems and the preservation of decay of the solution to the Benjamin equation and establish the propagation of the regularity theorem.When the initial data decay polynomially,the solution of the Benjamin equation will preserve the same decay rate.Finally,we summarize the research contents and methods of this paper and prospect to certain extent for the future work.