Large-time Behavior Of Several Types Of Hydrodynamic Equations

The fluid dynamics is an important branch of dynamics,and it aims to study the static state and motion state of fluid itself,as well as the interaction and flow law when there is relative motion between fluid and solid boundary wall.The main basis of fluid mechanics are Newton's law of motion and the law of conservation of mass,which often requires the knowledge of thermodynamics,and sometimes the basic laws of macroscopic electrodynamics and the relevant knowledge of physics.Many fluid mechanics system has established the corresponding mathematical model according to its physical background in order to better study the features of the fluid motion and the motion state.Many mathematicians and physicists have done a lot of research on the existence,uniqueness,regularity and large time behavior of solutions of various kinds of fluid mechanics equations.The large time behavior of several kinds of fluid mechanics equations studied in this paper is an important aspect of theoretical analysis in solving fluid mechanics problems.The main research objects of this paper are the following three kinds of hydrodynamic equations:compressible Navier-Stokes-Korteweg system,compressible magnetohydrodynamics(MHD)system,and two-phase flow system which coupled by compressible isothermal Euler equation and compressible isentropic Navier-Stokes equation.The contents of the paper include the following five parts:In Chapter one,we summarize the background of the three types of fluids dynamics equations and then we give some preliminary knowledge and notations in this paper.In Chapter two,we make full use of the properties of low-frequency and high-frequency decomposition of the function to study the lager time behavior of the solution to the Cauchy problem of 3D compressible Navier-Stokes-Korteweg systems.That is to say,it is proved that the decay rates of the 0-order to N-order spatial derivatives of the solution reach the optimal sense under the condition that the initial data??0+1?H4+?u0?H3 is sufficiently small.Specifically,our innovation is mainly reflected in the decay rates of the highest order(ie N+1-order)spatial derivatives of density and the decay rates of the high frequency part of the N-order spatial derivatives of density and velocity,which is(1+t)-3+2(N+1)/4.In Chapter three,we study the lager time behavior of solutions to the Cauchy problem of the 3D MHD system.It is proved that when is bounded,the decay rates of the first-order and second-order spatial derivatives of the solutions of MHD system are increased to(1+t)-5/4 through the pure energy method and Fourier splitting method,and then the decay rate of the second-order spatial derivatives of the solutions is improved to the optimal sense by using the low-frequency and high-frequency decomposition technique.Compared with previous results,our main innovation is to obtain the optimal decay rates of higher-order spatial derivatives.In Chapter four,we study the lager time behavior of solutions to the Cauchy problem of the 3D compressible two-phase fluid system.If the H3 norm of the initial data is small but its higher order Sobolev norm can be arbitrarily large,then the global existence and uniqueness of classical solutions are obtained by an ingenious energy method.Further,if the initial data norm of the H-s(0?s?3/2)or B2,?-s(0