Global Well-Posedness Of The Non-isentropic Euler Equations And Ferrohydrodynamics Equations

This thesis is concerned with the non-isentropic Euler equations and Ferrohydrody-namics equations.The global existence and uniqueness of the solutions to the equations and the large time behavior of the solutions to the equations are discussed in terms of mathematical theory.There are two parts in this thesis.The first part consists of Chapter 3 and Chapter 4,in which we discuss the global existence and decay for the solutions of the non-isentropic Euler equations with damping and heat conduction.The large time behavior of the non-isentropic Euler equations which coupled with Poisson equation is also studied.The second part consists of Chapter 5.The global existence,uniqueness and asymptotic behavior of the solutions of the compressible Ferrohydrodynamics equa-tions are discussed when the spin magnetic moment is magnetized and diffused under the action of an external magnetic field.In Chapter 1,we briefly introduce the research background of Euler equations and Ferrohydrodynamics equations,and review the research results of related mathematical models.The research contents and results of this paper are listed briefly.In Chapter 2,we recall some symbolic representations and definitions,and review some essential conclusions and crucial lemmas,which will be used in the subsequent chapters.We show part of details of the argument of conclusions.In Chapter 3,we consider the global existence,uniqueness and decay of the smooth solutions near a constant equilibrium to the compressible non-isentropic Euler equations in R3.With the method of energy estimates,we show the existence and uniqueness of the global classical solutions.Notice that we only assume that the H3 norm of initial data is small.However,When we derive the higher-order energy estimates of speed and temperature,the higher-order derivatives of temperature arises in the process of eliminating each other among linear term.So we have to assume that the H4 norm of initial data is small when we demonstrate the decay rate.But we do not assume that the Lp norm of initial data is small.Unlike Euler-Poisson equations,we only obtained the algebraic decay rate for the Euler equations because of the lack of Poisson equation and relaxation term of temperature.In Chapter 4,The existence,uniqueness and large time behavior of the global classical solution to the compressible non-isentropic Euler-Poisson equations near a non-constant steady state in R3 are discussed.Differing from the isentropic equations,the difficuties here are caused by the temperature.Using some concise interpolation tricks and energy estimates,we relaxed the regularity of the initial temperature in the previous result.It shows the existence and uniqueness of the global classical solution.We can find that the Poisson equation restored the lower-order dissipation of the density in the process of proving.Therefore,it is proved that the solutions converges to the stationary solution exponentially fast when time tends to infinity.In Chapter 5,we consider the large time behavior of the global solution of the Cauchy problem to the compressible Ferrouydrodynamics equations in R3 under the action of an external magnetic field,and the diffusion of the spin magnetic moment.The global existence and uniqueness of the solutions is proved by assuming that the lower-order norm of the initial data and a expression of the external magnetic field are small.Because both the H-s norm and the B2,?-s norm of solutions are preserved along time evolution,we use a negative Sobolev or Besov space to discuss the large time behavior of solutions.It is proved the time decay rates of the solutions and its higher-order derivatives when the divergence of the external magnetic field equals to zero.In order to obtain our results,we only assume that the H3 norm of initial data is small,while the smallness of the higher-order derivatives and Lp norm of the initial data are not necessary.We here claim that the decay results of density and velocity are optimal in the sense that they are consistent with those in the linearized case.But the decay results of angular velocity and magnetization are not optimal.We obtained faster decay results of angular velocity and magnetization by estimating directly the angular momentum equation and magnetization equation when we assume that the higher-order norm of initial data is small.