Well-posedness For Two Kinds Of Nonlinear Evolution Equations

In this paper,we mainly study two kinds of nonlinear evolution equations:the nonlin-ear Schrodinger equation and magnetohydrodynamic(MHD)equations.The Schrodinger equation describes the law that the state of microscopic particles changes with time.The magnetohydrodynamic equations are used to describe the interaction and influence be-tween the electrically conducting fluids(such as plasma)and electromagnetic field.This paper mainly is divided into four parts.In the first part,as the introduction,we mainly introduce the background and research progress of the Schrodinger equation and Magnetohydrodynamic equations.In the second part,we mainly study the derivative nonlinear Schrodinger equation under the case of no auxiliary space,proving the uncondi-tional uniqueness for the derivative nonlinear Schrodinger equation in the low regularity spaces below H1(R).In the third part,we first study large time existence of regular solutions for the compressible magnetohydrodynamic equation in T2 when initial data strictly away from vacuum:Then we show that the compressible magnetohydrodynamic equations converge to those of the incompressible magnetohydrodynamic equations when the volume viscosity tends to infinity,and we obtain the convergence rate of solution for the compressible magnetohydrodynamic equations.In the fourth part,we mainly study the global well-posedness of the incompressible magnetohydrodynamic equation in T2 in which the initial density does't have any regularity and the initial value does't satisfy any compatibility condition under allowance of the vacuum.In Chapter 2,we investigate the Cauchy problem for the following derivative non-linear Schrodinger equation and prove the uncondition uniqueness of derivative nonlinear Schrodinger equation(0-5)in C([0,T];HS(R))for any s ?(2/3,1].That is,without adding any auxiliary space(such as Bourgain space),the solution of derivative nonlinear Schrodinger equation is uniqueness and existence in low regularity space C(0,T];Hs(R)).Firstly,we apply the gauge trans-formation for derivative nonlinear Schrodinger to obtain better nonlinearity;Secondly,according to norms argument,resonance decomposition and Bourgain space argument,we can prove the initial regularity and improve gradually the regularity,then we can achieve the highest regularity by using the iteration method;Finally,the uniqueness of the solution can be obtained by using the contraction mapping principle.In Chapter 3,we mainly study the following compressible magnetohydrodynamic equationsFirstly,when the initial data strictly away from vacuum,for any regular initial data in T2 and any fixed time T>0.there exists v0 such that for any v?v0,we prove the large time existence of the regular solution.More precisely,the first step is to establish the L2 and H1 estimations for the compressible magnetohydrodynamic equations;The second step is to give the hypothesis condition Aq(T)?c0(2 ?q),we obtain the constraint estimation of ||??||L?(0,T;L2)by using the estimate with loss of integrability for solutions of the transport equation,the embedding theorem,and the maximal regularity estimation for parabolic system;The third step is to prove the reasonableness of the hypothesis condition by using the maximal regularity estimation for parabolic system and the bootstrap argument,then we obtain the large time existence.Secondly,the projection operator is used to derive the incompressible form from(0-6)when the volume viscosity v tends to infinity,and then we show by using the energy estimate that the compressible magnetohydrodynamic equations(0-6)tends to the following incompressible magnetohydrodynamic equations and the convergence rate between the solution of compressible and incompressible equa-tions is proportional to the volume viscosity v.In Chapter 4,we mainly study the following incompressible magnetohydrodynamic equationswhen the density p.velocity field u and magnetic field b satisfy the following cases 0????*,M:=?T2 ?0dx>0,u0,b0 ? H1(T2),and the density has neither any regularity nor positive lower bound,we prove that the solution for the inhomogeneous incompressible magnetohydrodynamic equations(0-8)is globally well-posed.More precisely,once the initial data does not satisfy any compati-bility conditions,the material derivative u=ut+u·?u should be introduced to obtain the boundedness of ||u||L2.On the other hand,the uniqueness of solution is hard to obtain under the Eulerian coordinates system due to the lack of regularity of the density.Therefore,to overcome the aforementioned difficulty,the Lagrangian approach is utilized to obtain the uniqueness of the solution.