The Global Solutions And Their Properties For Camassa-Holm Equations And Ginzburg-Landau Equations

In this paper, we study the existence of global solution and its property of the initial boundary value problem for Camassa-Holm equations and Ginzburg-Landau equations. There are three sections in this paper.The first section: We consider the initial boundary value problem on half line and bounded interval, with Kato's method for abstract quasi-linear evolution equations and a prior estimates of. solution, we get the existence of global smooth solution and the Blow-up of solution in finite time under some conditions. At some time, we also research the existence of global smooth solution of the initial boundary value problem for a class of generalized Camassa-Holm equations.The second section: Under the conditions of nonlinear boundary controbility, we consider the initial boundary value problem of Camassa-Holm equations with dissipative. By using the contractive mapping fixed point theorem and a priori estimates, the existence ofglobal smooth s olution, global attractor in H² , t ime p eriodic s olution or almost-periodic solution and the global exponential stability are proved.Finally, in the third section, by constructing some functional which similar to the conservation law of evolution equation and the technical estimates, we prove that in the inviscid limit the solution of generalized derivative Ginzburg-Landau equation (GGL equation)converges to the solution of derivative nonlinear Schrodinger equation correspondently in one-dimension; The existence of global smooth solution for a class of generalized derivative Ginzburg-Landau equation are proved in two-dimension, in some special case, we prove that the solution of GGL equation converges to the weak solution of derivative nonlinear Schrodinger equation; In general case, by using some integral identitiesof solution for generalized Ginzburg-Landau equations with inhomogeneous boundary condition and the estimates for the L² normon boundary of normal derivative and H¹'norm of solution, we provethe existence of global weak solution of the inhomogeneous boundary value problem for generalized Ginzburg-Landau equations.