Camassa-Holm方程和Ginzburg-Landau方程的整体解及其性质
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摘要
本文研究了Camassa—Holm方程和Ginzburg—Landau方程初边值问题整体解的存在性及其性质。全文共分三个部分。
第一部分：考虑了Camassa—Holm方程在半无界和有界区域上的初边值问题，用Kato关于拟线性演化方程的初值问题的理论及先验估计的方法，证明了整体解的存在性，及在一定条件下，解在有限时间内的Blow—up。同时还研究了一类广义Camassa—Holm方程初值问题整体解的存在性。
第二部分：在非线性控制边界条件之下，对于带耗散项的Camassa—Holm方程的初边值问题，用压缩映射不动点原理及先验估计方法，证明了整体光滑解的存在性、整体解的指数稳定性、H~2空间中整体吸引子的存在性以及时间周期解和殆时间周期解的存在性。
第三部分：在一维情形，我们考虑了一类带导数项的Ginzburg—Landau方程，通过构造一些类似于发展方程守恒律的泛函及巧妙的积分估计，证明了当粘性系数趋于零时，Ginzburg—Landau方程的解逼近相应的带导数项的Schr(?)dinger方程的解，并给出了最优收敛速度估计；在二维情形，我们证明了一类带导数项的广义Ginzburg—Landau方程整体光滑解的存在性，以及在某种特殊情形下，GL方程的解趋近于相应的带导数项的Schr(?)dinger方程的弱解；在一般情形下，我们讨论了一类Ginzburg—Landau方程的非齐次边值问题，通过几个积分恒等式，同时估计解的H~1模及法向导数在边界上的模，证明了整体弱解的存在性。
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 of
global smooth s olution, global attractor in H~(2) , 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 identities
of solution for generalized Ginzburg-Landau equations with inhomogeneous boundary condition and the estimates for the L~(2) norm
on boundary of normal derivative and H~(1)'norm of solution, we prove
the existence of global weak solution of the inhomogeneous boundary value problem for generalized Ginzburg-Landau equations.
第一部分：考虑了Camassa—Holm方程在半无界和有界区域上的初边值问题，用Kato关于拟线性演化方程的初值问题的理论及先验估计的方法，证明了整体解的存在性，及在一定条件下，解在有限时间内的Blow—up。同时还研究了一类广义Camassa—Holm方程初值问题整体解的存在性。
第二部分：在非线性控制边界条件之下，对于带耗散项的Camassa—Holm方程的初边值问题，用压缩映射不动点原理及先验估计方法，证明了整体光滑解的存在性、整体解的指数稳定性、H~2空间中整体吸引子的存在性以及时间周期解和殆时间周期解的存在性。
第三部分：在一维情形，我们考虑了一类带导数项的Ginzburg—Landau方程，通过构造一些类似于发展方程守恒律的泛函及巧妙的积分估计，证明了当粘性系数趋于零时，Ginzburg—Landau方程的解逼近相应的带导数项的Schr(?)dinger方程的解，并给出了最优收敛速度估计；在二维情形，我们证明了一类带导数项的广义Ginzburg—Landau方程整体光滑解的存在性，以及在某种特殊情形下，GL方程的解趋近于相应的带导数项的Schr(?)dinger方程的弱解；在一般情形下，我们讨论了一类Ginzburg—Landau方程的非齐次边值问题，通过几个积分恒等式，同时估计解的H~1模及法向导数在边界上的模，证明了整体弱解的存在性。
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 of
global smooth s olution, global attractor in H~(2) , 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 identities
of solution for generalized Ginzburg-Landau equations with inhomogeneous boundary condition and the estimates for the L~(2) norm
on boundary of normal derivative and H~(1)'norm of solution, we prove
the existence of global weak solution of the inhomogeneous boundary value problem for generalized Ginzburg-Landau equations.
引文
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