摘要
本文主要研究了一类非线性浅水波方程---耗散Camassa-Holm方程的Cauchy问题的局部适定性、精细的爆破机制、强解的爆破与整体存在性以及耗散MKdV方程的整体吸引子与数值计算。
在第一章中,我们首先介绍了方程的物理背景以及最新的研究进展,给出了与本文相关的一些定义与记号。在第二章,我们研究了一个广义弱耗散CH方程的Cauchy问题。首先我们利用Kato定理得到了方程在Hs(R),s>3/2上的局部适定性,然后利用一些先验估计和几个有用的引理给出方程精细的爆破机制和强解爆破的结果,最后证明了强解的爆破率是-2,分析结果表明：弱耗散项对于爆破率没有影响,但是耗散程度的强弱明显对水波的爆破产生影响。在第三章中,研究了一个弱耗散双组份CH方程的Cauchv问题。对于任意的初值z0=(u0,ρ0)∈Hs(R)×Hs-1(R),s≥2,我们利用Kato定理证明了系统Cauchy问题的局部适定性,然后同样给出了方程精细的爆破机制、强解的爆破结果和爆破率。第四章,我们研究了一个带有自由参数σ的弱耗散双组份CH方程的Cauchy问题。首先得到了方程在Hs(R)×Hs-1(R),s≥2上的局部适定性,然后给出了方程爆破的充要条件和几个强解爆破的结果以及爆破率。第五章我们研究了一个周期的弱耗散双组份CH方程。首先对于任意的初值z0=(u0,ρ0)∈Hs(R)×Hs-1(R),s≥2,我们证明了系统Cauchy问题的局部适定性,然后给出了方程爆破的充要条件和几个强解爆破的结果以及爆破率,最后利用构造Lyapunov函数的方法得到了当0<σ<2时方程整体解存在的一个充分条件。在第六章,对无界域上的含四阶耗散项的MKdV方程利用Sovolev插值不等式以及关于时间t的先验估计证明了该方程在无界域上解的存在性,然后利用算子分解技巧以及Kuratowskiiα-非紧测度讨论了解的光滑性并且得到了该方程在H2(R)上存在整体吸引子,最后利用启发性分析法及耗散守恒格式理论证明了所给的普遍性二时间层差分格式的计算稳定性并且做了相应的数值模拟,从图形中得到了与理论证明一致的结论。
In this doctoral dissertation, we study the local well-posedness, precise blow-up scenario and blow-up phenomena, global existence of strong solutions to the Cauchy problem for a class of nonlinear shallow water wave equations and the global attractor, numerical simulation of MKdV equation.
The dissertation is divided into six chapters. In the first chapter, we first introduce the physical background and the latest research advances of the equations. We then give some related definitions and notations of the dissertation. In the second chapter, we study the Cauchy problem of a generalized dissipative CH equation. We first establish the local well-posedness for the equation on Hs(R),s>3/2by Kato's theory. Then we present a precise blow-up scenario of strong solutions to the equation, and give several blow-up results by using some priori estimates and several useful lemmas. Finally, we prove that the blow-up rate of strong solutions to the equation is-2. This fact shows that the blow-up rate is not affected by the weakly dissipative term, but the occurrence of blow-up is affected by the dissipative parameter A. In the third chapter, we study a weakly dissipative two-component CH equation. We investigate the local well-posedness of the Cauchy problem for any initial data z0=(u0,ρ0)∈Hs(R)×Hs-1(R), s≥2by Kato's theory. Then we also present a precise blow-up scenario, several blow-up results and the blow-up rate of strong solutions to the equation. In fourth chapter, we study the Cauchy problem of a dissipative two-component CH equation with free parameter σ. We obtain the local well-posedness, precise blow-up scenario, blow-up phenomena and the blow-up rate of the equation. In fifth chapter, we study a weakly dissipative periodic two-component CH equation. We first obtain the local well-posedness, precise blow-up scenario, blow-up phenomena and the blow-up rate of the equation. Then we obtain a sufficient condition for the existence of global solution of the equation when0<σ<2by using Lyapunov function. In last chapter, the Sovolev interpolation inequality and prior estimate on time t are applied to show the existence of solution in unbounded domain. Moreover, operator decomposed technique and Kuratowskii α-noncompacted measure are applied to study the smooth property of the solution, then the existence of the global attractor of a dissipative MKdV equation in H2(R) is proved. Finally, we use nature analysis approach and dissipative conservative scheme theory to prove the computational stability of universal double time differential format and the accordant conclusion with theoretical proof is gained by numerical simulation.
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