Camassa-Holm型方程解的整体存在性和爆破性研究

This project is devoted to the study on the local well-posedness， global existence , blow-up of strong solutions and the existence of the weak solutions to the Cauchy problems for the Camassa-Holm-type equations which arise in the water waves. The nonlinear terms of these equations are much more complicated than those of Camassa-Holm equation. Thus we will pay more attention to the structure of those equations and the intrinsic relationship among nonlinear terms to overcome the difficulties of the systems. We will use the Kato’s theory in the semigroups of operators, the theory in the Sobolev spaces, the theory in the Besov spaces , the interpolation theory, Littlewood-Paley decomposition and the theory in the transport equation to study the local well-posedness of strong solutions to these Camassa-Holm-type equations; by using the diffeomorphism techniques and searching for some invariant properties as well as the commutator estimates， we will establish blow-up criteria; then applying conserved laws and blow-up criteria, we will obtain blow-up result and global existence result of strong solutions. We will also use Helly''s Theorem, the compensated compactness method, Radon measures and approximation methods to study the global existence of the weak solutions.

本项目主要研究某些具有物理背景的Camassa-Holm型方程的Cauchy问题的强解的局部适定性，整体适定性，爆破性以及弱解的存在性。所研究方程的非线性项比Camassa-Holm方程的非线性项复杂的多，因此要充分考虑到方程的结构以及各项内在的关系从而克服复杂的非线性项带来的困难。本项目拟使用算子半群中的Kato理论，Sobolev空间理论，Besov空间理论，内插理论，Littlewood-Paley分解以及运输方程理论去研究某些Camassa-Holm型方程在Sobolev空间以及在Besov空间中的强解的局部适定性；使用微分同胚技巧和一些不变性以及交换子估计建立爆破准则；然后使用一些守恒律以及爆破准则建立强解的爆破结果和整体存在性结果；拟使用Helly定理，补偿紧理论，Radon测度以及逼近的思想去研究弱解的全局存在性。

本项目主要使用弱解的知识，Littlewood-Paley分解，输运方程理论，Besov空间理论和内插定理以及Osgood引理去研究推广的Camassa-Holm方程的Cauchy问题、 二元Camassa-Holm方程的Cauchy问题和推广的Novikov方程的Cauchy问题以及推广的Degasperis-Procesi方程的Cauchy问题。取得成果如下：1、推广的Camassa-Holm方程的尖波存在性，解映射的不连续性以及在Besov空间中的局部适定性和爆破性；2、二元Camassa-Holm方程在Besov空间中的局部适定性和爆破性；3、推广的Novikov方程在Besov空间中的局部适定性和持续性；4、推广的Degasperis-Procesi方程的尖波存在性，解映射的不连续性以及在Besov空间中的局部适定性和爆破性。