关于多分支Camassa-Holm系统Cauchy问题的适定性以及强解的爆破与整体存在性的研究

The proposed project aims at studying two systems of important nonlinear evolution equations. One is a two-component Camassa-Holm (CH) system with quadratic and cubic nonlinearity, and the other is a multi-component Camassa-Holm system in higher dimensions. The former is the first completely integrable two-component system that admits peakon and weak kink solutions. In our proposed work, we will investigate the global existence and blow-up phenomena of strong solutions to the Cauchy problem for the two-component CH system by means of the conservation laws and blow-up scenario of strong solutions to the system. The latter is an extension model of the classical CH equation in higher dimensions but in multi-component, which has strong physical backgrounds. However, the development on the multi-component CH system shows very little in the literature. In this project, through utilizing the Littlewood-Paley decomposition, Morse type inequalities, transport equation theory, and commutator estimates, we will establish the local well-posedness for the Cauchy problem of the multi-component CH system in higher dimesional Besov spaces, derive a precise blow-up scenario for strong solutions, and finally construct its blow-up and global strong solutions.

本项目研究如下两类重要的非线性发展方程：具有二次与三次非线性项的两个分支Camassa-Holm（CH）系统和高维多分支的Camassa-Holm系统。前者是第一个具有尖峰孤立子解与弱扭结解的完全可积的两个分支系统，我们将利用系统的守恒律及其强解的爆破机制来研究其Cauchy问题强解的整体存在性与爆破现象。后者是经典CH方程在高维数与多分支中的推广模型，具有较强的物理背景，但是国内外对此系统的研究尚不多见。我们将利用Littlewood-Paley分解，Morse型不等式，输运方程理论与交换子估计等方法来研究高维多分支CH系统的Cauchy问题在Besov空间中的局部适定性及其强解准确的爆破机制，并构造其爆破强解与整体强解。

本项目主要研究了带二次或三次非线性项的双分支浅水波系统、高维多分支的Camassa-Holm(CH)系统以及三维不可压的Navier-Stokes (NS) 方程，这些模型都具有重要的物理意义。我们取得了如下研究成果：1）我们对具有尖峰孤立子解且带三次非线性项的双分支CH系统与b族系统建立了它们Cauchy问题的局部适定性，研究了其相应强解的整体存在与爆破现象。2）我们证明了修正的双分支CH系统整体守恒弱解的Lipschitz连续性。3）得到了双分支的Degasperis-Procesi系统解的无限传播速度及其在无穷远的渐近性态。4）在Besov空间中建立了高维多分支CH系统Cauchy问题的局部适定性及其强解的爆破机制。5）在临界的伸缩不变Besov-Sobolev空间得到了三维不可压NS方程的整体适定性。因此，我们较好地完成了本项目的预定目标。