具有尖峰解和高次非线性项的非线性色散波方程Cauchy问题的研究

In this project, we will study local well-posedness，precise blow-up scenario and blow-up phenomena, global existence of strong solutions, existence and uniqueness of global weak solutions, global existence of conservative solutions.and dissipative solutions to the Cauchy problem of several classes of nonlinear dispersive wave equations with peakon solutions and high-order nonlinearities arising from modern mechanics and physics. It is a new topic, and also a meaningful topic in mathematical theory and applied physics. The topic we will study is very challenging. Furthermore, the topic is the current focus in this field. We will study this topic by employing some new methods and new ideas of nonlinear dispersive wave equations, combining with semigroup theory, harmonic analysis, theory for dfferential equations in Banach spaces and the bifurcation theory. We expect the research will provide mathematical tools and theoretical bases for the corresponding applied subjects. Innovations in the mathematical theory are also expected through these studies. We hope that it is going to push both the research of nonlinear dispersive wave equations and our academic level to a higher one.

本项目将研究来源于现代力学和物理学的几类具有尖峰解和高次非线性项的非线性色散波方程Cauchy问题的局部适定性，强解的爆破，强解的整体存在性，整体弱解的存在性唯一性，守恒解和耗散解的整体存在性等问题。这是一个崭新的研究课题，也是在数学理论和物理学应用两面都有重要研究价值的课题，既具有相当的难度，又是该领域的热点研究问题。我们将研究应用色散波方程的一些新思想，新方法，结合半群理论，调和分析，Banach空间上的微分方程理论及其分叉理论等来研究该课题。通过对这些具体问题的研究，使我们在研究所需要的数学理论和方法上有所创新和突破，提升色散方程的研究层次，提高我们的研究水平。

本项目基本上是按原计划进行研究，我们研究了来源于现代力学和物理学的几类具有尖峰解和高次非线性项的非线性色散波方程Cauchy问题的局部适定性，强解的爆破，强解的整体存在性，整体弱解的存在性唯一性，守恒解和耗散解的整体存在性等问题。我们应用色散波方程的一些新思想，新方法，结合半群理论，调和分析，Banach空间上的微分方程理论及其分叉理论等来研究该课题，我们取得了一系列成果。