几类非线性色散波方程解的适定性与渐近行为研究

本项目主要研究来源于流体力学、弹性力学、生物海洋工程、自然灾害预测等领域中的非线性色散波方程。在前期工作的基础上，首先研究Benjamin-Ono型方程以及Camassa-Holm型方程解的局部适定性、整体存在性、爆破模式和大时间渐近行为等定性理论；进一步，深入细致研究解关于初值的一致连续依赖性与拓扑意义下的Hölder连续性以及解发生破裂现象后的性态等难点问题。由于所研究的模型具有复杂的高阶项，且会出现各分支间相互作用和奇异现象，促使我们必须结合方程的内在结构改进和创造新的研究工具和方法。对这些国际上极为关注的模型的理论研究有助于认识和理解尖峰孤子解以及破裂等重要自然现象，对偏微分方程理论的完善和发展起促进作用。

In this project, we mainly study several nonlinear dispersive wave equations and chemotaxis models, which come from fluid mechanics, elastic mechanics bio-ocean and disaster prediction. Based on our previous research works, we firstly consider the local well-posedness, global existence, blow-up profile and large time behavior for the solutions of Benjamin-Ono type equations and Camassa-Holm type equations. Moreover, we investigate the uniformly dependence and Hölder continuous of the data-to-solution map, and the continuation of the solution beyond wave breaking. The equations researched here always possess higher order of nonlinearities and derivatives, coupled with the interaction of each component and the singularity, so, we shall consider the inner structure of these equations to find new methods. The theoretical research of these models will help us to understand peakons and wave breaking, also promote the development of the theories for Partial Differential Equations.