角形区域水波问题及带状区域多重孤立波解

不可压欧拉方程组加上自由边界条件被称为水波问题，这是流体力学理论研究的核心问题之一。本项目拟研究的主要问题正是角形区域上带表面张力的水波问题的适定性，相比于经典的水波问题，这是一片全新的领域，对应带粘性情况下的contact line problem。最近几年内，角形区域水波问题理论研究才开始有所发展，其中一些工作是申请人及合作者的成果，包括角形区域椭圆问题奇性分解以及小角度带表面张力水波问题的先验估计和局部适定性。研究拟包含以下内容:1)一般角度的加权椭圆估计;2)一般角度的角形区域水波问题先验估计以及解的适定性等等。区别于经典水波问题的工作，本项目具有以下特色和创新之处: 1)本项目问题的能量框架是耗散的; 2)具有不同的变分结构，增加了角点耗散的势能函数;3)角形区域带来的奇性是水波问题领域全新的问题。另外，本项目还拟研究带状区域上的弱或不带表面张力的水波问题的多重孤立波解。

Incompressible Euler equations with free sufaces is called the water waves problem, which is one of the essential problems in theoretical fluid dynamics. Based on our previous work, the main goal of this project is to study the water waves problem on corner domains with surface tension, which is a completely new field compared to the classical water waves and corresponds to the contact line problem in the case of viscosity. In recent years, the study for water waves problem in corner domains just begins, and some of the works were done by the applicant and co-authors, including the singularity decomposition for related elliptic systems in corner domains, and a priori estimate and local-well posedness for the capillary water waves problem with constrained contact angles. Our study will focus on the following points: 1) weighted elliptic estimates for general angles; 2) a priori estimate for general contact angles and well-posedness etc.. Being different from classical water waves problem, our project has the following new ingredients: 1) The energy is dissipative; 2) The variation formulation is different due to the dissipation potential at the contact point; 3) Singularities take place near the corner, which are completely new in this field. Besides, this project also plans to investigate the existence of multi-soliton solution to the water waves problem in a strip domain with weak surface tension or without surface tension.