Mathematical Analysis of the Water Wave Motion

PI将继续她对水波运动的研究。重点讨论了水波运动中的奇异性以及自由界面与固定刚性边界的相互作用。其目标是了解波峰成角度的现象背后的机制，并了解波浪与固体边界(如海岸)的相互作用。这项研究将导致更好地理解方程的结构，这最终将导致更好的建模和更有效的现象的数值模拟。该项目将涉及研究生和博士后研究人员。在她早期的工作中，PI已经证明了二维和三维水波问题在光滑区域是局部适定的；她还得到了对于小的、光滑的和局部的初值，二维和三维水波方程的几乎整体和整体适定性。最近，她一直致力于了解水波运动中的奇点。她构造了一类二维水波方程的自相似解，并证明了一类含角波峰界面的水波解的局部时间存在性。本项目中存在的问题包括：解的唯一性；具有倾斜波峰的水波的长时间动力学；将二维结果推广到三维空间；表面张力在非光滑水波中的作用；以及自由界面与固定边界的相互作用。除了偏微分方程和调和分析理论中的经典工具，PI还计划开发新的技术来研究提出的问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The PI will continue her study of the motion of water waves. The focus is on the singularities in the water waves motion and the interaction of the free interface with a fixed rigid boundary. The goal is to understand the mechanism behind the phenomena of waves with angled crests, and to understand the interactions of waves with solid boundaries, such as the coast. This investigation will lead to better understandings of the structures of the equations, which will eventually lead to better modeling and more efficient numerical simulations of the phenomena. The project will involve graduate students and post-doctoral researchers. In her earlier work, the PI has shown that the two and three dimensional water wave problems are locally wellposed in smooth regime; she has also obtained the almost global and global wellposedness of the two and three dimensional water wave equations for small, smooth and localized initial data. Recently, she has been focusing on understanding the singularities in the water waves motion. She has constructed a class of self-similar solutions for the two dimensional water wave equation, and proved the local in time existence of solutions in a class of water waves that include interfaces with angled crests. Problems in the current project include the uniqueness of the solutions; the long time dynamics for water waves with angled crested interfaces; the extension of the two dimensional results to three space dimensions; the role of surface tension in non smooth water waves; and the interaction of the free interface with a fixed boundary. Besides classical tools from theories of partial differential equation and harmonic analysis, and from earlier work, the PI plans to develop new techniques to study the proposed problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.