Mathematical Analysis of Water Waves

尽管波动现象无处不在，但对这种自由边界问题的动力学的数学分析仍然对研究界提出了巨大的挑战，理解奇点，波破碎和流氓波的机制对科学和工程非常重要。PI建议继续研究水波的演变。 在此期间的重点是了解奇异性以及水的波动的长期行为。 PI将使用严格的分析工具来解决问题。通过让研究生和博士后参与，该项目将为年轻一代的研究人员提供培训基地。 水波的数学问题涉及在重力的影响下，将无粘性、不可压缩、无旋流体从二维和三维空间中的零密度区域分离的界面的运动，忽略表面张力。在之前的资助期间，PI证明了泰勒符号条件始终成立，并获得了任意（光滑）数据的二维和三维水波问题的局部适定性。PI证明了水波问题的非线性性质是三次或更高阶的，并得到了二维水波问题的几乎整体适定性和三维水波问题的整体适定性。 此外，为了理解水波运动中的奇异行为，PI构造了一类对流占主导地位的区域中的二维水波方程的自相似解。PI建议继续她的研究奇异性以及水波运动的长期行为。特别的问题包括稳定性的自相似解决方案，她构造和长期演变的更广泛的一类初始数据。其目的是实现良好的理解的基本类型的波-波与成角度的波峰，其在长时间的水波运动的作用，并获得更好的理解的色散和水波方程的结构。

Despite the ubiquity of wave phenomena, the mathematical analysis of the dynamics of such free boundary problems still presents great challenges of significant current interest to the research community, understanding the mechanisms of singularities, wave breaking and rogue waves is of great importance to science and engineering. The PI proposes to continue her study on the evolution of water waves. The focus in this funding period is on understanding singularities as well as long time behaviors of the water wave motion. The PI will use rigorous analytical tools to tackle the problems. Through involving graduate students and post doctors, the project will provide a training ground for a younger generation of researchers. The mathematical problem of water waves concerns the motion of the interface separating an inviscid, incompressible, irrotational fluid, under the influence of gravity, from a region of zero density in two and three space dimensions, neglecting surface tension. In prior funding periods, the PI showed that the Taylor sign condition always holds, and obtained the local well-posedness of the two and three dimensional water wave problems for arbitrary (smooth) data. And the PI showed that the nature of the nonlinearity of the water wave problems are of cubic or higher orders, and obtained the almost global well-posedness for the two dimensional and global well-posedness for the three dimensional water wave problems for small and smooth initial data. Furthermore, to understand singular behaviors in the water wave motion, the PI constructed a class of self-similar solutions for the two dimensional water wave equation in the regime where convection is in dominance. The PI proposes to continue her study of singularities as well as the long time behaviors of the water wave motion. Particular problems include the stability of the self-similar solution she constructed and the long time evolution for a broader class of initial data. The objective is to achieve good understanding of a basic type of waves--waves with angled crests, and its role in the long time water wave motion, and to obtain better understanding of dispersion and the structure of the water wave equations.