Bifurcation, Stability, and Non-uniqueness in Ideal Fluids

该项目致力于促进和推进自由表面水波和可压缩流体的数学理论，它们包含了广泛的有趣物理现象，其动力学受欧拉方程支配。主要目的是发展新的或扩展现有的分析技术，以建立定常水波的存在性和稳定性理论，并研究一维可压缩气体系统弱解的唯一性。这个项目的进展将加强我们对理想流体数学的理解，并开发新的数学工具，可以深入了解偏微分方程式中真正的非线性现象。这项研究还将包括与研究生和博士后研究人员的培训和合作。该项目将带来新的视角和新的方法，以在理想流体研究的一些基本问题上取得进展。具体地说，PI将使用一种新的全局分叉理论机制来构造存在来自水下物体或海底地形的局部扰动时的大振幅孤立水波。这个项目的第二个主题是关于孤立水波的稳定性。PI将研究多峰重力-毛细内孤立波的频谱稳定性，并证明重力-毛细内孤立波的非线性横向不稳定性。该项目的最后一步是研究可压缩等熵欧拉系统的非唯一性。这项研究涉及的主要内容和技术包括分叉方法、复分析、稳定性分析、凸积分机制和双曲守恒律理论。该奖项反映了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。

This project seeks to promote and advance the mathematical theory of free-surface water waves and compressible fluids, which host a wide range of interesting physical phenomena and whose dynamics is governed by the Euler equations. The main goals are to develop new or extend existing analytic techniques to establish existence and stability theory for steady water waves, and to investigate uniqueness properties for weak solutions to one-dimensional system of compressible gases. Progress in this project will enhance our understanding of the mathematics of ideal fluids and develop novel mathematical tools that can provide insight into truly nonlinear phenomena in partial differential equations. This research will also involve training and collaboration with graduate students and postdoctoral researchers.This project will bring new perspectives and develop novel approaches to make progress on some fundamental problems in the study of ideal fluids. Specifically, the PI will use a novel global bifurcation theoretic machinery to construct large-amplitude solitary water waves in presence of localized disturbances coming from either submerged objects or the bottom topography. The second topic of this project concerns stability of solitary water waves. The PI will study the spectral stability of multimodal gravity-capillary internal solitary waves and prove nonlinear transverse instability of gravity-capillary internal solitary waves. The final track of the project consists of the study of the non-uniqueness of entropy solutions to the compressible isentropic Euler system. The main ingredients and techniques involved in the study include bifurcation method, complex analysis, stability analysis, convex integration machinery, and the theory of hyperbolic conservation laws.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.