Stability of solitons and long-term dynamics of fluids

该项目的主要目标是研究几个重要的偏微分方程解。这些解描述了日常生活中可以观察到的物理现象，如流体的动力学、水波的运动和引力。我们的目标是严格分析这些解，并恢复关于它们行为的定量和定性信息作为数学定理。该研究项目涉及研究生和博士后助理，将偏微分方程、数值分析、几何和调和分析的思想和方法结合在一起。PI将研究几个重要演化方程的解的长期动力学，如二维和三维的欧拉方程、水波模型和广义相对论的爱因斯坦场方程。六个感兴趣的问题被明确确定为项目的一部分。这些问题涉及2D Euler方程和广义SQG方程的剪切流和涡旋的全局稳定性，2D和3D大盒子中水波的长期动力学，以及有意义的爱因斯坦方程解的全局稳定性。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。

The main goal of the project is to study solutions of several important partial differential equations. These solutions describe physical phenomena, such as the dynamics of fluids, the motion of water waves, and gravitation, which can be observed in daily life. We aim to analyze these solutions rigorously and to recover quantitative and qualitative information about their behavior as mathematical theorems. The research project involves graduate students and postdoctoral associates, and connects ideas and methods in partial differential equations, numerical analysis, geometry, and harmonic analysis.The PI will study the long term dynamics of solutions of several important evolution equations, such as the Euler equations in 2 and 3 dimensions, water wave models, and the Einstein field equations of General Relativity. Six problems of interest are explicitly identified as part of the project. These problems concern the global stability of shear flows and vortices for the 2D Euler and the generalized SQG equations, the long term dynamics of water waves in a large box in 2D and 3D, and the global stability of solutions of the Einstein equations with matter.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.