Qualitative Properties of Solutions to Fluids Equations

本项目研究流体模型的偏微分方程解。主要目标是了解不可压缩流体的演化。特别是，PI将研究涉及具有自由界面的流体的物理问题。这包括有或没有表面张力的水波的运动，以及流固耦合问题。PI将介绍一些技术，这些技术将促进我们对流体的了解，并促进在科学和工程中的进一步应用。研究生将参与项目的各个方面。对于流体结构相互作用系统，研究将寻求导致解的局部和长期存在的先验估计，并构造满足这些性质的解。PI还将研究解的定性性质，如解析正则性和Sobolev正则性的持久性以及渐近行为。此外，对于具有自由界面的欧拉方程的解，本研究试图回答类似的问题，重点是解的局部存在性和唯一性。PI将致力于与流体动力学相关的其他重要模型的解的性质，如2D和3D Boussinesq方程、活动标量方程和Prandtl方程。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 addresses the solutions of partial differential equations modeling fluids. The principal goal is understanding the evolution of an incompressible fluid. In particular, the PI will study physical problems involving fluids with a free interface. This includes the motion of water waves, with or without surface tension, and a fluid-structure interaction problem. The PI will introduce techniques which shall advance our knowledge of fluids and promote further applications in science and engineering. Graduate students will be involved in all aspects of the project.For the fluid structure interaction system, the research will seek a priori estimates leading to the local and long-time existence of solutions and the construction of solutions satisfying these properties. The PI will also study the qualitative properties of solutions, such as the persistence of analytic and Sobolev regularity, along with asymptotic behavior. Furthermore, the research seeks to answer similar questions for the solutions of the Euler equations with a free interface, focusing on the local existence and uniqueness of solutions. The PI will work on properties of solutions of other important models in connection with fluid dynamics, such as the 2D and 3D Boussinesq equations, active scalar equations, and the Prandtl equations.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.