Long Term Regularity of Solutions of Fluid Models

本研究课题研究一类描述水波、等离子体、引力等物理现象的偏微分方程。 这些方程的解的复杂性质通常排除了完整的数学分析，并且应用依赖于数值近似。 该项目严格研究这些方程的解，并将其行为的定量和定性信息恢复为数学定理。该项目涉及研究生和博士后研究人员，并促进与相关领域研究人员的合作，包括物理和工程。本项目研究几个准线性色散偏微分方程的解，包括二维和三维空间中的水波模型，更一般的两流体界面情况，以及广义相对论的爱因斯坦-克莱因-戈登方程。研究的主要问题涉及这些方程的平衡解的全局稳定性和在有限时间内成为奇异的解决方案的建设。该项目旨在为这一充满活力的领域提供新的方法和工具。

This research project studies a class of partial differential equations that describe physical phenomena including water waves, plasmas, and gravitation. The complicated nature of the solutions to such equations often precludes complete mathematical analysis, and applications rely on numerical approximations. This project studies the solutions of these equations rigorously and recovers quantitative and qualitative information about their behavior as mathematical theorems. The project involves graduate students and postdoctoral associates in the research and fosters collaborations with researchers in related fields, including physics and engineering. This project studies the solutions of several quasilinear dispersive partial differential equations, including water-wave models in two and three spatial dimensions, the more general case of two-fluid interfaces, and the Einstein-Klein-Gordon equations of general relativity. The main problems under study concern the global stability of the equilibrium solutions of these equations and the construction of solutions that become singular in finite time. The project aims to contribute new methods and tools to this dynamic field.