Analytical Description of an Incompressible Flow

该项目研究流体力学中的Navier-Stokes方程、Euler方程、本原方程和其他重要方程的解的定性性质。将研究解的空间和时间行为，例如解的大小、空间和时间的复杂性、定量的唯一连续性、解析和Gevrey正则性以及长时间渐近性。此外，还将讨论由Navier-Stokes方程、弹性方程和其他方程耦合而成的复杂方程组，重点讨论解的存在性、唯一性、正则性和长时间性态。了解解的行为在工程学、气象学、医学和海洋学中具有重要意义。该项目将讨论溶液的定性性质，特别强调流动中包括漩涡和振荡在内的精细结构的性质。该项目还将解决流体的大尺度运动和小尺度运动之间的关系。这项研究将开发更好地理解湍流流动的工具，并旨在找到理论和数值模拟之间的关系。

The project addresses the qualitative properties of solutions to the Navier-Stokes, Euler, primitive, and other important equations in fluid dynamics. The spatial and temporal behavior of solutions, such as the size of solutions, spatial and temporal complexity, quantitative unique continuation, analytic and Gevrey regularity, and long time asymptotics, will be studied. The complex systems of equations involving coupling of the Navier-Stokes equations with elasticity and other equations will also be considered with emphasis on existence, uniqueness, regularity, and long time behavior of solutions.The Navier-Stokes and Euler equations are principal models for a fluid motion. The understanding of behavior of the solutions is of fundamental importance in engineering, meteorology, medicine, and oceanography. The project will address qualitative properties of solutions with special emphasis on the properties of fine structures in the flow including vortices and oscillations. The project will also address the relationship between large and small scale motion of fluids. The research will develop tools for better understanding of a turbulent fluid flow and will aim at finding relations between theory and numerical simulations.