Mathematical Questions in Classical and Relativistic Fluids and Applications

数学流体动力学研究用于描述流体运动的方程的数学性质。这门学科历史悠久，但仍有许多重要问题有待解决。本项目主要研究可压缩流动和相对论性流体的自由边界问题。大气、恒星和气态行星可以用具有自由表面的可压缩流来描述。自由边界流体的方程提出了许多挑战，直到最近我们才开始了解它们的数学性质。所谓相对论性流体，是指处于爱因斯坦相对论不容忽视的状态下的流体。中子星和夸克-胶子等离子体（在重离子碰撞中形成的一种奇异物质状态）可以用粘性相对论流体来描述。相对论流体动力学方程具有丰富的数学结构，是近年来深入研究的源泉。该项目研究了经典流体和相对论流体中的各种问题，这些问题是非常活跃的研究领域的一部分。本项目将研究自由边界欧拉方程，重点关注可压缩流体的情况，这是一个比不可压缩流体发展得少得多的主题。该项目将探讨存在性、唯一性、长期行为、不可压缩极限以及表面张力和旋转在流动中的作用等问题。对于相对论性流体，重点将主要放在具有粘度的流体上。这是一个近年来物理界非常活跃的领域，但其数学基础还不发达。这个项目将有助于把相对论粘性流体理论建立在坚实的数学基础上，并利用这些知识来解决与物理直接相关的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematical fluid dynamics investigates the mathematical properties of the equations employed to describe the motion of fluids. This subject has a long history but many important questions remain open. This project focuses on free-boundary problems for compressible flows and relativistic fluids. The atmosphere, stars, and gaseous planets can be described by compressible flows with a free surface. The equations of free-boundary fluids present many challenges and only recently we have begun to understand their mathematical properties. By relativistic fluids, one means fluids in a regime where Einstein's theory of relativity cannot be neglected. Neutron stars and quark-gluon plasma (an exotic state of matter that forms in heavy-ion collisions) can be described by viscous relativistic fluids. The equations of relativistic fluid dynamics present a rich mathematical structure that has been the source of intensive investigation in recent years.This project investigates a variety of problems in classical and relativistic fluids that are part of very active research areas. This project will investigate the free-boundary Euler equations, with focus on the case of compressible fluids, a subject much less developed than its incompressible counterpart. This project will investigate questions of existence, uniqueness, long-time behavior, the incompressible limit, and the role of surface tension and rotation in the flow. For relativistic fluids, the focus will be primarily in the case of fluids with viscosity. This is an area that has witnessed a great deal of activity in physics in recent years but whose mathematical foundations are underdeveloped. This project will contribute to put the theory of relativistic viscous fluids on a firm mathematical basis and use this knowledge to solve problems of direct relevance to physics.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.