Regularity, stability, and singular limits in fluid dynamics

流体动力学的规律性、稳定性和奇异极限

• 批准号: 1211828
• 负责人: Vlad Vicol
• 金额: $ 13.14万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2014-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211828&HistoricalAwards=false
• 关键词: Regularity; stability; singular; limits; fluid

VicolDMS-1211828 The investigator analyzes several nonlinear nonlocal partial differential equations that arise in mathematical fluid mechanics. The goal is to develop new mathematical tools towards progress on various open problems, such as the global regularity versus finite time blow-up of classical solutions, the behaviour of solutions in singular limit regimes, and other qualitative properties. The first part of the project is devoted to studying the regularity of active scalar equations. The work is mostly geared towards the analysis of the inviscid and the supercritically-dissipative surface quasi-geostrophic equation, porous media equation, and the magneto-geostrophic equation. For all these models the a priori controlled quantities are too weak to guarantee the global existence of smooth solutions. The investigator and collaborators seek to identify analytic and geometric mechanisms that preclude the possible formation of singularities. The regularity of supercritical drift-diffusion equations is also considered. The second part of the project addresses the inviscid limit of the Navier-Stokes equations on a bounded domain. The goal is to establish new well-posedness results for the Prandtl boundary layer equations under weak matching conditions at the top of the boundary layer, and analyze the inviscid limit of the Navier-Stokes equations in these regimes. The equations describing incompressible Newtonian fluids, such as the Euler and the Navier-Stokes equations, account for interactions between a very broad range of space and time scales, in a highly nonlinear fashion. Due to this intrinsic complexity, the accuracy needed to fully resolve the underlying phenomena via numerical computations is out of reach for the foreseeable future. This project seeks to further advance the rigorous analysis of these equations, which is vital to a finer understanding and validation of the models, and to an accurate interpretation of numerical simulations.

VicolDMS-1211828 研究人员分析了数学流体力学中出现的几个非线性非局部偏微分方程。 我们的目标是开发新的数学工具，以取得进展的各种开放的问题，如全球正则性与有限时间爆破的经典解决方案，在奇异极限制度的解决方案的行为，和其他定性性质。 本项目的第一部分致力于研究活动标量方程的正则性。 主要工作是分析无粘和超临界耗散的表面准地转方程、多孔介质方程和磁地转方程。 所有这些模型的先验控制量太弱，无法保证光滑解的整体存在性。 研究人员和合作者试图确定排除可能形成奇点的分析和几何机制。 本文还讨论了超临界漂移扩散方程的正则性。 该项目的第二部分解决了有界域上的Navier-Stokes方程的无粘极限。 我们的目标是建立新的适定性结果的Prandtl边界层方程弱匹配条件下的边界层顶部，并分析在这些制度的Navier-Stokes方程的无粘极限。 描述不可压缩牛顿流体的方程，如欧拉方程和纳维尔-斯托克斯方程，以高度非线性的方式解释了非常广泛的空间和时间尺度之间的相互作用。 由于这种内在的复杂性，通过数值计算完全解决潜在现象所需的准确性在可预见的未来是遥不可及的。 该项目旨在进一步推进对这些方程的严格分析，这对于更好地理解和验证模型以及准确解释数值模拟至关重要。