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: 1348193
• 负责人: Vlad Vicol
• 金额: $ 10.36万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-01-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1348193&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研究者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。 目的是开发新的数学工具，以在各种开放问题上进行进步，例如全球规律性与经典解决方案的有限时间爆破，在奇异极限制度中的解决方案的行为以及其他定性属性。 该项目的第一部分致力于研究主动标量方程的规律性。 这项工作主要旨在分析无粘性和超级遗传性表面准斑块方程，多孔培养基方程和磁性地球形方程。 对于所有这些模型，先验控制的数量太弱，无法保证全球平滑解决方案的存在。 研究者和合作者试图确定排除奇异性可能形成的分析和几何机制。 还考虑了超临界漂移扩散方程的规律性。 该项目的第二部分解决了有限域上Navier-Stokes方程的无关极限。 目的是在边界层顶部的弱匹配条件下为PrandTL边界层方程建立新的良好性结果，并分析这些机制中Navier-Stokes方程的无关极限。 描述不可压缩的牛顿流体的方程式，例如Euler和Navier-Stokes方程，以高度非线性的方式解释了非常广泛的时空和时间尺度之间的相互作用。 由于这种内在的复杂性，在可预见的将来，通过数值计算完全解决基本现象所需的精度是无法实现的。 该项目旨在进一步推进对这些方程式的严格分析，这对于对模型的理解和验证以及对数值模拟的准确解释至关重要。