Regularity, stability, and singular limits in fluid dynamics

VICOLDMS-1211828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组。其目标是开发新的数学工具，以便在各种开放问题上取得进展，例如经典解的全局正则性与有限时间爆破、解在奇异极限区域中的行为，以及其他定性性质。该项目的第一部分致力于研究活动标量方程的正则性。这项工作主要用于分析无粘性和超临界耗散的地表准地转方程、多孔介质方程和磁地转方程。对于所有这些模型，先验控制量太弱，不能保证全局光滑解的存在。研究人员和合作者试图确定排除奇点可能形成的解析和几何机制。讨论了超临界漂移扩散方程的正则性。该项目的第二部分讨论了有界域上的Navier-Stokes方程的无粘性极限。目的是建立Prandtl边界层方程在边界层顶部弱匹配条件下的新适定性结果，并分析在这些条件下Navier-Stokes方程的无粘性极限。描述不可压缩牛顿流体的方程，如欧拉方程和纳维-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，在可预见的未来，通过数值计算完全解决潜在现象所需的准确性是遥不可及的。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对数值模拟的准确解释至关重要。

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.