Dynamics and Non-Dissipative Approximations of Nonlinear Nonlocal Fluid Equations

流体力学和其他场论的建模通常从对现象的简单描述开始——以偏微分方程的形式——然后添加校正项以更好地解释潜在的物理。这种情况的原型是在不可压缩流的欧拉方程中加入粘度，从而得到纳维-斯托克斯方程。这些修正的增加通常会产生深远的影响，例如使方程的解表现得更好，即正则化，并且在物理上更现实，但由于引入非局部效应和额外的时空尺度而进一步增加复杂性，例如，在边界层的发展中表现出来。本项目通过研究在实际应用中出现的各种正则化方法对水动力模型的数学结果来解决这些问题，例如地球物理流体动力学和电化学。本文研究了正则化效应较弱时的有效近似，并利用它们寻找新的近似方法来计算这些问题在这些情况下的解。该项目还将为研究生和博士后提供培训机会。该项目旨在为流体动力学方程的临界非耗散Kelvin-Voigt （KV）近似建立全局规则。考虑的模型包括表面准地转方程、无粘多孔介质方程、Darcy-Boussinesq方程以及非牛顿介质和多孔介质中的电对流方程。成功地解决这些问题需要引入新的思想和分析工具。该项目旨在研究模型及其KV近似解的长期行为，包括研究特定稳态的非线性稳定性和不稳定性，以及研究小尺度和爆炸的形成。本文讨论了方程中消失KV近似极限的有效性。该项目引入了Navier-Stokes方程的特定部分KV正则化，旨在建立存在边界的零粘度极限、它们的普朗特展开和相关的普朗特方程。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。

The modeling of hydrodynamic and other field theories usually starts with simpler description of the phenomena – in the form of partial differential equations – and then adds correction terms to better account of the underlying physics. Prototypical of this situation is the addition of viscosity to the Euler equations for an incompressible flow, resulting in the Navier-Stokes equations. The addition of these corrections often have profound consequences, such as making the solutions of the equations better behaved, i.e., regularized, and physically more realistic, but also add further complexities due to the introduction of nonlocal effects and additional spatiotemporal scales and manifested, for example, in the development of boundary layers. This project addresses these issues by investigating the mathematical consequences of various regularization approaches on hydrodynamical models arising in practical applications, such as geophysical fluid dynamics and electrochemistry. The study includes the formulation of effective approximations when the regularization effects are weak, and their use to find new approximation methods to compute the solutions to these problems in those regimes. The project will also provide training opportunities for graduate students and postdocs. The project is aimed at establishing global regularity for critical, non-dissipative Kelvin-Voigt (KV) approximations of hydrodynamic equations. The models considered include the surface quasigeostrophic equation, the inviscid porous medium equation, Darcy-Boussinesq equations, and electroconvection equations in non-Newtonian and porous media. Successful resolution of these problems requires the introduction of novel ideas and analytical tools. The project is to investigate the long-time behavior of solutions of the models and of their KV approximations, including studies of nonlinear stability and instability of specific steady states, and studies of formation of small scales and blow up. The project addresses the validity of the limit of vanishing KV approximation in the equations. The project introduces specific partial KV regularizations of the Navier-Stokes equations, aiming to establish their zero-viscosity limit in the presence of boundaries, their Prandtl expansions and associated Prandtl equations.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.