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基础物理。这种情况的典型例子是在不可压缩流的欧拉方程中加入粘性，从而得到纳维-斯托克斯方程。这些修正的增加通常具有深远的影响，例如使方程的解表现得更好，即，正则化，物理上更现实，但也增加了进一步的复杂性，由于引入非局部效应和额外的时空尺度，并表现在，例如，在边界层的发展。该项目通过调查实际应用中产生的流体动力学模型的各种正则化方法的数学后果来解决这些问题，例如地球物理流体动力学和电化学。这项研究包括制定有效的近似时，正规化的影响是弱的，并使用它们来寻找新的近似方法来计算这些问题的解决方案，在这些制度。该项目还将为研究生和博士后提供培训机会。该项目旨在建立流体动力学方程的临界、非耗散Kelvin-Voigt（KV）近似的全局正则性。考虑的模型包括表面准地转方程，无粘多孔介质方程，达西-Boussinesq方程，和电对流方程在非牛顿和多孔介质。成功解决这些问题需要引入新的想法和分析工具。该项目是调查的长期行为的解决方案的模型和他们的KV近似，包括研究的非线性稳定性和不稳定性的特定的稳定状态，以及研究的形成小尺度和爆破。该项目解决了消失KV近似在方程中的极限的有效性。该项目引入了Navier-Stokes方程的特定部分KV正则化，旨在建立边界条件下的零粘度极限，其普朗特展开式和相关的普朗特方程。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.