Dynamics and Non-Dissipative Approximations of Nonlinear Nonlocal Fluid Equations

非线性非局部流体方程的动力学和非耗散近似

基本信息

批准号：
2204614
负责人：
Mihaela Ignatova
金额：
$ 18.47万
依托单位：
Temple University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204614&HistoricalAwards=false
关键词：
Dynamics Non Dissipative Approximations Nonlinear

项目摘要

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.

流体动力学和其他场论的建模通常从对现象的更简单描述（以偏微分方程的形式）开始，然后添加修正项以更好地解释基础物理现象，这种情况的典型例子是在欧拉方程中添加粘度。不可压缩流的方程，产生纳维-斯托克斯方程。这些修正的添加通常会产生深远的影响，例如使方程的解表现更好，即正则化，并且物理上更现实，但由于引入了非局域效应和额外的时空尺度，并增加了进一步的复杂性，例如，在边界层的开发中，该项目通过研究实际应用中出现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。该研究包括在正则化效应较弱时制定有效近似，以及使用它们来寻找新的近似方法来计算这些情况下这些问题的解决方案。该项目还将为研究生和博士后提供培训机会，旨在建立流体动力学方程的关键非耗散开尔文-沃格特（KV）近似的全局规律，所考虑的模型包括表面准地转方程、无粘性多孔方程。非牛顿和多孔介质中的介质方程、达西-布辛斯克方程和电对流方程的成功解决需要引入新的方法。该项目旨在研究模型及其 KV 近似解的长期行为，包括特定稳态的非线性稳定性和不稳定性研究，以及小尺度和爆炸项目的形成研究。解决了方程中消失 KV 近似极限的有效性该项目引入了纳维-斯托克斯方程的特定部分 KV 正则化，旨在在存在边界的情况下建立其零粘度极限，他们的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。