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理论的建模通常以对现象的更简单描述（以部分微分方程的形式）开始，然后添加校正项以更好地说明基础物理。这种情况的原型是在不可压缩流中向Euler方程添加粘度，从而导致Navier-Stokes方程。这些更正的添加通常会产生深远的后果，例如使方程式的解决方案更好地表现，即定期和物理上更现实，但由于引入了非局部效应和其他空间时间尺度，因此增加了进一步的复杂性，例如在边界层的发展中表现出来。该项目通过研究在实际应用中（例如地球物理动力学和电化学）中引起的流体动力学模型的各种受管制方法的数学后果来解决这些问题。该研究包括当调节效应较弱时的有效近似值的制定，并用于寻找新的近似方法来计算这些问题中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。该项目旨在建立全球规律性，以实现流体动力学方程的关键，非缺血性开尔文 - voigt（KV）近似。所考虑的模型包括表面的quasigeotrophic方程，无粘性多孔培养基方程，darcy-Boussinesq方程以及非牛顿和多孔培养基中的电交流方程。成功解决这些问题需要引入新颖的思想和分析工具。该项目是研究模型解决方案及其KV近似的长期行为，包括对特定稳态的非线性稳定性和不稳定性的研究，以及对小尺度的形成和爆炸的研究。该项目解决了方程中消失的KV近似极限的有效性。该项目介绍了Navier-Stokes方程的特定部分KV调节，旨在在存在边界，其printTL扩展和关联的PrintTL方程的情况下建立其零粘度限制。该奖项反映了NSF的法定任务，并通过使用该基金会的知识分子优点和广泛的影响来评估NSF的法定任务，并被认为是珍贵的支持。