Dynamics and Non-Dissipative Approximations of Nonlinear Nonlocal Fluid Equations

Temple University, Philadelphia PA

Investigators

Abstract

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.