Vanishing diffusion limits and long time behaviour of a class of forced active scalar equations

TL;DR

This paper studies the long-term behavior and diffusion limits of a broad class of active scalar equations involving fractional Laplacians, with applications to geophysical fluid dynamics.

Contribution

It introduces new existence and convergence results for these equations under various parameter regimes and analyzes their long-term dynamics, including the existence of a global attractor.

Findings

Existence of solutions in certain parameter regimes

Convergence results as diffusion parameters vary

Existence of a unique global attractor for the system

Abstract

We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour of solutions to the general problem and prove the existence of a unique global attractor. We apply results to two particular active scalar equations arising in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.