Quasi-stationary solutions of the surface quasi-geostrophic equation

TL;DR

This paper finds explicit quasi-stationary solutions to the surface quasi-geostrophic equation with dissipation, providing insights into the behavior of solutions across different dissipation regimes, especially in the supercritical case.

Contribution

It introduces exact, time-evolving solutions for the surface quasi-geostrophic equation applicable to various dissipation effects, aiding understanding of the supercritical regime.

Findings

Exact solutions exist for all dissipation parameters

Solutions evolve in quasi-stationary states over time

Provides explicit examples in the supercritical regime

Abstract

In the present study, we find that the surface quasi-geostrophic equation admits exact solutions, which evolve with time in quasi-stationary states. The solutions presented are available for any dissipation effect $\kappa (-\Delta)^\alpha$ ($\kappa >0$, $0\le \alpha <1$), involved in the equation. When the equation is supercritical ($0\le \alpha <\frac12$), the problem on the existence of large global regular solutions remains open. This study, however, provides explicit sample solutions for the understanding of the uncertain problem.