Quasi-stationary solutions of the surface quasi-geostrophic equation

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 κ(−∆) α (κ > 0, 0 ≤ α < 1), involved in the equation. When the equation is supercritical (0 ≤ α < 12 ), 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.