Spatial-decay of solutions to the quasi-geostrophic equation with the critical and the super-critical dissipation

TL;DR

This paper investigates the long-range behavior of solutions to the 2D dissipative quasi-geostrophic equation with critical and supercritical fractional dissipation, providing insights into their asymptotic decay patterns.

Contribution

It offers new analysis of the far field asymptotics for solutions in critical and supercritical dissipation regimes, extending understanding of their decay properties.

Findings

Derived asymptotic decay rates for solutions

Characterized the influence of fractional Laplacian on solution behavior

Extended previous results to supercritical dissipation cases

Abstract

The initial value problem for the two dimensional dissipative quasi-geostrophic equation derived from geophisical fluid dynamics is studied. The dissipation of this equation is given by the fractional Laplacian. It is known that the half Laplacian is a critical dissipation for the quasi-geostrophic equation. In this paper, far field asymptotics of solutions are given in the critical and the supercritical cases.