Time periodic solutions to the $2$D quasi-geostrophic equation with the supercritical dissipation

TL;DR

This paper proves the existence of unique time periodic solutions for the 2D supercritical dissipative quasi-geostrophic equation with external forcing, introducing a novel approach that bypasses the contraction mapping principle.

Contribution

It establishes a new method for analyzing time periodic solutions in supercritical regimes without relying on contraction mappings.

Findings

Existence of unique time periodic solutions proven.

New approach applicable to supercritical dissipation cases.

Overcomes limitations of traditional contraction mapping methods.

Abstract

We consider the $2$D dissipative quasi-geostrophic equation with the time periodic external force and prove the existence of a unique time periodic solution in the case of the supercritical dissipation. In this case, the smoothing effect of the semigroup generated by the dissipation term is too weak to control the nonlinearity in the Duhamel term of the correponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.