Existence of asymmetric vortex patch for the generalized SQG equations

TL;DR

This paper proves the existence of asymmetric vortex patches for the generalized SQG equations with higher singularity, extending previous results to a broader range of the parameter .

Contribution

It constructs asymmetric co-rotating and traveling patches for  in [1,2), filling a gap in the existence theory for more singular velocities.

Findings

Existence of asymmetric vortex patches for  in [1,2).

Extension of previous results to more singular regimes.

Use of desingularization and implicit function theorem methods.

Abstract

This paper aims to study the existence of asymmetric solutions for the two-dimensional generalized surface quasi-geostrophic (gSQG) equations of simply connected patches for $\alpha\in[1,2)$ in the whole plane, where $\alpha=1$ corresponds to the surface quasi-geostrophic equations (SQG). More precisely, we construct non-trivial simply connected co-rotating and traveling patches with unequal vorticity magnitudes. The proof is carried out by means of a combination of a desingularization argument with the implicit function theorem on the linearization of contour dynamics equation. Our results extend recent ones in the range $\alpha\in[0,1)$ by Hassainia-Hmidi (DCDS-A, 2021) and Hassainia-Wheeler (SIAM J. Math. Anal., 2022) to more singular velocities, filling an open gap in the range of $\alpha$.