# Global solutions for the generalized SQG patch equation

## TL;DR

This paper proves the global stability of a specific patch solution for the generalized SQG equation with a more singular velocity, marking the first such stability result in this setting.

## Contribution

It establishes the first global stability result for patch solutions of the generalized SQG equation with lpha ter 1, extending understanding of stable solutions beyond known rotating V-states.

## Key findings

- Proves global stability of half-plane patch stationary solutions.
- First construction of stable global solutions for gSQG patches.
- Identifies the velocity's increased singularity for lpha ter 1.

## Abstract

We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter $\alpha \in (1,2)$. The cases $\alpha = 0$ and $\alpha = 1$ correspond to 2d Euler and SQG respectively, and our choice of the parameter $\alpha$ results in a velocity more singular than in the SQG case. Our main result concerns the global stability of the half-plane patch stationary solution, under small and localized perturbations. Our theorem appears to be the first construction of stable global solutions for the gSQG-patch equations. The only other nontrivial global solutions known so far in the patch setting are the so-called V-states, which are uniformly rotating and periodic in time solutions.