# Global weak solutions for generalized SQG in bounded domains

## TL;DR

This paper proves the existence of global weak solutions for a family of generalized SQG equations in bounded domains, addressing more singular nonlocal operators than standard SQG.

## Contribution

It introduces a method to establish global weak solutions for generalized SQG equations with more singular operators in bounded domains.

## Key findings

- Existence of global $L^2$ weak solutions proven.
- New commutator representations developed for bounded domains.
- Applicable to a broader class of SQG-like equations.

## Abstract

We prove the existence of global $L^2$ weak solutions for a family of generalized inviscid surface-quasi geostrophic (SQG) equations in bounded domains of the plane. In these equations, the active scalar is transported by a velocity field which is determined by the scalar through a more singular nonlocal operator compared to the SQG equation. The result is obtained by establishing appropriate commutator representations for the weak formulation together with good bounds for them in bounded domains.