Existence of Non-Decaying Solutions to the Generalized Surface Quasi-Geostrophic Equations

TL;DR

This paper proves the short-time existence and uniqueness of non-decaying solutions to the generalized Surface Quasi-Geostrophic equations in specific function spaces, advancing understanding of their mathematical properties.

Contribution

It establishes the existence and uniqueness of solutions in Hölder-Zygmund and uniformly local Sobolev spaces, which was previously unconfirmed.

Findings

Solutions exist and are unique in $C^r( ^2)$ for $r>1$

Solutions exist and are unique in $H_{ul}^s( ^2)$ for $s>2$

Solutions do not decay at infinity

Abstract

We establish the short-time existence and uniqueness of non-decaying solutions to the generalized Surface Quasi-Geostrophic equations in H\"older-Zygmund spaces $C^r(\mathbb{R}^2)$ for $r>1$ and uniformly local Sobolev spaces $H_{ul}^s(\mathbb{R}^2)$ for $s>2$.