Existence of Solutions to Fluid Equations in H\"{o}lder and Uniformly Local Sobolev Spaces

TL;DR

This paper proves short-time existence of solutions for the surface quasi-geostrophic and 3D Euler equations in specific function spaces, expanding the understanding of these equations' well-posedness in less traditional settings.

Contribution

It establishes short-time existence results in Hölder and uniformly local Sobolev spaces for these fluid equations, which are novel in these functional frameworks.

Findings

Existence of solutions in Hölder spaces for the surface quasi-geostrophic equation.

Existence of solutions in uniformly local Sobolev spaces for both equations.

Extension of methods to 3D Euler equations in these spaces.

Abstract

We establish short-time existence of solutions to the surface quasi-geostrophic equation in both the H\"{o}lder spaces $C^r(\mathbb{R}^2)$ for $r>1$ and the uniformly local Sobolev spaces $H^s_{ul}(\mathbb{R}^2)$ for $s\geq 3$. Using methods similar to those for the surface quasi-geostrophic equation, we also obtain short-time existence for the three-dimensional Euler equations in uniformly local Sobolev spaces.