Existence of Solutions to Fluid Equations in Hölder and Uniformly Local Sobolev Spaces

We establish short-time existence of solutions to the surface quasi-geostrophic equation in both the Hölder spaces $C^r({\mathbb{R}}^2)$ for $r>1$ and the uniformly local Sobolev spaces $H_{ul}^s({\mathbb{R}}^2)$ for $s\ge 3$. 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.