Existence of solutions to fluid equations in Hölder and uniformly local Sobolev spaces

Abstract

We establish short-time existence of solutions to the surface quasi-geostrophic (SQG) equation in the Hölder spaces Cr(R2) for r>1; to avoid an integrability assumption (such as membership of the data in an Lq space) we introduce a generalization of the SQG constitutive law. As an application of the Hölder theory, we use these solutions when forming an approximation sequence in the proof of existence of solutions of SQG in another class of non-decaying function spaces, the uniformly local Sobolev spaces Huls(R2) for s≥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.