Existence of solutions to fluid equations in H?lder and uniformly local Sobolev spaces

We establish short-time existence of solutions to the surface quasi-geostrophic (SQG) equation in the Holder spaces Cr (][82) 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 Holder 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 Hsul(][82) 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. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This article establishes the short-time existence of solutions to the surface quasi-geostrophic (SQG) equation in the Holder spaces Cr (][82) for r > 1. To avoid an integrability assumption, a generalization of the SQG constitutive law is introduced. Using these solutions, the existence of solutions of SQG in another class of non-decaying function spaces, the uniformly local Sobolev spaces Hsul(][82) for s >= 3, is proven. The short-time existence of the three-dimensional Euler equations in uniformly local Sobolev spaces is also obtained using similar methods as the SQG equation. (c) 2023 Elsevier Inc. All rights reserved.