Holder Regularity up to the Boundary for Critical SQG on Bounded Domains

Abstract

We consider the dissipative SQG equation in bounded domains, first introduced by Constantin and Ignatova in 2016. We show global Holder regularity up to the boundary of the solution, with a method based on the De Giorgi techniques. The boundary introduces several difficulties. In particular, the Dirichlet Laplacian is not translation invariant near the boundary, which leads to complications involving the Riesz transform.

arXiv commentadded proof of global existence, made several expositional changes based on referee report

References in corpus (7)

• An extension problem related to the fractional Laplacian
• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• A Parabolic Problem with a Fractional-Time Derivative
• Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation
• Local and global strong solutions for SQG in bounded domains
• Regularity for a Local-Nonlocal Transmission Problem
• Global in Time Classical Solutions to the 3D Quasi-geostrophic System for Large Initial Data