A Frequency Localized Maximum Principle Applied to the 2D Quasi-Geostrophic Equation

Abstract

In this paper, we prove a maximum principle for a frequency localized transport-diffusion equation. As an application, we prove the local well-posedness of the supercritical quasi-geostrophic equation in the critical Besov spaces \({\mathring{B}^{1-\alpha}_{\infty,q}}\), and global well-posedness of the critical quasi-geostrophic equation in \({\mathring{B}^{0}_{\infty,q}}\) for all 1 ≤ q < ∞. Here \({\mathring{B}^{s}_{\infty,q} }\) is the closure of the Schwartz functions in the norm of \({B^{s}_{\infty,q}}\).