Non-decaying weak solutions to the 2D quasi-geostrophic equations

Abstract

We investigate weak solutions to the two-dimensional quasi-geostrophic equations without dissipation. We establish global existence of weak solutions for temperature bounded and lacking spatial decay and velocity in the space $L_{ul}^2({\mathbb{R}}^2)$. Our methods rely on a spectral Serfati identity, which we use to establish uniform $L_{ul}^2$ bounds on a sequence of velocities satisfying the dissipative equations. These bounds, combined with a maximum principle on the scalar temperature, allow us to pass to the zero-dissipation limit, giving global-in-time weak solutions.

34 pages