Nondecaying Weak Solutions to the 2D Quasi-Geostrophic Equations
Abstract
Abstract. We investigate weak solutions to the two-dimensional quasi-geostrophic equa-
tions without dissipation. We establish global existence of weak solutions for temperature
bounded and lacking spatial decay and velocity in the space $L^2_{ul}(R^2)$. Our methods rely on a spectral Serfati-type identity, which we use to establish uniform $L^2_{ul}$ 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.
tions without dissipation. We establish global existence of weak solutions for temperature
bounded and lacking spatial decay and velocity in the space $L^2_{ul}(R^2)$. Our methods rely on a spectral Serfati-type identity, which we use to establish uniform $L^2_{ul}$ 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.
