Global solutions to the dissipative quasi-geostrophic equation with dispersive forcing

Abstract

We consider the initial value problem for the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing and prove the global existence of a unique solution in the scaling subcritical Sobolev spaces $H^{s}(\mathbb{R}^{2})$ ($s > 2 - \alpha$) and the scaling critical space $H^{2-\alpha}(\mathbb{R}^2)$. More precisely, for the scaling subcritical case, we establish a unique global solution for a given initial data $\theta_{0} \in H^{s}(\mathbb{R}^{2})$ ($s > 2 - \alpha$) if the size of dispersion parameter is sufficiently large and also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. For the scaling critical case, we find that the size of dispersion parameter to ensure the global existence is determined by each subset $K \subset H^{2-\alpha}(\mathbb{R}^{2})$, which is precompact in some homogeneous Sobolev spaces.