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 weak dissipation term $\kappa(-\Delta)^{\alpha/2}\theta\ (0<\alpha\leqslant 1)$ and dispersive forcing term $Au_2$. We establish a unique global solution for a given initial data $\theta_0$ which belongs to the scaling subcritical Sobolev space $H^s(\mathbb{R}^2)\ (s>2-\alpha)$ if the size of dispersion parameter is sufficiently large. This phenomenon is so-called the global regularity. We also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. Moreover, we show the global regularity in the scaling critical Sobolev space $H^{2-\alpha}(\mathbb{R}^2)$ and 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.