Decay Rates for Mild Solutions of the Quasi-Geostrophic Equation with Critical Fractional Dissipation in Sobolev-Gevrey Spaces

Abstract

We study the existence of mild solutions for the Quasi-geostrophic equation with critical fractional dissipation in Sobolev-Gevrey spaces. In order to be more specific, by assuming that the initial data $\theta _{0}\in \dot{H}_{a,\sigma }^{s}(\mathbb{R}^{2})$ (with $a>0$, $\sigma > 1$, $s\in [0,1)$) is small enough, we prove that there is a unique global in time (mild) solution

$$ \theta \in \widetilde{L^{\infty }}(\mathbb{R}^{+};\dot{H}_{a,\sigma }^{s}( \mathbb{R}^{2}))\cap L^{2}(\mathbb{R}^{+};\dot{H}^{s+\frac{1}{2}}_{a, \sigma }(\mathbb{R}^{2})) $$

for this equation. Furthermore, as a consequence, we establish some decay rates for this same solution as time goes to infinity; more precisely, this work also determines the following superior limit:

$$ \displaystyle \limsup _{t\rightarrow \infty } t^{\kappa }\|\theta (t) \|_{\dot{H}_{a,\sigma }^{\kappa }(\mathbb{R}^{2})}=0, $$

for all $\kappa \geq 0$ if $s=0$, and for all $\kappa >0$ whether $s\in (0,1)$.