Random Attractor Associated with the Quasi-Geostrophic Equation

We study the long time behavior of the solutions to the 2D stochastic quasi-geostrophic equation on \({\mathbb {T}}^2\) driven by additive noise and real linear multiplicative noise in the subcritical case (i.e. \(\alpha >\frac{1}{2}\)) by proving the existence of a random attractor. The key point for the proof is the exponential decay of the \(L^p\)-norm and a boot-strapping argument. The upper semicontinuity of random attractors is also established. Moreover, if the viscosity constant is large enough, the system has a trivial random attractor.