Random attractor associated with the quasi-geostrophic equation

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We study the long time behavior of the solutions to the 2D stochastic quasi-geostrophic equation on T² driven by additive noise and real linear multiplicative noise in the subcritical case (i.e. α>1/2) by proving the existence of a random attractor. The key point for the proof is the exponential decay of the Lᵖ-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.