Sub- and supercritical stochastic quasi-geostrophic equation

Details Sub- and supercritical stochastic quasi-geostrophic equation Sub- and supercritical stochastic quasi-geostrophic equation

2011-10-10

arxiv.org/abs/1110.1984v3

Mathematics

Probability

Scientific paper

In this paper we study the 2D stochastic quasi-geostrophic equation on $\mathbb{T}^2$ for general parameter $\alpha\in (0,1)$ and multiplicative noise. We prove the existence of weak solutions for additive noise, the existence of martingale solutions and Markov selections for multiplicative noise and under some condition pathwise uniqueness for all $\alpha\in (0,1)$ . In the subcritical case $\alpha>1/2$, we prove existence and uniqueness of (probabilistically) strong solutions. In particular, we prove ergodicity provided the noise is non-degenerate for $\alpha>2/3$. In this case, the convergence to the (unique) invariant measure is exponentially fast.

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