Sub and supercritical stochastic quasi-geostrophic equation. (English) Zbl 1322.60121
The quasi-geostrophic equation is an important model in geophysical fluid dynamics. This paper investigates the following stochastic quasi-geostrophic equation on the two-dimensional torus \(\mathbb T^2\):
\[
\dot \theta(t) =-u(t)\cdot\nabla \theta(t) -\kappa (-\Delta)^\alpha\theta(t) +G(\theta)\eta(t),\;\;\theta(0): \mathbb T^2\to \mathbb R,
\]
where \(\alpha>0\) is a constant, \(\eta\) is a Gaussian random field with white noise in time, \(u\) is determined by \(\theta\) via
\[
u= (u_1,u_2) = (-R_2 \theta, R_1\theta),
\]
where \(R_j\) is the \(j\)-th periodic Riesz transform. The existence of a weak solution and the Markov selection are derived for \(\alpha\in (0,1)\), the existence and uniqueness (i.e. well-posedness) of strong solutions is proved for \(\alpha\in (\frac 1 2,1)\). Moreover, sufficient conditions for the ergodicity, exponential ergodicity and the law of large numbers for the solution are presented.
Reviewer: Feng-Yu Wang (Swansea)
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 60G60 | Random fields |
| 60G15 | Gaussian processes |
| 60F15 | Strong limit theorems |
| 35Q86 | PDEs in connection with geophysics |
Keywords:
stochastic quasi-geostrophic equation; well-posedness; Gaussian random field; Markov selection; strong Feller property; ergodicity; law of large numbersReferences:
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