Stochastic quasi-geostrophic equation. (English) Zbl 1248.60074
The authors announce some results on 2D stochastic quasi-geostrophic equation in \(\mathbb T^{2}\) for general parameter \(\alpha \in (0, 1)\) and multiplicative noise. They prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all \(\alpha \in (0, 1)\). In the subcritical case \(\alpha > 1/2\), they prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for \(\alpha > 2/3\) provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast. In the general case, they prove the existence of Markov selections.
Reviewer: Zenghu Li (Beijing)
MSC:
60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
60H30 | Applications of stochastic analysis (to PDEs, etc.) |
35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
stochastic quasi-geostrophic; well posedness; martingale problem; Markov property; strong Feller property; Markov selections; ergodicityReferences:
[1] | DOI: 10.1007/s00028-006-0254-y · Zbl 1110.35110 · doi:10.1007/s00028-006-0254-y |
[2] | DOI: 10.1017/CBO9780511662829 · Zbl 0849.60052 · doi:10.1017/CBO9780511662829 |
[3] | DOI: 10.1007/BF01192467 · Zbl 0831.60072 · doi:10.1007/BF01192467 |
[4] | DOI: 10.1007/s00440-007-0069-y · Zbl 1133.76016 · doi:10.1007/s00440-007-0069-y |
[5] | DOI: 10.1016/j.jfa.2004.12.009 · Zbl 1078.60049 · doi:10.1016/j.jfa.2004.12.009 |
[6] | DOI: 10.1016/j.spa.2008.08.009 · Zbl 1177.60060 · doi:10.1016/j.spa.2008.08.009 |
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