Sub-critical and critical stochastic quasi-geostrophic equations with infinite delay. (English) Zbl 1471.35274
Summary: In this paper, we investigate a stochastic fractionally dissipative quasi-geostrophic equation driven by a multiplicative white noise, whose external forces contain hereditary characteristics. The existence and uniqueness of both local martingale and local pathwise solutions are established in \(H^s\) with \(s\geq 2-2\alpha\), where \(\alpha\in(\frac{1}{2},1)\). For the critical case \(\alpha=\frac{1}{2}\), we obtain the similar results in \(H^s\) with \(s>1\).
MSC:
| 35Q86 | PDEs in connection with geophysics |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60H40 | White noise theory |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 86A05 | Hydrology, hydrography, oceanography |
| 86A10 | Meteorology and atmospheric physics |
| 35R07 | PDEs on time scales |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35R11 | Fractional partial differential equations |
Keywords:
stochastic quasi-geostrophic equation; fractional Laplacian; infinite delay; martingale solution; pathwise solutionReferences:
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