Ergodicity of the 2D Navier-Stokes equations with degenerate multiplicative noise |
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| Authors: | Zhao?Dong,Xu-hui?Peng mailto:pengxuhui@amss.ac.cn" title=" pengxuhui@amss.ac.cn" itemprop=" email" data-track=" click" data-track-action=" Email author" data-track-label=" " >Email author |
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| Affiliation: | 1.RCSDS, Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing,China;2.Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science,Hunan Normal University,Changsha,China;3.School of Mathematics Sciences,University of Chinese Academy of Sciences,Beijing,China |
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| Abstract: | ![]()
Consider the two-dimensional, incompressible Navier-Stokes equations on torus T2 = [?π, π]2 driven by a degenerate multiplicative noise in the vorticity formulation (abbreviated as SNS): dw t = νΔw t dt + B(Kw t ,w t )dt + Q(w t )dW t . We prove that the solution to SNS is continuous differentiable in initial value. We use the Malliavin calculus to prove that the semigroup {P t }t≥0 generated by the SNS is asymptotically strong Feller. Moreover, we use the coupling method to prove that the solution to SNS has a weak form of irreducibility. Under almost the same Hypotheses as that given by Odasso, Prob. Theory Related Fields, 140: 41–82 (2005) with a different method, we get an exponential ergodicity under a stronger norm. |
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