Ergodicity of the 2D Navier-Stokes Equations with Degenerate Multiplicative Noise  

作　　者：Zhao DONG Xu-hui PENG 

机构地区：[1]RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 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 410081. China [3]School of Mathematics Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

出　　处：《Acta Mathematicae Applicatae Sinica》2018年第1期97-118,共22页应用数学学报（英文版）

基　　金：supported by the National Natural Science Foundation of China(No.11371041,11431014);the Key Laboratory of Random Complex Structures and Data Science,Academy of Mathematics and Systems Science,Chinese Academy of Sciences(No.2008DP173182);supported by NSFC(No.11501195);a Scientific Research Fund of Hunan Provincial Education Department(No.17C0953);the Youth Scientific Research Fund of Hunan Normal University(No.Math140650);the Construct Program of the Key Discipline in Hunan Province

摘　　要：Consider the two-dimensional, incompressible Navier-Stokes equations on torus T^2= [-π, π]^2 driven by a degenerate multiplicative noise in the vorticity formulation（abbreviated as SNS）： dwt = ν？w_tdt ＋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.Consider the two-dimensional, incompressible Navier-Stokes equations on torus T^2= [-π, π]^2 driven by a degenerate multiplicative noise in the vorticity formulation（abbreviated as SNS）： dwt = ν？w_tdt ＋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.

关 键 词：tochastic Navier-Stokes equation asymptotically strong Feller property ERGODICITY 

分 类 号：O1[理学—数学]