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

Zhao DONG^{1,3}, Xu-hui PENG^{2}

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

Abstract 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):dw_{t}=ν△w_{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.

Fund:Zhao Dong was supported by the National Natural Science Foundation of China (No.11371041, 11431014) and the Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (No.2008DP173182). Xuhui Peng was 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) and the Construct Program of the Key Discipline in Hunan Province.

Albeverio, S., Debussche, A., Xu, L. Exponential mixing of the 3d stochastic navier-stokes equations driven by mildly degenerate noises. Applied Mathematics and Optimization, 66(2):273-308(2012)

[2]

Brzézniak, Z., Liu, W., Zhu, J.H. Strong solutions for spde with locally monotone coefficients driven by lévy noise. Nonlinear Analysis Real World Applications, 17(6):283-310(2014)

[3]

Brzézniak, Z., Hausenblas, E., Zhu, J.H. 2d stochastic navier-stokes equations driven by jump noise. Nonlinear Analysis, 79(1):122-139(2013)

[4]

Constantin, P., Foias, C. Navies-Stokes equations. University of Chicago Press, Chicago, 1988

[5]

Dong, Z., Xie, Y. Ergodicity of stochastic 2d navier-stokes equation with lévy noise. Journal of Differential Equations, 251(1):196-222(2011)

[6]

Dong, Z., Xu, L., Zhang, X. Invariant measures of stochastic 2d navier-stokes equation, driven by α-stable processes. Electronic Communications in Probability, 16(2011)

[7]

E, W., Mattingly, J.C. Ergodicity for the navier-stokes equation with degenerate random forcing:finitedimensional approximation. Communications on Pure and Applied Mathematics, 54(11):1386-1402(2010)

[8]

E, W., Mattingly, J.C., Sinai, Y. Gibbsian dynamics and ergodicity for the stochastically forced navierstokes equation. Communications in Mathematical Physics, 224(1):83-106(2001)

[9]

Goldys, B., Maslowski, B. Exponential ergodicity for stochastic burgers and 2d navier-stokes equations. Journal of Functional Analysis, 226(1):230-255(2005)

[10]

Gawarecki, L., Mandrekar, V. Stochastic Differential Equations in Infinite Dimensions. Springer, Berlin, Heidelberg, 2010

[11]

Hairer, M., Mattingly, J.C. Ergodicity of the 2d navier-stokes equations with degenerate stochastic forcing. Annals of Mathematics, 164(3):993-1032(2006)

[12]

Hairer, M., Mattingly, J.C. Spectral gaps in wasserstein distances and the 2d stochastic navier-stokes equations. Annals of Probability, 36(6):2050-2091(2006)

[13]

Hairer, M., Mattingly, J.C. A theory of hypoellipticity and unique ergodicity for semilinear stochastic pdes. Electronic Journal of Probability, 16(2):658-738(2008)

[14]

Kuksin, S., Shirikyan, A. Coupling approach to white-forced nonlinear pdes. Journal De Mathématiques Pures Et Appliquées, 81(6):567-602(2002)

[15]

Kuksin, S., Shirikyan, A. A coupling approach to randomly forced nonlinear pde's. I. Communications in Mathematical Physics, 221(2):351-366(2001)

[16]

Komorowski, T., Peszat, S., Szarek, T. On ergodicity of some markov processes. Annals of Probability, 38(4):1401-1443(2010)

[17]

Mattingly, J.C. Exponential convergence for the stochastically forced navier-stokes equations and other partially dissipative dynamics. Communications in Mathematical Physics, 230(3):421-462(2002)

[18]

Mattingly, J.C. The dissipative scale of the stochastics navier-stokes equation:regularization and analyticity. Journal of Statistical Physics, 108(5):1157-1179(2002)

[19]

Mohammed, S., Zhang, T. Dynamics of stochastic 2d navier-stokes equations. Journal of Functional Analysis, 258(10):3543-3591(2010)

[20]

Nualart, David. Malliavin Calculus and Related Topics. Springer, Berlin, Heidelberg, 2006

[21]

Odasso, C. Exponential mixing for the 3d stochastic navier-stokes equations. Communications in Mathematical Physics, 270(1):109-139(2007)

[22]

Odasso, C. Exponential mixing for stochastic pdes:the non-additive case. Probability Theory and Related Fields, 140(1):41-82(2005)

[23]

Röckner, M., Zhang, X. Stochastic tamed 3d navier-stokes equations:existence, uniqueness and ergodicity. Probability Theory and Related Fields, 145(1):211(2010)

[24]

Xu, L., Zegarlinski, B. Existence and exponential mixing of infinite white α-stable systems with unbounded interactions. Electronic Journal of Probability, 15(28):1994-2018(2009)