Hòa Trộn Hàm Mũ của Phương Trình Navier-Stokes Ngẫu Nhiên 3D Do Tạp Âm Phần Chế Động Nhẹ Kích Hoạt

Bricmont, J., Kupiainen, A., Lefevere, R.: Exponential mixing of the 2D stochastic Navier-Stokes dynamics. Commun. Math. Phys. 230(1), 87–132 (2002). MR MR1930573 (2003i:76056) Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-dimensional Systems. London Mathematical Society Lecture Note Series, vol. 229. Cambridge University Press, Cambridge (1996). MR MR1417491 (97k:60165) Da Prato, G., Debussche, A.: Ergodicity for the 3D stochastic Navier-Stokes equations. J. Math. Pures Appl. 82(8), 877–947 (2003). MR MR2005200 (2004m:60133) Debussche, A., Odasso, C.: Markov solutions for the 3D stochastic Navier-Stokes equations with state dependent noise. J. Evol. Equ. 6(2), 305–324 (2006). MR MR2227699 (2007f:35214) Weinan, E., Mattingly, J.C., Sinai, Ya.: Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Commun. Math. Phys. 224(1), 83–106 (2001). Dedicated to Joel L. Lebowitz. MR MR1868992 (2002m:76024) Weinan, E., Mattingly, J.C.: Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation. Commun. Pure Appl. Math. 54(11), 1386–1402 (2001). MR MR1846802 (2002g:76075) Eckmann, J.-P., Hairer, M.: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Commun. Math. Phys. 219(3), 523–565 (2001). MR MR1838749 (2002d:60054) Elworthy, K.D., Li, X.-M.: Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125(1), 252–286 (1994). MR MR1297021 (95j:60087) Flandoli, F., Romito, M.: Markov selections for the 3D stochastic Navier-Stokes equations. Probab. Theory Relat. Fields 140(3–4), 407–458 (2008). MR MR2365480 (2009b:76033) Folland, G.B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn. Pure and Applied Mathematics (New York). Wiley, New York (1999). A Wiley-Interscience publication. MR MR1681462 (2000c:00001) Hairer, M., Mattingly, J.C.: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. 164(3), 993–1032 (2006). MR MR2259251 (2008a:37095) Hairer, M., Mattingly, J.C.: A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Electron. J. Probab. 16, 658–738 (2011) Kuksin, S., Shirikyan, A.: A coupling approach to randomly forced nonlinear PDE’s. I. Commun. Math. Phys. 221(2), 351–366 (2001). MR MR1845328 (2002e:35253) Kuksin, S., Shirikyan, A.: Ergodicity for the randomly forced 2D Navier-Stokes equations. Math. Phys. Anal. Geom. 4(2), 147–195 (2001). MR MR1860883 (2002i:35147) Kuksin, S.B.: Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2006). MR MR2225710 (2007h:60055) Mattingly, J.C.: Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230(3), 421–462 (2002). MR MR1937652 (2004a:76039) Nualart, D.: The Malliavin Calculus and Related Topics, 2nd edn. Probability and its Applications (New York). Springer, Berlin (2006). xiv+382 pp Norris, J.: Simplified Malliavin calculus, Séminaire de Probabilités, XX, 1984/1985. Lecture Notes in Math., vol. 1204, pp. 101–130. Springer, Berlin (1986). MR MR942019 (89f:60058) Odasso, C.: Exponential mixing for the 3D stochastic Navier-Stokes equations. Commun. Math. Phys. 270(1), 109–139 (2007). MR MR2276442 (2008e:60190) Odasso, C.: Exponential mixing for stochastic PDEs: the non-additive case. Probab. Theory Relat. Fields 140(1–2), 41–82 (2008) Röckner, M., Zhang, X.: Stochastic tamed 3D Navier-Stokes equations: existence, uniqueness and ergodicity. Probab. Theory Relat. Fields 145(1–2), 211–267 (2009). MR MR2520127 Romito, M., Xu, L.: Ergodicity of the 3d stochastic Navier-Stokes equations driven by mildly degenerate noise. Stoch. Process. Appl. 121(4), 673–700 (2011) Shirikyan, A.: Qualitative properties of stationary measures for three-dimensional Navier-Stokes equations. J. Funct. Anal. 249(2), 284–306 (2007). MR MR2345334 (2008i:35183)