Ergodicity in randomly forced Rayleigh–Bénard convection. (9th September 2016)

• Record Type: Journal Article
• Authors: Földes, J; Glatt-Holtz, N E; Richards, G; Whitehead, J P
• Is Part Of: Nonlinearity. Volume 29:Number 11(2016:Nov.)
• Page Start: 3309
• Page End: 3345
• Publication Date: 2016-09-09
• Subjects: ergodicity -- turbulent convection -- stochastic partial differential equations -- upper bound theory -- asymptotic coupling; 60H15 -- 76D06 -- 37L55; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/11/3309
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We consider the Boussinesq approximation for Rayleigh–Bénard convection perturbed by an additive noise and with boundary conditions corresponding to heating from below. In two space dimensions, with sufficient stochastic forcing in the temperature component and large Prandtl number Pr > 0, we establish the existence of a unique ergodic invariant measure. In three space dimensions, we prove the existence of a statistically invariant state, and establish unique ergodicity for the infinite Prandtl Boussinesq system. Throughout this work we provide streamlined proofs of unique ergodicity which invoke an asymptotic coupling argument, a delicate usage of the maximum principle, and exponential martingale inequalities. Lastly, we show that the background method of Constantin and Doering (1996 Nonlinearity 9 1049–60) can be applied in our stochastic setting, and prove bounds on the Nusselt number relative to the unique invariant measure.