Ergodicity in randomly forced Rayleigh–Bénard convection

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 P r > 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-Doering [CD96] can be applied in our stochastic setting, and prove bounds on the Nusselt number relative to the unique invariant measure.

Invariant Measures for Nonlinear Conservation Laws Driven by Stochastic Forcing

“…In particular, Fehrman-Gess [48] investigated the well-posedness and continuous dependence of the stochastic degenerate parabolic equations of porous medium type, including the cases with fast diffusion and heterogeneous fluxes. By using the methods developed in [57,64,65,74] and developing the probabilistic Gronwall inequality based on delicate reasoning about a stopping time, such MHD equations driven by additive noise of zero spatial average in the vanishing Rossby number and vanishing magnetic Reynold's number limit were also shown to have a unique invariant measure (that is necessarily ergodic) in [56].…”

Generalized couplings and ergodic rates for SPDEs and other Markov models

“…Finally, we consider Boussinesq approximation for the Rayleigh-Bénard convection perturbed by additive noise. The physical motivation behind the model as well as the relevance of the model for fluid dynamics are explained in detail in [16]. Consider the following system of equations evolving on a domain D :…”

“…It was shown in [15, Theorem 1.1] that if the velocity u and temperature T satisfies periodic boundary conditions, then the process (u, T ) is exponentially ergodic. However, as was noted in [16], "using periodic boundary conditions in the vertical directions is not appropriate from the physical point of view". Therefore, following [16], we equip the system (4.45)-(4.46) with mixed periodic and non-homogeneous Dirichlet boundary conditions that are physically relevant.…”

“…We use the coupling construction, which is a small modification of the construction from [16,Section 5.2]. Fix x, y ∈ H. We take X x,y := (u x , Θ x ) and let Y x,y := ( u, Θ) be the solution to the same system of equations with the initial condition y and the additional control terms:…”

Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations

“…More generally, note that stochastic perturbations acting in the temperature equation as in (5.49) below has a significant physical motivation as a model for radiogenic heating and other volumetric heat sources driving turbulent convection. See [SH77,STO01,FGHRW16,FGHRW17].…”