Convergence of stochastic 2D inviscid Boussinesq equations with transport noise to a deterministic viscous system

Abstract

The inviscid 2D Boussinesq system with thermal diffusivity and multiplicative noise of transport type is studied in the L 2-setting. It is shown that, under a suitable scaling of the noise, weak solutions to the stochastic 2D Boussinesq equations converge weakly to the unique solution of the deterministic viscous Boussinesq system. Consequently, the transport noise asymptotically regularises the inviscid 2D Boussinesq system and enhances dissipation in the limit.

LDP and CLT for SPDEs with transport noise

“…The scaling limit of (1.1) to (1.2) was first observed in [52] for linear transport equation and has then been extended to various nonlinear PDEs, including stochastic 2D Euler equation in vorticity form [37], stochastic mSQG equation [73] and stochastic 2D inviscid Boussinesq system [71]; see also [38] for a large class of nonlinearities which include as particular cases the 3D Keller-Segel system and the 2D Kuramoto-Sivashinsky equation. All of the aforementioned results are set on the torus T d = R d /Z d ; some results on stochastic heat equations in bounded domain or infinite channel can be found in [39,46].…”

Global existence for the stochastic Boussinesq equations with transport noise and small rough data

“…In [39], together with Yue, the third author of this paper proved almost-sure global existence of weak solutions to the Boussinesq equations using random data approach. Luo [34] recently considered 2D Boussinesq equations with transport noise and Alonso-Orán and Bethencourt de León proved global well-posedness with transport noise and Sobolev initial data [6]. See also [25] for the stochastic fractional Boussinesq equations.…”

Stochastic inviscid Leray-$α$ model with transport noise: convergence rates and CLT

“…In recent years, there have been intensive investigations on scaling limits of SPDEs with transport noise to deterministic limit equations with an additional viscous term, see e.g. [30,19,37] and also the earlier work [24] where the limit equation is driven by additive space-time white noise. The enhanced dissipation was used in [25,20] to suppress possible blow-up of solutions to some nonlinear equations.…”