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

TL;DR

This paper demonstrates that, under specific noise scaling, stochastic inviscid 2D Boussinesq equations converge to a deterministic viscous system, revealing noise's regularizing and dissipative effects.

Contribution

It establishes the weak convergence of stochastic inviscid Boussinesq solutions to a viscous system, highlighting noise-induced regularization and dissipation enhancement.

Findings

Weak solutions converge to viscous Boussinesq system

Transport noise regularizes the inviscid system

Noise enhances dissipation in the limit

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 regularizes the inviscid 2D Boussinesq system and enhances dissipation in the limit.