Scaling Limits for Stochastic SQG Equations and 2D Inviscid Critical Boussinesq Equations with Transport Noises

TL;DR

This paper investigates how solutions to stochastic SQG and Boussinesq equations with transport noise behave under scaling, showing convergence to deterministic dissipative equations, thus revealing the noise's limiting effects.

Contribution

It establishes weak convergence of solutions with transport noise to deterministic dissipative equations under specific scaling limits, extending understanding of stochastic fluid models.

Findings

Solutions converge to deterministic equations under scaling

Weak convergence results for stochastic SQG and Boussinesq equations

Noise effects diminish in the scaling limit

Abstract

We study stochastic SQG equations on the torus $\mathbb{T}^2$ with multiplicative transport noise in the $L^2$-setting. Under a suitable scaling of the noise, we show that the solutions converge weakly to the unique solution to the deterministic dissipative SQG equation. A similar scaling limit result is proved also for the stochastic 2D inviscid critical Boussinesq equations.