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

Quantitative convergence rates for scaling limit of SPDEs with transport noise

“…• The Stratonovich formulation (1.17), together with the divergence free structure of the noise and the variational approach, are the right tools to derive a priori estimates on ω; this step is a mixture of analytic and probabilistic techniques and has already been developed for several PDEs in [15,12,22,23,13].…”

“…Second, for some nonlinear inviscid fluid equations with multiplicative transport noise, we already know that, under a suitable scaling of the noise, the stochastic equations converge weakly to some deterministic viscous equation, see e.g. [15,12,22]. These results, similarly, may be interpreted as the emergence of an eddy viscosity, in a fluid with small scale turbulence.…”

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

“…For instance, linear transport equation perturbed by transport noise was considered in [22], where the limit equation is a second order deterministic parabolic equation. Later on, similar results are proved for stochastic 2D Euler, mSQG and inviscid Boussinesq equations with transport noises, see [14,28,27]. This idea has also been applied to some dissipative systems, showing the phenomenon of dissipation enhancement, see [19] for the vorticity form of 3D Navier-Stokes equations and [15] for some general nonlinear equations whose solutions might explode for initial data above some threshold.…”

“…Our next result is concerned with the 2D inviscid Boussinesq equations describing the evolution of an incompressible fluid, subject to a vertical force which is proportional to some scalar field, such as the temperature. This part is motivated by [27], where one can find some references on various well posedness results on the equations. In the current paper, we consider the critical case as in Subsection 1.1.…”

“…Analogous to the discussion of Theorem 3.8, the similar result can be obtained for the scaling limit of stochastic Boussinesq equations (cf. Theorem 1.2 in [27]). For (ξ 0 , ω 0 ) ∈ (L(T 2 )) 2 , define (ξ 0 , ω 0 ) L 2 = ξ 0 L 2 ∨ ω 0 L 2 ; moreover, given θ N ∈ ℓ 2 , we denote by C θ N (ξ 0 , ω 0 )= collection of laws of weak solutions (ξ, ω) to (1.7) with initial data (ξ 0 , ω 0 ).…”