2D Euler Equations with Stratonovich Transport Noise as a Large-Scale Stochastic Model Reduction

Abstract

The limit from an Euler-type system to the 2D Euler equations with Stratonovich transport noise is investigated. A weak convergence result for the vorticity field and a strong convergence result for the velocity field are proved. Our results aim to provide a stochastic reduction of fluid-dynamics models with three different time scales.

Stochastic Modelling of Small-Scale Perturbation

“…Therefore, it is possible to deduce a convergence result for the vorticity ω L (and as a consequence also for the velocity u L ) from a convergence result at the level of characteristics. To be precise, in [22] it is proved the following. Theorem 1.…”

“…Nevertheless, motivated by the works in [15,[19][20][21], we also point out as a possible alternative approach the so-called stochastic advection by Lie transport. According to this scheme, the evolution of u L is prescribed in a manner that is basically equivalent to the splitting of (2) at the level of vorticity, see, for instance, in [22] and Theorem 1 below.…”

“…Another strategy, that is specific for equations of transport type, may be the following. For simplicity, and having in mind Remark 1, we present the idea contained in [22] for 2D Euler equations on the two dimensional torus D = T 2 in vorticity form…”

Enhanced dissipation for stochastic Navier-Stokes equations with transport noise

“…According to arguments by separation of scales [32,33], stochastic 2D fluid equations driven by multiplicative transport noise in Stratonovich form are suitable models in fluid dynamics, see also [37,17] for variational considerations. Here, the transport noise is assumed to be spatially divergence free, replacing the incompressible flows mentioned above.…”

From additive to transport noise in 2D fluid dynamics

“…Equations (1.1) above aim to represent the small-scale component of a twodimensional incompressible fluid [1], with the additive noise and damping on the right-hand-side modelling the influence on the fluid of a possibly irregular boundary or topography. The choice of the parameter ǫ −1 in front of both noise and damping is appropriate when looking at the system with respect to the point of view of a large-scale observer, see [24] and subsection 1.2 for details. In view of this, it makes sense to couple (1.1) with a large-scale scalar dynamics:…”