Well-posedness for a stochastic 2D Euler equation with transport noise

Abstract

We prove the existence of a unique global strong solution for a stochastic two-dimensional Euler vorticity equation for incompressible flows with noise of transport type. In particular, we show that the initial smoothness of the solution is preserved. The arguments are based on approximating the solution of the Euler equation with a family of viscous solutions which is proved to be relatively compact using a tightness criterion by Kurtz.

A pathwise parameterisation for stochastic transport

“…for any p ≥ 2, with γ as above. We used here the fact that the 2D Euler equation ( 3) has a unique global solution in W k,2 (T 2 ) for k ≥ 2, as proven in [9]. Moreover ξ is deterministic and time-independent so…”

“…and (W i ) i∈N is a sequence of independent Brownian motions. Global well-posedness for equation (1) has been studied in [9] and the numerical and data assimilation perspective has been studied in [4] and [5]. In [9] the authors have proven that equation (1) admits a uniques pathwise solution which lives in the Sobolev space…”

A Pathwise Parameterisation for Stochastic Transport

“…Global wellposedness for Eq. ( 1) has been proven in [8] and the numerical and data assimilation perspective has been studied in [4] and [5]. In [8] the authors have shown that Eq.…”

“…( 1) has been proven in [8] and the numerical and data assimilation perspective has been studied in [4] and [5]. In [8] the authors have shown that Eq. ( 1) admits a unique pathwise solution which belongs to the Sobolev space…”

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

“…[7,40,41] for some early works; recently there are some rigorous justifications of such noise in fluid equations, see [27,28] for results in 2D case and also [15] where the examples include 3D Navier-Stokes equations and primitive equations. There are several papers [8,23,9,35] on the vorticity form of stochastic 2D Euler equations with transport noise, establishing well-posedness for initial data of various regularity. 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.…”