Closed Estimates of Leray Projected Transport Noise and Strong Solutions of the Stochastic Euler Equations

TL;DR

This paper establishes the existence of local strong solutions for stochastic Euler equations with transport noise on a 3D torus, overcoming challenges posed by the Leray projector in energy estimates.

Contribution

It introduces novel energy control techniques for stochastic Euler equations with transport noise, leading to the first proof of local strong solutions in this setting.

Findings

Existence of local strong solutions for stochastic Euler equations.

Energy estimates controlled despite Leray projector complications.

Solutions exist until potential blow-up in specific Sobolev norms.

Abstract

We consider the incompressible Euler and Navier-Stokes equations on the three dimensional torus, in velocity form, perturbed by a transport or transport-stretching Stratonovich noise. Closed control of the noise contributions in energy estimates are demonstrated, for any positive integer ordered Sobolev Space and the equivalent Stokes Space; difficulty arises due to the presence of the Leray Projector disrupting cancellation of the top order derivative. This is particularly pertinent in the case of a transport noise without stretching, where the vorticity form cannot be used. As a consequence we obtain, for the first time, the existence of a local strong solution to the corresponding stochastic Euler equation. Furthermore, smooth solutions are shown to exist until blow-up in $L^1\left([0,T];W^{1,\infty}\right)$.