Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise

TL;DR

This paper constructs global-in-time solutions to 3D Euler equations with transport noise, demonstrating non-uniqueness and prescribing energy, using a flow transformation and convex integration adapted for stochastic equations.

Contribution

It introduces a novel method combining flow transformation and convex integration to construct strong solutions and show non-uniqueness for stochastic 3D Euler equations.

Findings

Existence of global-in-time H"older continuous solutions with prescribed energy.

Non-uniqueness of solutions from different initial conditions.

Development of a new approach for stochastic Euler equations.

Abstract

We construct H\"older continuous, global-in-time probabilistically strong solutions to 3D Euler equations perturbed by Stratonovich transport noise. Kinetic energy of the solutions can be prescribed a priori up to a stopping time, that can be chosen arbitrarily large with high probability. We also prove that there exist infinitely many H\"older continuous initial conditions leading to non-uniqueness of solutions to the Cauchy problem associated with the system. Our construction relies on a flow transformation reducing the SPDE under investigation to a random PDE, and convex integration techniques introduced in the deterministic setting by De Lellis and Sz\'ekelyhidi, here adapted to consider the stochastic case. In particular, our novel approach allows to construct probabilistically strong solutions on $[0,\infty)$ directly.