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

Abstract

We construct Hölder 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 Hölder 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 Székelyhidi, here adapted to consider the stochastic case. In particular, our novel approach allows to construct probabilistically strong solutions on $[0,\infty )$ directly.

49 pages

References in corpus (5)

• Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics
• 2D Euler equations with Stratonovich transport noise as a large scale stochastic model reduction
• From additive to transport noise in 2D fluid dynamics
• Burst of Point Vortices and Non-Uniqueness of 2D Euler Equations
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Cited by in corpus (2)

• Global existence and non-uniqueness for the Cauchy problem associated to 3D Navier-Stokes equations perturbed by transport noise
• No blow-up by nonlinear Itô noise for the Euler equations