Non-uniqueness of Leray--Hopf solutions for the $3D$ fractional Navier--Stokes equations perturbed by transport noise

TL;DR

This paper demonstrates the non-uniqueness of Leray--Hopf solutions for 3D fractional Navier--Stokes equations with transport noise, using convex integration and a flow transformation, without additional deterministic forces.

Contribution

It proves the first non-uniqueness result for fractional Navier--Stokes equations with stochastic perturbations in the unforced case.

Findings

Existence of infinitely many solutions starting from the same initial data.

Solutions are globally defined, Hölder continuous, and satisfy energy inequalities.

Non-uniqueness holds for any stochastic perturbation without external forcing.

Abstract

For the $3D$ fractional Navier--Stokes equations perturbed by transport noise, we prove the existence of infinitely many H\"older continuous analytically weak, probabilistically strong Leray--Hopf solutions starting from the same deterministic initial velocity field. Our solutions are global in time and satisfy the energy inequality pathwise on a non-empty random interval $[0,\tau]$. In contrast to recent related results, we do not consider an additional deterministic suitably chosen force $f$ in the equation. In this unforced regime, we prove the first result of Leray--Hopf nonuniqueness for fractional Navier--Stokes equations with any kind of stochastic perturbation. Our proof relies on convex integration techniques and a flow transformation by which we reformulate the SPDE as a PDE with random coefficients.