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

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.