Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier–Stokes equations: Existence and nonuniqueness

Abstract

We are concerned with the three-dimensional incompressible Navier–Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in ${\mathit{L}}^2$ we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result, in particular, implies nonuniqueness in law. Second, we prove nonuniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain nonuniqueness of the associated semiflows.