Non-uniqueness in law of stochastic 3D Navier--Stokes equations

Abstract

We consider the stochastic Navier--Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear multiplicative and a nonlinear noise of cylindrical type, all driven by a Wiener process. In these settings, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval $[0,\infty )$. Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, non-uniqueness in law holds on an arbitrary time interval $[0,T]$, $T>0$.

80 pages

References in corpus (1)

• On ill- and well-posedness of dissipative martingale solutions to stochastic 3D Euler equations

Cited by in corpus (3)

• Remarks on the non-uniqueness in law of the Navier-Stokes equations up to the J.-L. Lions' exponent
• On ill- and well-posedness of dissipative martingale solutions to stochastic 3D Euler equations
• Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier--Stokes equations: existence and non-uniqueness