Nonuniqueness in law for stochastic hypodissipative Navier-Stokes equations

TL;DR

This paper demonstrates the nonuniqueness of probabilistic weak solutions for stochastic hypodissipative Navier-Stokes equations with dissipation exponent less than 1/2, using convex integration and Beltrami waves.

Contribution

It introduces a novel convex integration approach employing Beltrami waves to construct nonunique solutions in a stochastic setting for hypodissipative Navier-Stokes equations.

Findings

Existence of initial conditions with multiple solutions.

Construction of solutions violating energy inequalities.

First use of Beltrami waves in stochastic convex integration.

Abstract

We study the incompressible hypodissipative Navier-Stokes equations with dissipation exponent $0 < \alpha < \frac{1}{2}$ on the three-dimensional torus perturbed by an additive Wiener noise term and prove the existence of an initial condition for which distinct probabilistic weak solutions exist. To this end, we employ convex integration methods to construct a pathwise probabilistically strong solution, which violates a pathwise energy inequality up to a suitable stopping time. This paper seems to be the first in which such solutions are constructed via Beltrami waves instead of intermittent jets or flows in a stochastic setting.