On the 3D Navier-Stokes Equations with a Linear Multiplicative Noise and Prescribed Energy

TL;DR

This paper constructs weak and strong solutions to the 3D Navier-Stokes equations with linear multiplicative noise and prescribed energy, demonstrating non-uniqueness through convex integration techniques.

Contribution

It introduces a novel convex integration approach to generate solutions with prescribed energy in stochastic Navier-Stokes equations, highlighting non-uniqueness.

Findings

Existence of weak and strong solutions with prescribed energy

Solutions are valid up to large stopping times

Non-uniqueness of solutions under certain conditions

Abstract

For a prescribed deterministic kinetic energy we use convex integration to construct analytically weak and probabilistically strong solutions to the 3D incompressible Navier-Stokes equations driven by a linear multiplicative stochastic forcing. These solutions are defined up to an arbitrarily large stopping time and have deterministic initial values, which are part of the construction. Moreover, by a suitable choice of different kinetic energies which coincide on an interval close to time 0, we obtain non-uniqueness.