Existence and non-uniqueness of weak solutions with continuous energy to the 3D deterministic and stochastic Navier-Stokes equations

TL;DR

This paper demonstrates the existence of infinitely many weak solutions with continuous energy profiles for the 3D Navier-Stokes equations, both deterministic and stochastic, using a novel backward convex integration approach.

Contribution

It introduces a new backward convex integration scheme and constructs solutions with continuous energy, addressing non-uniqueness in 3D Navier-Stokes equations.

Findings

Existence of infinitely many weak solutions with continuous energy profiles.

Construction of probabilistically strong solutions in the stochastic case.

Introduction of a new time-dependent frequency truncated NSE for solution approximation.

Abstract

The continuity of the kinetic energy is an important property of incompressible viscous fluid flows. We show that for any prescribed finite energy divergence-free initial data there exist infinitely many global in time weak solutions with smooth energy profiles to both the 3D deterministic and stochastic incompressible Navier-Stokes equations. In the stochastic case the constructed solutions are probabilistically strong. Our proof introduces a new backward convex integration scheme with delicate selections of initial relaxed solutions, backward time intervals, and energy profiles. Our initial relaxed solutions satisfy a new time-dependent frequency truncated NSE, different from the usual approximations as it decreases the large Reynolds error near the initial time, which plays a key role in the construction.