Non-uniqueness in law of three-dimensional Navier-Stokes equations diffused via a fractional Laplacian with power less than one half

TL;DR

This paper proves non-uniqueness in law for three-dimensional Navier-Stokes equations with fractional Laplacian diffusion of power less than one-half, using novel regularity and stochastic techniques.

Contribution

It establishes non-uniqueness in law for Navier-Stokes with fractional diffusion below half power, employing new regularity and stochastic analysis methods.

Findings

Non-uniqueness in law holds for fractional Laplacian power less than one-half.

Constructed solutions have Hölder regularity with small exponent.

Non-uniqueness persists at Leray-Hopf regularity level for small powers.

Abstract

Non-uniqueness of three-dimensional Euler equations and Navier-Stokes equations forced by random noise, path-wise and more recently even in law, have been proven by various authors. We prove non-uniqueness in law of the three-dimensional Navier-Stokes equations forced by random noise and diffused via a fractional Laplacian that has power between zero and one half. The solution we construct has H$\ddot{\mathrm{o}}$lder regularity with a small exponent rather than Sobolev regularity with a small exponent. For the power sufficiently small, the non-uniqueness in law holds at the level of Leray-Hopf regularity. In particular, in order to handle transport error, we consider phase functions convected by not only a mollified velocity field but a sum of that with a mollified Ornstein-Uhlenbeck process if noise is additive and a product of that with a mollified exponential Brownian motion if noise is multiplicative.