Global-in-time probabilistically strong solutions to stochastic power-law equations: existence and non-uniqueness

TL;DR

This paper proves the existence of infinitely many global solutions to stochastic power-law fluid equations in dimensions three and higher for certain power indices, demonstrating non-uniqueness in law and establishing sharp conditions for uniqueness.

Contribution

It establishes the existence of multiple global solutions for stochastic power-law fluids with specific indices, highlighting non-uniqueness and sharpness of conditions in three dimensions.

Findings

Existence of infinitely many solutions for certain power indices

Non-uniqueness in law for the stochastic power-law equations

Sharp threshold for uniqueness when r ≥ (3d+2)/(d+2)

Abstract

We are concerned with the power-law fluids driven by an additive stochastic forcing in dimension $d\geq3$. For the power index $r\in(1,\frac{3d+2}{d+2})$, we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in $L^p_{loc}([0,\infty);L^2)\cap C([0,\infty);W^{1,\max\{1,r-1\}}),p\geq1$ for every divergence free initial condition in $L^2\cap W^{1,\max\{1,r-1\}}$. This result in particular implies non-uniqueness in law. Our result is sharp in the three dimensional case in the sense that the solution is unique if $r\geq \frac{3d+2}{d+2}$.