Existence and Non-Uniqueness of Ergodic Leray-Hopf Solutions to the Stochastic Power-Law Flows

TL;DR

This paper demonstrates the existence of infinitely many ergodic Leray-Hopf solutions for stochastic shear-thinning fluid flows with power-law indices in a specific range, using convex integration techniques.

Contribution

It introduces a new energy functional and applies convex integration to construct multiple solutions, including ergodic ones, for stochastic shear-thinning fluids.

Findings

Existence of infinitely many Leray-Hopf solutions.

Construction of ergodic solutions with probabilistic strength.

First such solutions for this class of stochastic fluids.

Abstract

We study long time behavior of shear-thinning fluid flows in $d \geq 3$ dimensions, driven by additive stochastic forcing of trace class, with power-law indices ranging from $1$ to $ \frac{2d}{d+2}$. We particularly focus on Leray-Hopf solutions, i.e. on analytically weak solutions satisfying energy inequality. Introducing a new kind of energy related functional into the technique of convex integration enables the construction of infinitely many such solutions that are probabilistically strong for a certain initial value. Furthermore, we provide global i time estimates which lead to the existence of infinitely many stationary and even ergodic Leray--Hopf solutions. These results represent the first construction of Leray-Hopf solutions in the framework of stochastic shear-thinning fluids within this range of power-law indices.