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arXiv · 2412.08622

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

Stefanie Elisabeth Berkemeier
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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.

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Stefanie Elisabeth Berkemeier. 2026-01-22. Existence and Non-Uniqueness of Ergodic Leray-Hopf Solutions to the Stochastic Power-Law Flows. https://arxiv.org/abs/2412.08622

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