Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Stefanie Elisabeth Berkemeier

TL;DR

This work addresses the existence and non-uniqueness of Leray–Hopf solutions for stochastic power-law flows with shear-thinning behavior in dimensions $d\ge 3$, where the power-law index satisfies $\iota\in\left(1, \frac{2d}{d+2}\right)$. It develops a stochastic convex integration framework augmented by a novel auxiliary energy functional to enforce the energy inequality and achieve global-in-time control, yielding infinitely many probabilistically strong Leray–Hopf solutions and, via Krylov–Bogoliubov and Krein–Milman, infinitely many ergodic stationary solutions. The paper extends the non-uniqueness paradigm to stochastic non-Newtonian fluids in this intermediate index range, providing regularity, energy estimates, and ergodic properties for the constructed solutions. The results significantly advance understanding of long-time behavior, ergodicity, and energy dissipation for stochastic shear-thinning flows and open avenues for further exploration of ergodic theory in non-Newtonian SPDEs.

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.

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

TL;DR

This work addresses the existence and non-uniqueness of Leray–Hopf solutions for stochastic power-law flows with shear-thinning behavior in dimensions , where the power-law index satisfies . It develops a stochastic convex integration framework augmented by a novel auxiliary energy functional to enforce the energy inequality and achieve global-in-time control, yielding infinitely many probabilistically strong Leray–Hopf solutions and, via Krylov–Bogoliubov and Krein–Milman, infinitely many ergodic stationary solutions. The paper extends the non-uniqueness paradigm to stochastic non-Newtonian fluids in this intermediate index range, providing regularity, energy estimates, and ergodic properties for the constructed solutions. The results significantly advance understanding of long-time behavior, ergodicity, and energy dissipation for stochastic shear-thinning flows and open avenues for further exploration of ergodic theory in non-Newtonian SPDEs.

Abstract

We study long time behavior of shear-thinning fluid flows in dimensions, driven by additive stochastic forcing of trace class, with power-law indices ranging from to . 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.

Paper Structure

This paper contains 54 sections, 18 theorems, 236 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Motivation and Previous Works
  3. Main Result
  4. Strategy of the Proof
  5. Organization of the Paper
  6. Preliminaries
  7. Function Spaces
  8. Continuous Functions
  9. Bessel Potential Spaces
  10. Operators
  11. Leray Projection and Fourier cut-offs
  12. Right-Inverse of Divergence
  13. Non-Newtonian Tensor
  14. Existence & Non-Uniqueness of Leray--Hopf Solutions
  15. Factorization
  16. ...and 39 more sections

Key Result

Theorem 1.7

For a certain initial value, $\iota\in(1,\frac{2d}{d+2})$ and sufficiently small $\gamma \in (0,1)$ there exist infinitely many ergodic Leray--Hopf solutions to PLF, which belong $\mathcal{P}$-a.s. to the class

Figures (2)

  • Figure 1: Relationships among the components of the next iteration step $v_{q+1}$.
  • Figure 2: Construction of the diagonal sequence $\left( d_n\right)_{n \in \mathbb{N}}$, which consists of the elements in the pink boxes.

Theorems & Definitions (41)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 31 more