Existence of a martingale weak solution to the Equations of Non-Stationary Motion of Non-Newtonian Fluids with a stochastic perturbation

TL;DR

This paper proves the existence of a martingale weak solution for stochastic non-Newtonian fluid equations with shear-dependent viscosity in a bounded domain, using advanced stochastic analysis techniques.

Contribution

It establishes the existence of solutions for stochastic non-Newtonian fluid equations with shear-dependent viscosity, extending previous deterministic results to stochastic settings.

Findings

Existence of a martingale weak solution under specified growth conditions.

Application of pressure decomposition adapted to stochastic equations.

Use of stochastic compactness and truncation methods for proof.

Abstract

In this paper, we consider the stochastic %equations of incompressible non-Newtonian fluids driven by a cylindrical Wiener process $W$ with shear rate dependent on viscosity in a bounded Lipschitz domain $D\in \mathbb{R}^n$ during the time interval $(0,T)$. For $q>\frac{2n+2}{n+2}$ in the growth conditions (1.2), we prove the existence of a martingale weak solution with $\nabla\cdot u=0$ by using a pressure decomposition which is adapted to the stochastic setting, the stochastic compactness method and the $L^\infty$-truncation.