Stochastic power law fluids: Existence and uniqueness of weak solutions

TL;DR

This paper studies a stochastic PDE modeling non-Newtonian fluid flow with polynomial stress tensor and random forcing, establishing conditions for the existence and uniqueness of weak solutions.

Contribution

It provides the first rigorous proof of existence and uniqueness of weak solutions for this class of stochastic non-Newtonian fluid models.

Findings

Proved existence of weak solutions under certain conditions.

Established uniqueness of solutions in the stochastic setting.

Analyzed the impact of polynomial stress tensors and colored noise.

Abstract

We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the rate of strain tensor, while the colored noise is considered as a random force. We investigate the existence and the uniqueness of weak solutions to this SPDE.