Global existence of solutions of the stochastic incompressible non-Newtonian fluid models

Abstract

In this paper, we study the existence of solutions of stochastic incompressible non-Newtonian fluid models in $\mathbb{R}$. For the existence of solutions, we assume that the extra stress tensor $S$ is represented by $S(\mathbb{A})=\mathbb{F}(\mathbb{A})\mathbb{A}$ for $n\times n$ matrix $\mathbb{G}$. We assume that $\mathbb{F}(0)$ is uniformly elliptic matrix and \begin{align*} |{\mathbb F}({\mathbb G})|, \,\, | D {\mathbb F} ({\mathbb G})|, \,\, | D^2({\mathbb F} ({\mathbb G}) ){\mathbb G}| \leq c \quad \mbox{for all} \quad 0 < |{\mathbb G}| \leq r_0 \end{align*} for some $r_0>0$. Note that ${\mathbb{F}}_1$ and ${\mathbb{F}}_2$ for $d\in \mathbb{R}$, and ${\mathbb{F}}_3$ for $d\ge 3$ introduced in (1.2) satisfy our assumption.