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 $d \geq 3$ introduced in (1.2) satisfy our assumption.
Categories: math.AP