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

In this paper, we study the existence of solutions of stochastic incompressible non-Newtonian fluid models in R. For the existence of solutions, we assume that the extra stress tensor S is represented by S( A) = F ( A) A for n × n matrix G. We assume that F(0) is uniformly elliptic matrix and {align*} |{ F}({ G})|, \,\, | D { F} ({ G})|, \,\, | D^2({ F} ({ G}) ){ G}| ≤ c for all 0 < |{ G}| ≤ r_0 {align*} for some r₀ > 0. Note that F₁ and F₂ for d ∈ R, and F₃ for d ≥ 3 introduced in (1.2) satisfy our assumption.