Global existence of solutions of the stochastic incompressible non-Newtonian fluid models
Tongkeun Chang, Minsuk Yang
TL;DR
The paper analyzes the stochastic incompressible non-Newtonian fluid model on ${\mathbb R}^n$ with a stress structure ${\mathbb F}({\mathbf{D}}u){\mathbf{D}}u$ and random forcing. Using a Besov-space framework with time weights and a truncation-based fixed-point scheme, the authors prove a local existence and uniqueness result for small initial data and forcing data, obtaining ${\mathbb P}(\tau=\infty) \ge 1-\varepsilon$. The approach hinges on sharp stochastic Stokes estimates for the linear part and careful nonlinear bounds for the truncated nonlinearities, together with a stopping-time argument to convert local existence into high-probability global existence. This work extends deterministic non-Newtonian results to the stochastic setting in standard Besov spaces and provides a rigorous foundation for stochastic non-Newtonian flows in unbounded domains. The results are constructed in a framework that accommodates representative stress laws ${\mathbb F}_1, {\mathbb F}_2$, and ${\mathbb F}_3$ and enables a path toward broader stochastic well-posedness for complex fluids.
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 $d \geq 3$ introduced in (1.2) satisfy our assumption.