Cauchy problem for the non-newtonian viscous incompressible fluid

Abstract

We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor $\tau ^V(\mathbb{e}) = \tau (\mathbb{e}) - 2\mu _1 \Delta \mathbb{e}$, where the nonlinear function $\tau (\mathbb{e})$ satisfies $\tau _{ij}(\mathbb{e})e_{ij} \ge c|\mathbb{e}|^p$ or $\tau _{ij}(\mathbb{e})e_{ij} \ge c(|\mathbb{e}|^2+|\mathbb{e}|^p)$. First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p > 1$ for both models. Then, under vanishing higher viscosity $\mu _1$, the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p > \frac{3n}{n+2}$, its uniqueness and regularity for $p \ge 1 + \frac{2n}{n+2}$. In the case of the second model the existence of the weak solution is proved for $p>1$.