Global existence of non-Newtonian incompressible fluids in half space with nonhomogeneous initial-boundary data

Abstract

In this study, we investigate the global existence of weak solutions of non-Newtonian incompressible fluids governed by (1.1). When $u_0 \in \dot B^{\alpha-\frac{2}{p}}_{p,q}({\mathbb R}^{n}_+) \, \cap \,\dot B^{ 1 -\frac4{n+2}}_{\frac{n+2}2,\frac{n+2}2}({\mathbb R}^{n}_+) \,\cap \, \dot B^{1 +\frac{n}p}_{p,1} (\mathbb{R}_+)$ is given, we will find the weak solutions for the equation (1.1) in the function space $C_b ([ 0, \infty; \dot B^{\alpha -\frac2p}_{p,q} ({\mathbb R}^n_+)) \cap C_b (0, \infty; \dot B^{1 -\frac4{n+2}}_{\frac{n+2}2} (\mathbb{R}_+)) \cap L^\infty(0, \infty; \dot W^1_\infty(\mathbb{R}_+))$, $ n+2 < p < \infty, \,\, 1 \leq q \leq \infty, \,\, 1 + \frac{n+2}p < \alpha < 2$. We show the existence of weak solutions in the anisotropic Besov spaces $\dot B^{\alpha, \frac{\alpha}2}_{p,q} (\mathbb{R}_+ \times (0, \infty))$ (see Theorem (1.2)) and we show the embedding $\dot B^{\alpha, \frac{\alpha}2}_{p,q} (\mathbb{R}_+ \times (0, \infty) \subset C_b ([ 0, \infty; \dot B^{\alpha -\frac2p}_{p,q} ({\mathbb R}^n_+))$ (see Lemma (2.8)). For the global existence of solutions, we assume that the extra stress tensor $S$ is represented by $S({\mathbb A}) = {\mathbb F} ( {\mathbb A}) {\mathbb A}$, where ${\mathbb F}(0) $ is a uniformly elliptic matrix and $ {\mathbb F} \in C^2(B(0,1))$, where $B(0,1)$ is open ball in ${\mathbb R}^{n\times n}$ whose center is origin and radius. is $1$. Note that $S_1$, $S_2$ and $S_3$ introduced in (1.2) satisfy our assumptions.