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

In this study, we investigate the global existence of weak solutions of non-Newtonian incompressible fluids governed by (1.1). When $u_0\in {\dot{B}}_{p,q}^{\alpha -\frac{2}{p}}({\mathbb{R}}_{+}^n)\cap {\dot{B}}_{\frac{n+2}{2},\frac{n+2}{2}}^{1-\frac{4}{n+2}}({\mathbb{R}}_{+}^n)\cap {\dot{B}}_{p,1}^{1+\frac{n}{p}}({\mathbb{R}}_{+})$ is given, we will find the weak solutions for the equation (1.1) in the function space $C_b([0,\mathrm{\infty };{\dot{B}}_{p,q}^{\alpha -\frac{2}{p}}({\mathbb{R}}_{+}^n))\cap C_b(0,\mathrm{\infty };{\dot{B}}_{\frac{n+2}{2}}^{1-\frac{4}{n+2}}({\mathbb{R}}_{+}))\cap L^{\mathrm{\infty }}(0,\mathrm{\infty };{\dot{W}}_{\mathrm{\infty }}^1({\mathbb{R}}_{+}))$, $n+2<p<\mathrm{\infty },1\le q\le \mathrm{\infty },1+\frac{n+2}{p}<\alpha <2$. We show the existence of weak solutions in the anisotropic Besov spaces ${\dot{B}}_{p,q}^{\alpha ,\frac{\alpha }{2}}({\mathbb{R}}_{+}\times (0,\mathrm{\infty }))$ (see Theorem (1.2)) and we show the embedding ${\dot{B}}_{p,q}^{\alpha ,\frac{\alpha }{2}}({\mathbb{R}}_{+}\times (0,\mathrm{\infty })\subset C_b([0,\mathrm{\infty };{\dot{B}}_{p,q}^{\alpha -\frac{2}{p}}({\mathbb{R}}_{+}^n))$ (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.