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

TL;DR

This paper proves the global existence of weak solutions for non-Newtonian incompressible fluids in a half-space with nonhomogeneous initial-boundary data, using anisotropic Besov spaces and specific stress tensor conditions.

Contribution

It establishes the existence of weak solutions in anisotropic Besov spaces under certain conditions on initial data and stress tensor structure, extending previous results to non-Newtonian fluids.

Findings

Existence of weak solutions in anisotropic Besov spaces.

Embedding of Besov spaces into continuous bounded functions.

Conditions on stress tensor ensuring global solutions.

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.