论文标题
一半空间中具有非均匀初始数据的全球不可压缩流体的全球存在
Global existence of non-Newtonian incompressible fluids in half space with nonhomogeneous initial-boundary data
论文作者
论文摘要
在这项研究中，我们研究了由（1.1）控制的非牛顿不可压缩流体的弱解决方案的全球存在。当$ u_0 \ in \ dot b^{α-\ frac {2} {p}}} _ { - \ frac4 {n+2}} _ {\ frac {n+2} 2，\ frac {n+2} 2} 2}（{\ MathBb r}^{n}^{n} _+）\，\ cap \，\ dot b^dot b^dot b^{1+\ frac {1+\ freac {1+\ freac {1+\ freac {1} （\ Mathbb {r} _+）$，我们将在功能空间中的方程（1.1）找到弱解决方案$ c_b（[[0，\ infty; \ dot; \ dot b^{α-\ frac2p} _ { b^{1 - \ frac4 {n+2}} _ {\ frac {n+2} 2}（\ m马比布{r} _+））\ cap l^\ infty（0，\ infty; \ dot; \ dot w^1__ \ infty（ \，\，1 \ leq q \ leq \ infty，\，\，1 + \ frac {n + 2} p <α<2 $。我们显示了各向异性besov space $ \ dot b^{α，\fracα2} _ { \fracα2} _ { 对于全球解决方案的存在，我们假设额外的压力张量$ s $由$ s（{\ sathbb a}）= {\ Mathbb f}（{\ Mathbb a}）{\ Mathbb a}）{\ Mathbb a} $，其中$ {\ Mathbb f}（\ Mathbb f}（0）$ f}（0） c^2（b（0,1））$，其中$ b（0,1）$是$ {\ mathbb r}^{n \ times n} $的$ b（0,1）$，其中心是原点和半径。是$ 1 $。请注意，（1.2）中引入的$ S_1 $，$ S_2 $和$ S_3 $满足我们的假设。
In this study, we investigate the global existence of weak solutions of non-Newtonian incompressible fluids governed by (1.1). When $u_0 \in \dot B^{α-\frac{2}{p}}_{p,q}({\mathbb R}^{n}_+) \, \cap \,\dot B^{ 1 -\frac4{n+2}}_{\frac{n+2}2,\frac{n+2}2}({\mathbb R}^{n}_+) \,\cap \, \dot B^{1 +\frac{n}p}_{p,1} (\mathbb{R}_+)$ is given, we will find the weak solutions for the equation (1.1) in the function space $C_b ([ 0, \infty; \dot B^{α-\frac2p}_{p,q} ({\mathbb R}^n_+)) \cap C_b (0, \infty; \dot B^{1 -\frac4{n+2}}_{\frac{n+2}2} (\mathbb{R}_+)) \cap L^\infty(0, \infty; \dot W^1_\infty(\mathbb{R}_+))$, $ n+2 < p < \infty, \,\, 1 \leq q \leq \infty, \,\, 1 + \frac{n+2}p < α< 2$. We show the existence of weak solutions in the anisotropic Besov spaces $\dot B^{α, \fracα2}_{p,q} (\mathbb{R}_+ \times (0, \infty))$ (see Theorem (1.2)) and we show the embedding $\dot B^{α, \fracα2}_{p,q} (\mathbb{R}_+ \times (0, \infty) \subset C_b ([ 0, \infty; \dot B^{α-\frac2p}_{p,q} ({\mathbb R}^n_+))$ (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.
