Global existence of non-Newtonian incompressible fluid 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 ∈ B ̇ p , q α − 2 p ( R + n ) ∩ B ̇ n + 2 2 , n + 2 2 1 − 4 n + 2 ( R + n ) ∩ B ̇ p , 1 1 + n p ( R + n ) is given, we will find the weak solutions for the Eq. (1.1) in the function space C b [ 0 , ∞ ; B ̇ p , q α − 2 p ( R + n ) ∩ C b (0 , ∞ ; B ̇ n + 2 2 1 − 4 n + 2 ( R + n )) ∩ L ∞ (0 , ∞ ; W ̇ ∞ 1 ( R + n )) , n + 2 < p < ∞ , 1 ≤ q ≤ ∞ , 1 + n + 2 p < α < 2. We show the existence of weak solutions in the anisotropic Besov spaces B ̇ p , q α , α 2 ( R + n × (0 , ∞)) (see Theorem 1.2) and we show the embedding B ̇ p , q α , α 2 R + n × (0 , ∞) ⊂ C b [ 0 , ∞ ; B ̇ p , q α − 2 p ( R + n ) (see Lemma 2.8). For the global existence of solutions, we assume that the extra stress tensor S is represented by S (A) = F (A) A , where F satisfies the assumption (A). Note that S1, S2 and S3 introduced in (1.2) satisfy our assumptions. [ABSTRACT FROM AUTHOR]