Existence Theory for Generalized Newtonian Fluids

Appendix A

Function spaces

A.1 Function spaces involving the divergence

Given a Young function A, denote by $H^A(G)$ the Banach space of those vector-valued functions $\mathbf{u}:G\to {\mathbb{R}}^n$ such that the norm

${\Vert \mathbf{u}\Vert }_{H^A(G)}={\Vert \mathbf{u}\Vert }_{L^A(G,{\mathbb{R}}^n)}+{\Vert \mathrm{div}\mathbf{u}\Vert }_{L^A(G)}$

(A.1.1)

is finite. We also denote by $H_0^A(G)$ its subspace of those functions $\mathbf{u}\in H^A(G)$ whose normal component on ∂G vanishes, in the sense that ...