Weak Neumann implies Stokes

Consider a domain ω ⊂ℝ ⁿ with possibly non compact but uniform C ³-boundary and assume that the Helmholtz projection P exists on L ^p(W) for some 1 <p <∞. It is shown that the Stokes operator in L ^p(W) generates an analytic semigroup on L _σ ^p(ω) admitting maximal L ^q-L ^p-regularity. Moreover, for u ₀ ε L _σ ^p(W) there exists a unique local mild solution to the Navier-Stokes equations on domains of this form provided p > n.