The Navier–Stokes equations in exterior Lipschitz domains: L^p-theory

We show that the Stokes operator defined on L_σ^p(Ω) for an exterior Lipschitz domain Ω⊂Rⁿ (n≥3) admits maximal regularity provided that p satisfies |1/p−1/2|<1/(2n)+ε for some ε>0. In particular, we prove that the negative of the Stokes operator generates a bounded analytic semigroup on L_σ^p(Ω) for such p. In addition, L^p-L^q-mapping properties of the Stokes semigroup and its gradient with optimal decay estimates are obtained. This enables us to prove the existence of mild solutions to the Navier–Stokes equations in the critical space L^∞(0,T;L_σ³(Ω)) (locally in time and globally in time for small initial data).