On The Two-Dimensional Stokes Problem in Exterior Domains: The Maximum Modulus Theorem

Abstract The two-dimensional Stokes IBVP on $$(0,T)\times \Omega $$ ( 0 , T ) × Ω is investigated under the assumptions that $$\Omega \subset {\mathbb {R}}^2$$ ⊂ R 2 a smooth exterior domain, initial datum $$v_0$$ v belongs to $$L^\infty (\Omega )$$ L ∞ and $$(v_0,\nabla \phi )=0$$ ∇ ϕ = for all $$\phi \in L^1_{\ell oc}(\Omega ∈ ℓ o c 1 with $$\nabla L^1(\Omega . well-posedeness in ((0,T)\times maximum modulus theorem are achieved, particular one deduces semigroup bounded analytic semigroup.