On the existence of local strong solutions for the Navier-Stokes equations in completely general domains

There are only very few results on the existence of unique local in time strong solutions of the Navier-Stokes equations for completely general domains Ω⊆R³, although domains with edges and corners, bounded or unbounded, are very important in applications. The reason is that the L^q-theory for the Stokes operator A is available in general only in the Hilbert space setting, i.e., with q=2. Our main result for a general domain Ω is optimal in a certain sense: Consider an initial value and a zero external force. Then the condition is sufficient and necessary for the existence of a unique local strong solution u∈L⁸(0,T;L⁴(Ω)) in some interval 0,T), 0<T≤∞, with u(0)=u₀, satisfying Serrin’s condition . Note that Fujita-Kato’s sufficient condition u₀∈D(A^1/4) is strictly stronger and therefore not optimal.