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

Abstract

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 ⊆ R 3 , 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 u 0 ∈ L σ 2 ( Ω ) and a zero external force. Then the condition ∫ 0 ∞ ‖ e − t A u 0 ‖ 4 8 d t < ∞ is sufficient and necessary for the existence of a unique local strong solution u ∈ L 8 ( 0 , T ; L 4 ( Ω ) ) in some interval [ 0 , T ) , 0 < T ≤ ∞ , with u ( 0 ) = u 0 , satisfying Serrin’s condition 2 8 + 3 4 = 1 . Note that Fujita–Kato’s sufficient condition u 0 ∈ D ( A 1 / 4 ) is strictly stronger and therefore not optimal.