Optimal initial value conditions for local strong solutions of the Navier–Stokes equations in exterior domains

Abstract

Let u be a weak solution of the Navier–Stokes equations in an exterior domain Ω ⊆ ℝ 3 and a time interval [0, T [, 0 < T ≤ ∞, with initial value u 0 and external force f = div F . We address the problem to find the optimal (weakest possible) initial value condition in order to obtain a strong solution u ∈ L s (0, T ; L q (Ω)) in some time interval [0, T [, 0 < T < ∞, where s,q with 3 < q < ∞ and 2/ s + 3/ q = 1 are so-called Serrin exponents. Our main result states, for Serrin exponents s,q with 3 < q ≤ 8, a smallness condition on ∫ 0 T || e - ν τ A u 0 || q s d τ to imply existence of a strong solution u ∈ L s (0, T ; L q (Ω)); here A denotes the Stokes operator. Moreover, when 3 < q < ∞, we will prove the necessity of the condition ∫ 0 ∞ || e - ντ A u 0 || q s d τ < ∞ to get a strong solution u on [0, T [, 0 < T ≤ ∞.