On optimal initial value conditions for local strong solutions of the Navier-Stokes equations

Consider a smooth bounded Ω ⊂ ℝ3, and the Navier-Stokes system in [0,∞) × Ω with initial value u 0 ∈l Lσ2 (Ω) and external force f = div F, F ∈ L2(0, ∞ L2 (Ω) Ls/2 (0, ∞ Lq/2 (Ω) where 2 < s < ∞, 3, < q < ∞, 2/s+3/q = 1, are so-called Serrin exponents. It is an important question what is the optimal (weakest possible) initial value condition in order to obtain a unique strong solution u Ls(0,T; Lq(Ω) in some initial interval [0, T), 0 < T ≤ ∞. Up to now several sufficient conditions on u 0 are known which need not be necessary. Our main result, see Theorem 1.1, shows that the condition ∫0∞||e-t Au0|| qs dt < ∞, A denotes the Stokes operator, is sufficient and necessary for the existence of such a strong solution u. In particular, if ∫0 ∞ ||e-t Au0|| qs dt = ∞, u0 Lσ 2(Ω), then any weak solution u in the usual sense does not satisfy Serrin's condition u Ls(0,T; Lq(Ω) for each 0 < T ≤ ∞. © 2009 Università degli Studi di Ferrara.