Necessary and sufficient conditions on local strong solvability of the Navier–Stokes system

Abstract

Consider in a smooth bounded domain Ω ⊆ ℝ3 and a time interval [0, T), 0 < T ≤ ∞, the Navier–Stokes system with the initial value and the external force f = div F, F ∈ L 2(0, T; L 2(Ω)). As is well-known, there exists at least one weak solution on [0, T) × Ω in the sense of Leray–Hopf; then it is an important problem to develop conditions on the data u 0, f as weak as possible to guarantee the existence of a unique strong solution u ∈ L s (0, T; L q (Ω)) satisfying Serrin's condition with 2 < s < ∞, 3 < q < ∞ at least if T > 0 is sufficiently small. Up to now there are known several sufficient conditions, yielding a larger class of corresponding local strong solutions, step by step, during the past years. Our following result is optimal in a certain sense yielding the largest possible class of such local strong solutions: let E be the weak solution of the linearized system (Stokes system) in [0, T) × Ω with the same data u 0, f. Then we show that the smallness condition ‖E‖ L s (0,T; L q (Ω)) ≤ ϵ* with some constant ϵ* = ϵ*(Ω, q) > 0 is sufficient for the existence of such a strong solution u in [0, T). This leads to the following sufficient and necessary condition: Given F ∈ L 2(0, ∞; L 2(Ω)), there exists a strong solution u ∈ L s (0, T; L q (Ω)) in some interval [0, T), 0 < T ≤ ∞, if and only if E ∈ L s (0, T′; L q (Ω)) with some 0 < T′ ≤ ∞.