Initial Values for the Navier-Stokes Equations in Spaces with Weights in Time

Abstract

We consider the nonstationary Navier-Stokes system in a smooth bounded domain Ω ⊂ R3 with initial value u0 ∈ Lσ(Ω). It is an important question to determine the optimal initial value condition in order to prove the existence of a unique local strong solution satisfying Serrin’s condition. In this paper, we introduce a weighted Serrin condition that yields a necessary and sufficient initial value condition to guarantee the existence of local strong solutions u(·) contained in the weighted Serrin class ∫ T 0 (τ ‖u(τ)‖q) dτ < ∞ with 2s + 3 q = 1 − 2α, 0 < α < 1 2 . Moreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class. 2010 Mathematics Subject Classification: 35Q30; 76D05