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∈ L_σ²(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 ∫₀^T(τ^α||u(τ)||_q)^sdτ < ∞ with 2/s + 3/q = 1 − 2α, 0 < α < 1/2. Moreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class.