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

Reinhard Farwig, Yoshikazu Giga, Pen-Yuan Hsu
2016 Funkcialaj Ekvacioj  
We consider the nonstationary Navier-Stokes system in a smooth bounded domain Ω ⊂ R 3 with initial value u 0 ∈ L 2 σ (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
more » ... ed in the weighted Serrin class T 0 (τ α u(τ ) q ) s dτ < ∞ with 2 s + 3 q = 1 − 2α, 0 < α < 1 2 . Moreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class.
doi:10.1619/fesi.59.199 fatcat:dfpwnuax7raoxil435dt7qfr7q