Asymptotic Stability of Large Solutions with Large Perturbation to the Navier–Stokes Equations

Consider weak solutions w of the Navier–Stokes equations in Serrinʹs class w∈Lα(0, ∞; Lq(Ω)) for 2/α+3/q=1 with 3<q⩽∞, where Ω is a general unbounded domain in R3. We shall show that although the initial and external disturbances from w are large, every perturbed flow v with the energy inequality converges asymptotically to w as ‖v(t)−w(t)‖L2(Ω)→0, ‖∇v(t)−∇w(t)‖L2(Ω)=O(t−1/2) as t→∞.