Global solutions to 3D incompressible Navier–Stokes equations with some large initial data. (July 2022) - Yu, Yanghai; Li, Jinlu; Yin, Zhaoyang
Abstract: In this paper, we derive a new smallness hypothesis of initial data for the three-dimensional incompressible Navier–Stokes equations. More precisely, we prove that if ‖ u 0 1 + u 0 2 ‖ B ̇ p, 1 3 p − 1 + ‖ u 0 3 ‖ B ̇ p, 1 3 p − 1 ‖ u 0 1 ‖ B ̇ p, 1 3 p − 1 + ‖ u 0 2 ‖ B ̇ p, 1 3 p − 1 × exp C ( ‖ u 0 ‖ B ̇ ∞, 2 − 1 2 + ‖ u 0 ‖ B ̇ ∞, 1 − 1 ) is small enough, the Navier–Stokes equations have a unique global solution. As an application, we construct two examples of initial data satisfying the smallness condition, but whose B ̇ ∞, ∞ − 1 ( R 3 ) norm can be arbitrarily large.