Existence and asymptotic behavior to the incompressible nematic liquid crystal flow in the whole space

In this article, first of all, the global existence and asymptotic stability of solutions to the incompressible nematic liquid crystal flow is investigated when initial data are a small perturbation near the constant steady state (0, δ 0 ); here, δ 0 is a constant vector with | δ 0 |=1. Precisely, we show the existence and asymptotic stability with small initial data ( u 0 , d 0 − δ 0 ) ∈N ̇r , λ , ∞ − β ×N ̇r , λ , ∞ − β + 1for n ≥ 2 , 0 ≤ λ < n , 1 ≤ r < ∞ , r > n − λ , β = 1 − n − λ r . The initial data classN ̇r , λ , ∞ − β ×N ̇r , λ , ∞ − β + 1of us is not entirely included in the space B M O −1 × B M O and contains strongly singular functions and measures. As an application, we obtain a class of asymptotic existence of a basin of attraction for each self‐similar solution with homogeneous initial data. We also study global existence of a large class of decaying solutions and construct an explicit asymptotic formula for ∣ x ∣→ ∞ , relating the self‐similar profile ( U ( x ), D ( x )) to its corresponding initial data ( u 0 , d 0 ). In two dimensions, we obtain higher‐order asymptotics of ( u ( x ), d ( x )). Copyright © 2016 John Wiley & Sons, Ltd.