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Existence and asymptotic behavior to the incompressible nematic liquid crystal flow in the whole space
Author(s) -
Yang Minghua,
Sun Jinyi
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3605
Subject(s) - mathematics , compressibility , exponential stability , mathematical analysis , liquid crystal , space (punctuation) , constant (computer programming) , perturbation (astronomy) , flow (mathematics) , singular perturbation , mathematical physics , geometry , physics , thermodynamics , quantum mechanics , condensed matter physics , nonlinear system , linguistics , philosophy , computer science , programming language
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.