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On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces
Author(s) -
Liu Qiao,
Zhao Jihong
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.3846
Subject(s) - liquid crystal , mathematics , homogeneous , besov space , constant (computer programming) , compressibility , space (punctuation) , initial value problem , argument (complex analysis) , cauchy problem , mathematical analysis , physics , condensed matter physics , mechanics , combinatorics , chemistry , computer science , biochemistry , functional analysis , interpolation space , gene , programming language , operating system
In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global‐in‐time strong solutions to the Cauchy problem of the three‐dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data ( u 0 , d 0 −d ¯0 ) ∈B ̇p , 13 p − 1 ( R 3 ) ×B ̇p , 13 p( R 3 ) with 1 < p < ∞ and further assume that ( u 0 , d 0−d ¯0 ) ∈B ̇q , ∞ − s ( R 3 ) ×B ̇q , ∞ − s + 1 ( R 3 ) with 1 < q ≤ p and max 0 , 1 − 3 q≤ s < min 4 − 3 q , 1 + 3 p, then the global‐in‐time strong solution ( u , d ) to the nematic liquid crystal flows admits the following temporal decay rate:u ( t )B ̇p , 1 ℓ+d ( t ) −d ¯0B ̇p , 1 ℓ + 1≤ C ( 1 + t ) − ℓ + s 2 − 3 21 q − 1 pfor all − s − 31 q − 1 p< ℓ ≤ 3 p − 1 .Here,d ¯0 ∈ S 2is a constant unit vector. The highlight of our argument is to show that theB ̇q , ∞ − s ×B ̇q , ∞ − s + 1 ‐norms (with 0 ≤ s < min { 4 − 3 q , 1 + 3 p } ) of solution are preserved along time evolution. Copyright © 2016 John Wiley & Sons, Ltd.