Asymptotic Behavior of Solutions to the Liquid Crystal System in $H^m(\mathbb{R}^3)$

Abstract

Abstract. In this paper we study the large time behavior of regular solutions to a nematic liquid crystals system in Sobolev spaces H m (R 3 ) for m ≥ 0.We obtain optimal decay rates in H m (R 3 ) spaces, in the sense that the rates coincide with the rates of the underlying linear counterpart. The fluid under consideration has constant density and small initial data.

Strong solutions to the density-dependent incompressible nematic liquid crystal flows

“…Later, Lin and Liu [6,7] made some important analytic studies, such as the existence of weak/strong solutions and the partial regularity of suitable solutions. Recently, Dai, Qing and Schonbek [8] studied the large time behavior of solutions and gave the decay rate in the whole space in R 3 with small initial data. Grasselli and Wu [9] also considered the long-time behavior and obtained the estimates on the convergence rate for nematic liquid crystal flows with external force.…”

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

“…We refer to [17][18][19] for more explanations. The asymptotic behavior of solutions to the flow of nematic liquid crystals was studied in [31][32][33]. Now, we state our result as follows.…”

Time decay rate of global strong solutions to nematic liquid crystal flows in R+3

“…Most recently, Lin-Wang [24] have shown the existence of global weak solutions in dimension three under the assumption that the initial director field d 0 (Ω) ⊂ S 2 + . Concerning the long time asymptotical behavior of global strong solutions to (1.1) in R 3 , Liu-Xu [31] have established an optimal decay rate for (u, ∇d) H m (R 3 ) under the assumption that (u 0 , d 0 ) ∈ H m (R 3 ) × H m+1 (R 3 , S 2 ) (m ≥ 3) has sufficiently small (u 0 , ∇d 0 ) L 2 (R 3 ) -norm; while Dai, and her coauthors, has obtained in [4,5] optimal decay rates in H m (R 3 ) provided u 0 H 1 (R 3 ) + d − e 3 H 2 (R 3 ) is sufficiently small.…”