Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows

Xu, Xiang; Zhang, Zhifei

doi:10.1016/j.jde.2011.08.028

Cited by 115 publications

(80 citation statements)

References 12 publications

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“…Here we would like to mention a few of previous results. In dimensions two, Lin-Lin-Wang [26] have proved the existence of global Leray-Hopf type weak solutions to (1.1) with initial and boundary conditions, which is smooth away from finitely many possible singular times (see Hong [17] for (1.1) in Ω = R 2 , Hong-Xin [18] and Xu-Zhang [41] for other related works). Lin-Wang [29] have also proved the uniqueness for such weak solutions.…”

Section: Introductionmentioning

confidence: 99%

Finite Time Singularity of the Nematic Liquid Crystal Flow in Dimension Three

Huang

Lin

et al. 2016

Arch Rational Mech Anal

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In this paper, we consider the initial and boundary value problem of a simplified nematic liquid crystal flow in dimension three and construct two examples of finite time singularity. The first example is constructed within the class of axisymmetric solutions, while the second example is constructed for any generic initial data (u 0 , d 0 ) that has sufficiently small energy, and d 0 has a nontrivial topology.

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Section: Introductionmentioning

confidence: 99%

Finite Time Singularity of the Nematic Liquid Crystal Flow in Dimension Three

Huang

Lin

et al. 2016

Arch Rational Mech Anal

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“…For any bounded smooth domain in

{double-struckR}^{2}

, F. Lin, J. Lin, and C. Wang have proved the global existence of Leray–Hopf‐type weak solutions to in . The uniqueness of weak solutions in two dimensions was studied by the authors in . M. Hong and Z. Xin studied the global existence for general Ericksen–Leslie system in two dimensions .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Yang

Sun

2016

Math Methods in App Sciences

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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, ı 0 /; here, ı 0 is a constant vector with jı 0 j D 1. Precisely, we show the existence and asymptotic stability with small initial dataof us is not entirely included in the space BMO 1 BMO 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 j x j! 1, 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//.

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“…In dimension two, Lin-Lin-Wang [30] have proved the existence of global LerayHopf type weak solutions to initial and boundary value problem of (1.1) with finitely many possible singular times (see [15] for Ω = R 2 , [17,43,28] for more general systems, and [8,24,25,45] for some other related works). Lin-Wang [33], Wang-Wang-Zhang [44], and Li-Titi-Xin [27] have also proved the uniqueness for such weak solutions.…”

Section: Introductionmentioning

confidence: 99%