On standing waves and gradient-flow for the Landau–De Gennes model of nematic liquid crystals

Daniele Barbera, Vladimir Georgiev*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The article treats the existence of standing waves and solutions to gradient-flow equation for the Landau–De Gennes models of liquid crystals, a state of matter intermediate between the solid state and the liquid one. The variables of the general problem are the velocity field of the particles and the Q-tensor, a symmetric traceless matrix which measures the anisotropy of the material. In particular, we consider the system without the velocity field and with an energy functional unbounded from below. At the beginning we focus on the stationary problem. We outline two variational approaches to get a critical point for the relative energy functional: by the Mountain Pass Theorem and by proving the existence of a least energy solution. Next we describe a relationship between these solutions. Finally we consider the evolution problem and provide some Strichartz-type estimates for the linear problem. By several applications of these results to our problem, we prove via contraction arguments the existence of local solutions and, moreover, global existence for initial data with small L2-norm.

Original languageEnglish
Pages (from-to)672-699
Number of pages28
JournalEuropean Journal of Mathematics
Volume8
Issue number2
DOIs
Publication statusPublished - 2022 Jun

Keywords

  • Gradient flow
  • Least energy solution
  • Liquid crystals
  • Mountain Pass Theorem
  • Strichartz estimates

ASJC Scopus subject areas

  • Mathematics(all)

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