TY - JOUR

T1 - On standing waves and gradient-flow for the Landau–De Gennes model of nematic liquid crystals

AU - Barbera, Daniele

AU - Georgiev, Vladimir

N1 - Funding Information:
Daniele Barbera is supported in part by INDAM — Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni. Vladimir Georgiev is partially supported by Project 2017 “Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari” of INDAM, GNAMPA — Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni, by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, by Top Global University Project, Waseda University, the Project PRA 2018 49 of University of Pisa and by the project PRIN 2020XB3EFL funded by the Italian Ministry of Universities and Research.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/6

Y1 - 2022/6

N2 - 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.

AB - 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.

KW - Gradient flow

KW - Least energy solution

KW - Liquid crystals

KW - Mountain Pass Theorem

KW - Strichartz estimates

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U2 - 10.1007/s40879-022-00537-5

DO - 10.1007/s40879-022-00537-5

M3 - Article

AN - SCOPUS:85129334604

SN - 2199-675X

VL - 8

SP - 672

EP - 699

JO - European Journal of Mathematics

JF - European Journal of Mathematics

IS - 2

ER -