Communications in Mathematical Sciences

Volume 22 (2024)

Number 2

Uniqueness of global weak solutions to the frame hydrodynamics for biaxial nematic phases in $\mathbb{R}^2$

Pages: 461 – 485

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n2.a7

Authors

Sirui Li (School of Mathematics and Statistics, Guizhou University, Guiyang, China)

Chenchen Wang (School of Mathematics and Statistics, Guizhou University, Guiyang, China)

Jie Xu (Laboratory of Scientific & Engineering Computing (LSEC), Institute of Computational Mathematics & Scientific/Engineering Computing (ICMSEC), and Academy of Mathematics & Systems Science (AMSS), Chinese Academy of Sciences, Beijing, China)

Abstract

We consider the hydrodynamics for biaxial nematic phases described by a field of orthonormal frame, which can be derived from a molecular-theory-based tensor model. We prove the uniqueness of global weak solutions to the Cauchy problem of the frame hydrodynamics in dimension two. The proof is mainly based on the suitable weaker energy estimates within the Littlewood–Paley analysis. We take full advantage of the estimates of nonlinear terms with rotational derivatives on $SO(3)$, together with cancellation relations and dissipative structures of the biaxial frame system.

Keywords

liquid crystals, biaxial nematic phase, frame hydrodynamics, uniqueness of weak solutions, Littlewood–Paley theory

2010 Mathematics Subject Classification

35A02, 35Q35, 76A15

Full Text (PDF format)

Sirui Li is partially supported by the NSFC under grant No. 12061019 and by the Growth Foundation for Youth Science and Technology Talent of Educational Commission of Guizhou Province of China under grant No. [2021]087.Jie Xu is partially supported by the NSFC under grant Nos. 12288201, 12001524, and 12371414, and by National Key Research and Development Program of China (2023YFA1008802).

Received 31 May 2022

Received revised 1 July 2023

Accepted 13 July 2023

Published 1 February 2024