Communications in Mathematical Sciences

Volume 19 (2021)

Number 1

On dissipative solutions to a simplified hyperbolic Ericksen–Leslie system of liquid crystals

Pages: 175 – 192



Feng Cheng (Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, China)

Ning Jiang (School of Mathematics and Statistics, Wuhan University, Wuhan, China)

Yi-Long Luo (School of Mathematics, South China University of Technology, Guangzhou, China)


We study dissipative solutions to a 3D simplified hyperbolic Ericksen–Leslie system for liquid crystals with Ginzburg–Landau approximation. First, we establish a weak-strong stability principle, which leads to a suitable notion of dissipative solutions to the hyperbolic Ericksen–Leslie system. Then, we introduce a regularized system to approximate the original system, for which we can prove the existence of global-in-time weak solutions. Finally, we prove that there is at least one dissipative solution for this simplified hyperbolic Ericksen–Leslie system.


Ericksen–Leslie system, dissipative solution, weak strong uniqueness

2010 Mathematics Subject Classification

35L51, 76A15, 82D15, 82D25

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The second-named author Ning Jiang was supported by a grant from the National Natural Science Foundation of China under contract Nos. 11971360 and 11731008.

Received 8 March 2020

Accepted 12 August 2020

Published 24 March 2021