Weak Solutions to the Equations of Stationary Compressible Flows in Active Liquid Crystals

Weak Solutions to the Equations of Stationary Compressible Flows in Active Liquid Crystals

Year:    2023

Author:    Zhilei Liang, Apala Majumdar, Dehua Wang, Yixuan Wang

Communications in Mathematical Analysis and Applications, Vol. 2 (2023), Iss. 1 : pp. 70–114

Abstract

The equations of stationary compressible flows of active liquid crystals are considered in a bounded three-dimensional domain. The system consists of the stationary Navier-Stokes equations coupled with the equation of Q-tensors and the equation of the active particles. The existence of weak solutions to the stationary problem is established through a two-level approximation scheme, compactness estimates and weak convergence arguments. Novel techniques are developed to overcome the difficulties due to the lower regularity of stationary solutions, a Moser-type iteration is used to deal with the strong coupling of active particles and fluids, and some weighted estimates on the energy functions are achieved so that the weak solutions can be constructed for all values of the adiabatic exponent $\gamma>1.$

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/cmaa.2022-0021

Communications in Mathematical Analysis and Applications, Vol. 2 (2023), Iss. 1 : pp. 70–114

Published online:    2023-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    45

Keywords:    Active liquid crystals stationary compressible flows Navier-Stokes equations Q-tensor weak solutions weak convergence.

Author Details

Zhilei Liang

Apala Majumdar

Dehua Wang

Yixuan Wang