# Weak solutions to the equations of stationary compressible flows in active liquid crystals

Liang, Zhilei and Majumdar, Apala and Wang, Dehua and Wang, Yixuan (2022) Weak solutions to the equations of stationary compressible flows in active liquid crystals. Communications in Mathematical Analysis and Applications. ISSN 2790-1920 (In Press) (https://doi.org/10.48550/arXiv.2205.00358)

 Text. Filename: Liang_etal_arXiv_2022_Weak_solutions_to_the_equations_of_stationary_compressible_flows_in_active_liquid_crystals.pdf Accepted Author Manuscript Restricted to Repository staff only until 28 December 2023. License: Strathprints license 1.0 Download (575kB) | Request a copy

## 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 $\gamma>1$.

#### ORCID iDs

Liang, Zhilei, Majumdar, Apala ORCID: https://orcid.org/0000-0003-4802-6720, Wang, Dehua and Wang, Yixuan;
• Item type: Article 82769 DateEvent28 December 2022Published28 December 2022Accepted30 April 2022Submitted This has been accepted for publication in the 'Communications in Mathematical Analysis and Applications'. active liquid crystals, stationary compressible flows, Navier-Stokes equations, Q-tensor, weak solutions, weak convergence, Mathematics, Mathematics(all) Science > Mathematics Faculty of Science > Mathematics and Statistics Pure Administrator 13 Oct 2022 16:27 18 Jan 2023 11:46 https://strathprints.strath.ac.uk/id/eprint/82769