TY - JOUR

T1 - Weak solutions to the equations of stationary compressible flows in active liquid crystals

AU - Liang, Zhilei

AU - Majumdar, Apala

AU - Wang, Dehua

AU - Wang, Yixuan

N1 - This has been accepted for publication in the 'Communications in Mathematical Analysis and Applications'.

PY - 2023/3/30

Y1 - 2023/3/30

N2 - 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 $\gamma>1$.

AB - 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 $\gamma>1$.

KW - active liquid crystals

KW - stationary compressible flows

KW - Navier-Stokes equations

KW - Q-tensor

KW - weak solutions

KW - weak convergence

U2 - 10.48550/arXiv.2205.00358

DO - 10.48550/arXiv.2205.00358

M3 - Article

SN - 2790-1920

VL - 2023

SP - 70

EP - 114

JO - Communications in Mathematical Analysis and Applications

JF - Communications in Mathematical Analysis and Applications

IS - 2

ER -