The Lᵖ-Lq maximal regularity for the Beris-Edward model in the half-space

In this paper, we consider the model describing viscous incompressible liquid crystal flows, called the Beris-Edwards model, in the half-space.This model is a coupled system by the Navier-Stokes equations with the evolution equation of the director fields Q. The purpose of this paper is to prove that the linearized problem has a unique solution satisfying the maximal Lᵖ -Lq regularity estimates, which is essential for the study of quasi-linear parabolic or parabolic-hyperbolic equations. Our method relies on the R-boundedness of the solution operator families to the resolvent problem in order to apply operator-valued Fourier multiplier theorems. Consequently, we also have the local well-posedness for the Beris-Edwards model with small initial data.