Strong Well-Posedness of the Q-Tensor Model for Liquid Crystals: The Case of Arbitrary Ratio of Tumbling and Aligning Effects $$\xi $$

Abstract

AbstractThe Beris–Edwards model of nematic liquid crystals couples an equation for the molecular orientation described by the Q-tensor with a Navier–Stokes type equation with an additional non-Newtonian stress caused by the molecular orientation. Both equations contain a parameter $$\xi \in \mathbb {R}$$ ξ ∈ R measuring the ratio of tumbling and alignment effects. Previous well-posedness results largely vary on the space dimension n and the constraints of the parameter $$\xi \in \mathbb {R}$$ ξ ∈ R . This work addresses strong well-posedness of this model, first locally and then globally for small initial data, both in the $$L^p$$ L p -$$L^2$$ L 2 -setting for $$p > \frac{4}{4-n}$$ p > 4 4 - n , in the general cases, i.e., for $$n = 2, 3$$ n = 2 , 3 and without any restriction on $$\xi $$ ξ . The approach is based on methods from quasilinear equations and the fact that the associated linearized operator admits maximal $$L^p$$ L p -$$L^2$$ L 2 -regularity. The proof of the latter property relies on techniques from sectorial operators, Schur complements and $$\mathcal {J}$$ J -symmetry.