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

The 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 \(\xi \in \mathbb {R}\) 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 \(\xi \in \mathbb {R}\). This work addresses strong well-posedness of this model, first locally and then globally for small initial data, both in the \(L^p\)-\(L^2\)-setting for \(p > \frac{4}{4-n}\), in the general cases, i.e., for \(n = 2, 3\) and without any restriction on \(\xi \). The approach is based on methods from quasilinear equations and the fact that the associated linearized operator admits maximal \(L^p\)-\(L^2\)-regularity. The proof of the latter property relies on techniques from sectorial operators, Schur complements and \(\mathcal {J}\)-symmetry.