Well-posedness of a fully-coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data

TL;DR

This paper establishes short-time well-posedness and global weak solutions for a coupled Navier-Stokes and Q-tensor model describing nematic liquid crystals with inhomogeneous boundary conditions, using advanced regularity and fixed point techniques.

Contribution

It proves well-posedness and existence of solutions for the Beris--Edwards model with inhomogeneous boundary data, extending previous results to more general boundary conditions.

Findings

Solutions have higher regularity in time, enabling Lipschitz continuity of nonlinear terms.

Well-posedness is achieved via contraction mapping and linearized system isomorphism.

Global weak solutions exist under the given boundary conditions.

Abstract

We prove short-time well-posedness and existence of global weak solutions of the Beris--Edwards model for nematic liquid crystals in the case of a bounded domain with inhomogeneous mixed Dirichlet and Neumann boundary conditions. The system consists of the Navier-Stokes equations coupled with an evolution equation for the $Q$-tensor. The solutions possess higher regularity in time of order one compared to the class of weak solutions with finite energy. This regularity is enough to obtain Lipschitz continuity of the non-linear terms in the corresponding function spaces. Therefore the well-posedness is shown with the aid of the contraction mapping principle using that the linearized system is an isomorphism between the associated function spaces.