Dissipative solutions to a Beris-Edwards type model for compressible active nematic liquid crystals

TL;DR

This paper establishes the existence of dissipative solutions for a complex model of compressible active nematic liquid crystals, advancing mathematical understanding of their hydrodynamics with boundary effects and non-Newtonian stresses.

Contribution

It introduces a novel proof of dissipative solutions for a Beris-Edwards type model with non-Newtonian stress and boundary conditions, using a three-level approximation scheme.

Findings

Existence of dissipative solutions proven

Handles non-Newtonian stress tensor complexities

Addresses nonhomogeneous boundary conditions

Abstract

We study the hydrodynamics of compressible active nematic liquid crystals in a three-dimensional and bounded domain, with a nonlinear viscosity tensor and nonhomogeneous boundary data, in a Landau-de Gennes framework. We prove the existence of dissipative solutions within a Beris-Edwards type model for active nematodynamics, which are weak solutions satisfying the underlying equations modulo a defect measure. The proof follows from a three level approximation scheme -- the Galerkin approximation, the classical parabolic regularization of the continuity equation, and the convex regularization of the potential generating the viscous stress. New techniques are required to deal with non-Newtonian stress tensor, larger classes of admissible pressure potentials and nonhomogeneous boundary conditions.