# Suitable weak solutions for the co-rotational Beris-Edwards system in dimension three

## TL;DR

This paper proves the global existence of suitable weak solutions for the three-dimensional co-rotational Beris-Edwards system modeling nematic liquid crystals, demonstrating regularity except on a negligible singular set.

## Contribution

It establishes the existence of global suitable weak solutions for the 3D co-rotational Beris-Edwards system with specific bulk potentials, extending the mathematical understanding of liquid crystal flows.

## Key findings

- Solutions are smooth away from a set of zero parabolic Hausdorff measure.
- The system couples Navier-Stokes with a dissipative Q-tensor equation.
- Existence results hold for both Landau-De Gennes and Ball-Majumdar potentials.

## Abstract

In this paper, we establish the global existence of a suitable weak solution to the co-rotational Beris-Edwards $Q$-tensor system modeling the hydrodynamic motion of nematic liquid crystals with either Landau-De Gennes bulk potential in $\mathbb R^3$ or Ball-Majumdar bulk potential in $\mathbb{T}^3$, a system coupling the forced incompressible Navier-Stokes equation with a dissipative, parabolic system of Q-tensor $Q$ in $\mathbb R^3$, which is shown to be smooth away from a closed set $\Sigma$ whose $1$-dimensional parabolic Hausdorff measure is zero.