Existence of Global Weak Solutions to an Inhomogeneous Doi Model for Active Liquid Crystals

TL;DR

This paper proves the existence of global weak solutions for an extended inhomogeneous Doi model describing active liquid crystals, incorporating an additional stress tensor, under minimal initial data assumptions in 2D and 3D.

Contribution

It extends the Doi model for active particles by adding a stress tensor and establishes global weak solutions without restrictions on initial data or physical parameters.

Findings

Existence of global weak solutions in 2D and 3D.

Applicable to both passive and active particles.

No restrictions on Reynolds and Deborah numbers.

Abstract

In this paper, we consider an inhomogeneous Doi model which was introduced by W. E and P. Zhang [Meth. Appl. of Anal., 13 (2006), pp. 181 - 198]. We extend their model, which couples a Smoluchowski equation to a Navier-Stokes type equation, for active particles by introducing an additional stress tensor. Exploiting the energetic and entropic structure of the system, we establish the existence of global-in-time weak solutions in two and three space dimensions for both passive and active particles. In particular, our result holds for minimal regularity assumptions on the initial data and without restrictions on the Reynolds and Deborah number.