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

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