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

DOI10.1016/J.JDE.2023.01.006arXiv2006.16832MaRDI QIDQ6344114

Publication date: 30 June 2020

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.

Mathematics Subject Classification ID

Smoothness and regularity of solutions to PDEs (35B65) PDEs in connection with fluid mechanics (35Q35) Navier-Stokes equations for incompressible viscous fluids (76D05) Asymptotic methods, singular perturbations applied to problems in fluid mechanics (76M45) Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) (82D30) NLS equations (nonlinear Schrödinger equations) (35Q55) Liquid crystals (76A15) Existence problems for PDEs: global existence, local existence, non-existence (35A01) Weak solutions to PDEs (35D30)

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