Global weak solution and blow-up criterion of the general Ericksen–Leslie system for nematic liquid crystal flows

Abstract

In this paper we investigate the three dimensional general Ericksen-Leslie (E-L) system with Ginzburg-Landau type approximation modeling nematic liquid crystal flows. First, by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor, we prove existence of global-in-time weak solutions under physically meaningful boundary conditions and suitable assumptions on the Leslie coefficients, which ensures that the total energy of the E-L system is dissipated. Moreover, for the E-L system with periodic boundary conditions, we prove the local well-posedness of classical solutions under the so-called Parodi's relation and establish a blow-up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field.

Existence of weak solutions to the Ericksen‐Leslie model for a general class of free energies

“…14,17 The full Ericksen-Leslie model but still neglecting the second material derivative and equipped with the Dirichlet energy and double-well potential (5) was considered by Wu et al 16 They showed the existence of weak solutions under the condition that 4 is large enough. Cavaterra et al 18 proved the existence of weak solutions for the same system when 4 is only assumed to be positive. They added a regularizing p-Laplacian to the velocity equation.…”

“…To be able to use such a property, previous work had to invoke additional regularity in the approximation of the velocity field either by assuming 4 to be sufficiently large 16 or by introducing an additional regularization. 18 Due to the generalization with respect to the free energy considered in the present paper, the continuity of the solution operator to Equation (1b) is no longer at hand. Additionally, a simultaneous discretization is more suitable for a numerical approximation.…”

Global Strong Solutions of the Full Navier--Stokes and $Q$-Tensor System for Nematic Liquid Crystal Flows in Two Dimensions

“…We refer to [6,7,20,26,38,42] and the references therein for mathematical analysis of the general EricksenLeslie system either under the unit length constraint of the molecule director or with Ginzburg-Landau approximation of the free energy.…”

Pullback attractors of the two-dimensional non-autonomous simplified Ericksen–Leslie system for nematic liquid crystal flows

“…Therefore, it is interesting to study the existence of attractors of the Ericksen-Leslie system for nematic liquid crystal flows to explore the complexity of their geometric structure. There have been many works about the well-posedness of solutions and the long-time behaviour of solutions of the Ericksen-Leslie system for nematic liquid crystal flows(see [1,4,10,11,19,25,29,34]). For example, in [10], the authors proved the existence of trajectory attractors of the Sun-Liu model for nematic liquid crystals in 3D.…”