Global well-posedness of the two dimensional Beris–Edwards system with general Laudau–de Gennes free energy

Abstract

In this paper, we consider the Beris-Edwards system for incompressible nematic liquid crystal flows. The system under investigation consists of the Navier-Stokes equations for the fluid velocity u coupled with an evolution equation for the order parameter Q-tensor. One important feature of the system is that its elastic free energy takes a general form and in particular, it contains a cubic term that possibly makes it unbounded from below. In the two dimensional periodic setting, we prove that if the initial L ∞ -norm of the Q-tensor is properly small, then the system admits a unique global weak solution. The proof is based on the construction of a specific approximating system that preserves the L ∞ -norm of the Q-tensor along the time evolution.Here, a, b, c ∈ R are material-dependent and temperature-dependent constants. On the other hand, the elastic free energy characterizes the distortion effect of the liquid crystal and its density function F elastic gives the strain energy density due to spatial variations in the Q-tensor:( 1.4) for 1 i, j, k, l d. Throughout this paper, we use the Einstein summation convention over repeated indices. The coefficients L 1 , L 2 , L 3 , L 4 are material-dependent elastic constants. We note that F elastic consists of three independent terms associated with L 1 , L 2 , L 3 that are quadratic in the first order partial derivatives of the Q-tensor, plus a cubic term associated with the coefficient L 4 . In the literature, the case L 2 = L 3 = L 4 = 0 is usually called isotropic, otherwise anisotropic if at least one of L 2 , L 3 , L 4 does not vanish. In particular, the retention of the cubic term (i.e., L 4 = 0) is due to the physically relevant consideration that it allows a complete reduction of the Landau-de Gennes energy E(Q) to the classical Oseen-Frank energy for nematic liquid crystals [6,31] (see also [24, Appendix B]). There exists a vast recent literature on the mathematical study of the Landau-de Gennes theory, and we refer interested readers to [2-7, 9-15, 17-24, 30-32, 34, 35, 38-40] as well as the references cited therein.In this paper, we study a basic model for the evolution of an incompressible nematic liquid crystal flow, which was first proposed by Beris-Edwards [7]. The resulting PDE system consists of the Navier-Stokes equations for the fluid velocity u and nonlinear convection-diffusion equations of parabolic type for the Q-tensor (see, e.g., [2,10,34]). To simplify the mathematical setting, throughout this paper we confine ourselves to the two dimensional periodic case such that d = 2 and Ω = T 2 , where T 2 stands for the periodic box with period ℓ i in the i-th direction with O = (0, ℓ 1 ) × (0, ℓ 2 ) being the periodic cell. Without loss of generality, we simply set O = (0, 1) 2 . Then the coupled system we are going to study takes the following form:

Mathematical problems of nematic liquid crystals: between dynamical and stationary problems

“…For the BE system existence and regularity in the whole space was initially proved in [12,13] with subsequent developments in [25,[29][30][31][32][33][34][35]. In these studies, one often restricts to the corotational case ξ = 0 which, although not particularly relevant physically, does simplify the system significantly.…”

From the Landau-de Gennes theory to the Ericksen-Leslie theory in dimension two

“…There have been quite many works on the global existence of solutions to the system (1.3)- (1.5), see [39,40,3,33,12,1,44] and the references therein. In particular, the first existence of global weak solutions to the cauchy problem for (1.3)- (1.5) in the dimension two (2D) and dimension three (3D) is established by Paicu-Zarnescu in [39] under the conditions that ξ = 0 and…”

“…These results in [39,40] have been generalized to many interesting cases. In particular, for 2D periodic initial data, the global well-posedness of strong solutions to (1.3)-(1.5) was obtained in [3] under just condition (1.7), which was relaxed to allow L 2 , L 3 and L 4 being non-zero with some other minor conditions recently in [33]; and for initial-boundary value problems for 2D and 3D, the global existence of weak solutions to the system (1.3)-(1.5) has been proved in [12,1] under conditions that ξ = 0 and (1.7) holds. Similar results have been obtained in [44], where the bulk potential (1.2) is replaced by Ball-Majumdar type bulk potential.…”

“…), which will be given in our forthcoming paper. Meanwhile, it should be noted that this two-dimensional system (1.10) includes the two-dimensional system in [39,40,3,33,12,1,44], in which v : (0,…”