Weak solutions for an initial–boundary Q-tensor problem related to liquid crystals

Abstract

The coupled Navier-Stokes and Q-Tensor system is considered in a bounded three-dimensional domain under homogeneous Dirichlet boundary conditions for the velocity u and either nonhomogeneous Dirichlet or homogeneous Neumann boundary conditions for the tensor Q.The corresponding initial-value problem in the whole space R 3 was analyzed in [Paicu & Zarnescu, 2012].In this paper, three main results concerning weak solutions will be proved; the existence of global in time weak solutions (bounded up to infinite time), a uniqueness criteria and a maximum principle for Q. Moreover, we identify how to modify the system to deduce symmetry and traceless for Q, for any weak solution. The presence of a stretching term in the Q-system plays a crucial role in all the analysis.

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

“…Asymptotic behavior of the Cauchy problem in R 3 with ξ = 0 is recently discussed in [10]. Besides, initial boundary value problems subject to various boundary conditions for d = 2, 3 have been investigated by several authors in [2,16,17] under the assumption ξ = 0. In these works, they proved the existence of global weak solutions, the existence and uniqueness of local strong solutions, as well as some regularity criteria, etc.…”

“…The main difficulty in handling the current full coupled system with ξ ∈ R is due to the fact that for ξ = 0 the system (1.1)-(1.5) no longer enjoys certain maximum principle for the Q-equation (1.3), which is instead true in the case in which ξ = 0 (see, e.g., [17,Theorem 3]). Due to the loss of control on Q in L ∞ (0, T ; L ∞ ), extra difficulties arise in obtaining estimates for those highly nonlinear terms of the system (see Proposition 12).…”

“…Next, we recall that when ξ = 0 the system (1.1)-(1.5) enjoys a maximum principle for the Q-equation (1.3) (see, e.g., [17,Theorem 3]). However, since now the parameter ξ is allowed to be nonzero, the maximum principle property is no longer valid.…”

Weak Time Regularity and Uniqueness for a $Q$-Tensor Model

“…The existence of weak solutions and a uniqueness criteria have been already studied (see [11] for a Cauchy problem in the whole R 3 and [7] for a initial-boundary problem in a bounded domain Ω).…”

“…The existence of weak solutions and a uniqueness criteria have been already studied (see [11] for a Cauchy problem in the whole R 3 and [7] for a initial-boundary problem in a bounded domain Ω).Nevertheless, results on strong regularity have only been treated in [11] for a Cauchy problem in the whole R 3 .In this paper, imposing Dirichlet or Neumann boundary conditions, we show the existence and uniqueness of a local in time weak solution with weak regularity for the time derivative of the velocity and the tensor variables (u, Q). Moreover, we gives a regularity criteria implying that this solution is global in time.…”