[Paper Review] Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions

Helmut Abels, Georg Dolzmann+1 more|arXiv (Cornell University)|Dec 20, 2013

Navier-Stokes equation solutions19 citations

TL;DR

This paper establishes the existence and uniqueness of local strong solutions for the Beris-Edwards model of nematic liquid crystals with homogeneous Dirichlet boundary conditions on a bounded domain in ℝᵈ (d=2,3). By incorporating a Q-tensor-dependent viscosity and using linearization with Banach's fixed-point theorem, the authors prove local well-posedness in a suitable Sobolev framework, advancing the mathematical analysis of complex fluid-structure interactions in liquid crystal flows.

ABSTRACT

Existence and uniqueness of local strong solution for the Beris--Edwards model for nematic liquid crystals, which couples the Navier-Stokes equations with an evolution equation for the Q-tensor, is established on a bounded domain in the case of homogeneous Dirichlet boundary conditions. The classical Beris--Edwards model is enriched by including a dependence of the fluid viscosity on the Q-tensor. The proof is based on a linearization of the system and Banach's fixed-point theorem.

Motivation & Objective

• To establish the existence and uniqueness of local strong solutions for the Beris-Edwards model of nematic liquid crystals with homogeneous Dirichlet boundary conditions.
• To extend the classical Beris-Edwards model by incorporating a Q-tensor-dependent fluid viscosity, reflecting physical dependence of viscosity on molecular orientation.
• To provide a rigorous mathematical framework for the coupled Navier-Stokes and Q-tensor evolution system under physically relevant boundary and initial conditions.
• To address the challenge of nonlinear coupling between fluid velocity, pressure, and Q-tensor dynamics in a bounded domain with smooth boundary.
• To contribute to the understanding of well-posedness in complex fluid models with variable viscosity and tensor-valued order parameters.

Proposed method

• Formulate the coupled system: Navier-Stokes equations with variable viscosity ν(Q) and a parabolic evolution equation for the Q-tensor.
• Use a linearization scheme around a reference solution to transform the nonlinear system into a sequence of linear subproblems.
• Apply Banach's fixed-point theorem in a suitable function space (Sobolev spaces H¹₀ and H²) to prove existence and uniqueness of solutions.
• Employ energy estimates and Sobolev embeddings (e.g., L⁶ and L³ norms) to control nonlinear terms involving Q, u, and their gradients.
• Utilize Poincaré and Young’s inequalities to bound higher-order derivatives and ensure convergence of the iterative scheme.
• Incorporate the full dependence of viscosity ν(Q) on the Q-tensor, modeling physical viscosity variations due to molecular alignment.

Experimental results

Research questions

• RQ1 Does the Beris-Edwards model with Q-tensor-dependent viscosity admit a unique local strong solution under homogeneous Dirichlet boundary conditions?
• RQ2 How does the inclusion of Q-dependent viscosity affect the regularity and well-posedness of the fluid-tensor system?
• RQ3 Can the nonlinear coupling between Navier-Stokes and Q-tensor evolution be handled via a fixed-point argument in a high-order Sobolev space framework?
• RQ4 What are the necessary regularity assumptions on initial data (u₀, Q₀) to ensure local existence of strong solutions?
• RQ5 How do the nonlinear terms involving ∇Q, ΔQ, and their interactions affect the energy estimates and convergence of the iterative scheme?

Key findings

• Local strong solutions exist for the Beris-Edwards model with Q-tensor-dependent viscosity and homogeneous Dirichlet boundary conditions on a bounded C⁴ domain.
• The solution is unique in a suitable neighborhood of the initial data within a complete metric space of functions with sufficient Sobolev regularity.
• The proof relies on a linearization procedure and Banach’s fixed-point theorem applied to the coupled system in H¹₀(Ω) × H²(Ω) × H¹₀(Ω; S₀) spaces.
• Energy estimates involving ∇ΔQ, Au, and ΔQ are controlled via Sobolev embeddings and Young’s inequality, with constants depending on the initial Q-tensor.
• The viscosity ν(Q) is assumed to be smooth and bounded in terms of Q, ensuring the necessary regularity for the fixed-point argument.
• The system is well-posed locally in time, with the solution class satisfying u ∈ H¹(0,T;L²) ∩ L²(0,T;H²), Q ∈ H¹(0,T;H¹₀) ∩ L²(0,T;H²), and ∇ΔQ ∈ L²(Ω_T).