Beris–Edwards Hydrodynamic Theory

• Beris–Edwards hydrodynamic theory is a tensorial continuum framework that couples incompressible Navier–Stokes flow with a Q-tensor evolution to describe nematic liquid crystals.
• It captures energetic, hydrodynamic, and defect phenomena, enabling accurate modeling of phase transitions and spontaneous defect formation.
• The theory extends classical director models by incorporating elastic anisotropy, Q-dependent viscosity, and rigorous existence and regularity results.

The Beris–Edwards hydrodynamic theory provides a tensorial, continuum-level description of nematic liquid crystal dynamics by coupling incompressible Navier–Stokes equations for velocity with a dissipative, parabolic evolution for an orientational order parameter represented as a symmetric, traceless $3\times3$ $Q$-tensor. This framework unifies the energetic, hydrodynamic, and defect-formation features of nematic fluids, accommodating both phase transitions and vectorial or higher-order orientational phenomena, and resolves limitations inherent in director-based models. The theory rigorously admits suitable weak solutions with partial regularity in three dimensions and captures essential phenomena such as eigenvalue-range preservation, multistability, spontaneous defect formation, and convergence to classical limits such as the Ericksen–Leslie model (Du et al., 2019, Feng et al., 2021, Wu et al., 2017, Valani et al., 18 Nov 2025, Wang et al., 2013).

1. Mathematical Structure of the Beris–Edwards System

Let $Q(x,t)\in S_0(3)$, the space of symmetric, traceless $3\times3$ matrices, represent the orientational order, $u(x,t)$ the incompressible velocity field, and $P(x,t)$ the pressure. The coupled PDE system is:

$$\begin{cases} \partial_t u + u \cdot \nabla u + \nabla P = \mu \Delta u + \nabla \cdot \sigma, \qquad \nabla \cdot u = 0, \ \partial_t Q + u \cdot \nabla Q - S(\nabla u, Q) = \Gamma H, \end{cases}$$

where $$D_{ij} = \frac{1}{2}(\partial_i u_j + \partial_j u_i)$$ and $$\Omega_{ij} = \frac{1}{2}(\partial_i u_j - \partial_j u_i)$$. In the co-rotational (flow-aligning) regime ($\xi=0$), $S(\nabla u, Q) = \Omega Q - Q \Omega$. The elastic (symmetric) and reactive (antisymmetric) stresses enter via $\sigma^s$ and $\sigma^a$, and the system is closed via the molecular field $H = -\delta\mathcal{F}/\delta Q$ derived from a variational Landau–de Gennes or Ball–Majumdar energy functional. The energy has the “one-constant” form:

$$\mathcal{F}[Q] = \int_{\Omega} \Big( \frac{L}{2} |\nabla Q|^2 + f_{\text{bulk}}(Q) \Big) dx.$$

This structure admits several generalizations, e.g., four-constant elasticity (Feng et al., 2021), Q-dependent fluid viscosity (Abels et al., 2013), or more general geometric domains and boundary conditions (Barbera et al., 28 Jun 2024).(Du et al., 2019, Abels et al., 2013, Barbera et al., 28 Jun 2024, Feng et al., 2021)

2. Bulk Potentials and Energetic Landscape

Two primary choices for the bulk energy $f_{\text{bulk}}$ are prevalent:

• Landau–de Gennes potential:

$$f_{\text{LdG}}(Q) = \frac{a}{2} \operatorname{Tr}(Q^2) - \frac{b}{3} \operatorname{Tr}(Q^3) + \frac{c}{4} [\operatorname{Tr}(Q^2)]^2,$$

yielding

$$H = -L \Delta Q + a Q - b[Q^2-(\operatorname{Tr}Q^2)/3 I] + c\operatorname{Tr}(Q^2)Q.$$

• Ball–Majumdar potential (singular bulk):

$$f_{\text{BM}}(Q) = v\,G_{\text{BM}}(Q) - \kappa\,|Q|^2,$$

where $G_{\text{BM}}$ blows up as any eigenvalue of $Q$ approaches $\pm 1/3$, enforcing physicality; $H$ is given accordingly.

The elastic term may include generalized four-constant elastic energies, admitting coupled cubic and quartic invariants in gradients and $Q$ itself (Feng et al., 2021, Du et al., 2019). The parameter regimes select the active orientational phase (uniaxial/biaxial, nematic/isotropic) and define stability and defect core structure.

3. Solution Concepts and Regularity Theory

The analytical framework for the Beris–Edwards system distinguishes between weak, strong, and suitable weak solutions:

• Weak solutions: $(u,Q)$ lies in Bochner spaces

$$u \in L^{\infty}(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega)), \quad Q \in L^{\infty}(0,T;H^1(\Omega)) \cap L^2(0,T;H^2(\Omega)),$$

with the PDEs interpreted in the sense of distributions (Du et al., 2019).

• Suitable weak solutions: These satisfy, in addition, the local energy inequality at all test functions $\varphi \in C^\infty_c(\Omega \times [0,T))$, furnishing a local control of kinetic and orientational dissipation:

$$\int (|u|^2 + |\nabla Q|^2)\varphi + 2 \int_0^t \int (|\nabla u|^2 + |H|^2) \varphi \leq \ldots$$

Such solutions admit a partial regularity theory analogous to Caffarelli–Kohn–Nirenberg for Navier–Stokes: smoothness holds off a singular set $\Sigma$ of vanishing $1$-dimensional parabolic Hausdorff measure (Du et al., 2019).

A nontrivial aspect is the preservation of physical constraints, notably the eigenvalue-bounds on $Q$-tensors, under the co-rotational flow (via maximum principles). The presence of singular potentials (e.g., Ball–Majumdar) ensures that the physical domain $|Q|<1/3$ cannot be violated in global evolution, under suitable initialization, by convexity arguments and Yosida–Moreau regularization (Du et al., 2019, Wu et al., 2017).

4. Analytical Techniques and Existence Results

The existence and partial regularity proofs integrate several advanced techniques:

1. Retarded-time mollification: System is regularized by mollifying convective terms using time-lagged arguments; energy inequalities are propagated at the regularized level and passed to the limit (Du et al., 2019).
2. Maximum principles: $L^\infty$ bounds for $Q$ are maintained for all time, provided sufficiently small initial norm and where the singular potential is active (Liu et al., 2018, Wu et al., 2017).
3. $\varepsilon$-regularity and blow-up: Small energy on parabolic cylinders yields local smoothness via contradiction and rescaling arguments; any nontrivial blow-up solves the linearized system and must be smooth (Du et al., 2019).
4. Morrey-space bootstrap: On small spatial-temporal cylinders, local higher integrability is obtained, and estimates are closed via Duhamel-type and Oseen–kernel methods (Du et al., 2019).
5. Covering argument: The singular set is shown to be of vanishing parabolic 1-Hausdorff measure by estimating the dissipation and using Vitali covers (Du et al., 2019).

Global weak (suitable) solutions exist for any initial data, and their regularity is restricted only by the possible formation over time of points/curves of singular dissipation, a structure paralleling results for 3D incompressible Navier–Stokes (Du et al., 2019).

5. Scaling Limits and Reduction to Classical Nematic Models

The Beris–Edwards formalism rigorously contains director and phase-field models as singular limits:

• Ericksen–Leslie limit ($L \to 0$ or $\varepsilon \to 0$): In the limit of vanishing elastic constant, solutions collapse onto the manifold of uniaxial $Q$, $Q^*(x,t) = s_+ (d^*\otimes d^* - \frac{1}{3}I)$, and the system reduces to director-based hydrodynamics. Multiple works provide Hilbert-expansion derivations of the explicit Oseen--Frank elastic coefficients and Leslie viscosities in terms of $Q$-tensor, Landau–de Gennes, and Onsager parameters (Feng et al., 2021, Wang et al., 2013, Xin et al., 2021). The limiting director field may develop finite-energy singularities (defects) corresponding to observed topological defects.
• Sharp-interface and phase-separation regimes: For the diffuse interface model (phase transitions nematic-isotropic), using relative entropy methods, it has been shown that as the interfacial thickness parameter vanishes, the Beris–Edwards solution converges to a sharp-interface hydrodynamic limit with the mean-curvature flow governing interface dynamics; the nematic region reduces to Oseen–Frank evolution (Su, 30 Jan 2024).
• Hydrostatic and thin-domain limits: In thin geometry (strip domains), the system reduces to a hydrostatic Navier–Stokes model, with the $Q$-tensor driven to vanish everywhere except possibly for boundary layers, due to vertical diffusion dominance (Li et al., 2022).
• High Ericksen number decoupling: As the Ericksen number increases, a weakly coupled model is obtained; eigenvalue-range preservation is lost outside the co-rotational regime, and the $Q$-field is driven by reaction–vorticity balance (Wu et al., 2017).

6. Physical Phenomena: Defects, Multistability, and Spatiotemporal Complexity

The Beris–Edwards system provides a rigorous, dynamical account of nematic liquid crystal phenomena inaccessible to classical director or macroscopic theories:

• Defect formation and evolution: Both topologically necessary and dynamically emergent defects are captured. Vorticity-induced and phase-mismatch mechanisms drive high gradients in $Q$, leading to defect nucleation and motion as solutions to appropriately rescaled Beris–Edwards equations (Wu et al., 2017).
• Multistability and bifurcation structure: In confined geometries with strong boundary alignment, the model admits multiple coexisting equilibrium and flow states (e.g., Bowser and Dowser patterns), with transitions to oscillatory dynamics and spatiotemporal chaos under external driving (e.g., pressure gradients) (Valani et al., 18 Nov 2025).
• Phase transitions: The $Q$-tensor field provides a natural order parameter for nematic–isotropic interfaces, with interfacial thickness and dynamical law for the interface derivable as a singular limit (Su, 30 Jan 2024).
• Distinction from director-based models: Because the $Q$-tensor includes information about both scalar order and biaxiality, phenomena such as scalar order suppression in the bulk, defect-core regularization, and multistable structural landscapes are described at a level inaccessible to the Ericksen–Leslie model (Valani et al., 18 Nov 2025, Wang et al., 2013).

7. Numerical and Functional Analytic Foundations

The mathematical analysis of the Beris–Edwards system spans multiple functional and computational regimes:

• Maximal regularity and well-posedness: Local and global well-posedness results have been established in various spatial domains, including bounded domains, periodic boxes, and half-spaces, utilizing $\mathcal{R}$-boundedness of solution operators and operator-valued multiplier theory to derive maximal $L^p$-$L^q$ regularity and continuous dependence on data (Barbera et al., 28 Jun 2024, Abels et al., 2013, Liu et al., 2018).
• Numerical schemes: Unconditionally energy-stable, semi-implicit schemes—such as those based on the Invariant Energy Quadratization (IEQ) method—have been proven to converge to weak solutions, with the equivalence of auxiliary-reformulated systems rigorously established (Weber et al., 2023).
• Role of Q-dependent viscosity: Mathematical well-posedness extends to systems where fluid viscosity depends nonlinearly on $Q$, provided the viscosity remains strictly positive and regular as a function on $S_0$ (Abels et al., 2013).
• Approximation techniques: Local existence is often proved via linearization/fixed-point arguments, and global existence relies on a combination of energy dissipation, maximum principle, and compactness frameworks (Liu et al., 2018, Du et al., 2019, Li et al., 2022).

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Key references:

• Suitable weak solutions and partial regularity: (Du et al., 2019)
• Four-constant elasticity and uniaxial-biaxial reduction: (Feng et al., 2021)
• Dynamics and defect mechanisms: (Wu et al., 2017)
• Multistability and chaos in confined nematics: (Valani et al., 18 Nov 2025)
• Reduction to Ericksen–Leslie director theory: (Wang et al., 2013, Xin et al., 2021)
• Sharp interface and phase transitions: (Su, 30 Jan 2024)
• Maximal $L^p$-$L^q$ regularity: (Barbera et al., 28 Jun 2024)
• Well-posedness results in 2D and Q-dependent viscosity: (Liu et al., 2018, Abels et al., 2013)