Landau–de Gennes Flow Dynamics

• Landau–de Gennes flow is a variational evolution of a nematic Q-tensor field driven by a free energy that captures isotropic, uniaxial, and biaxial states.
• It employs diverse elastic energy models, from one-constant to anisotropic and quartic formulations, alongside singular bulk potentials to enforce physical admissibility.
• Finite element discretizations and implicit time-stepping schemes yield energy stability and convergence, accurately modeling phase transitions and defect dynamics.

Landau–de Gennes flow is the evolution of a nematic $Q$-tensor field driven by a Landau–de Gennes free energy. The order parameter is a symmetric, traceless matrix field,

$$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$

or, in equivalent notation, $$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$. Uniaxial states have the form $Q=s(n\otimes n-I/3)$, the isotropic state is $Q=0$, and biaxial states have three distinct eigenvalues. In the PDE literature, the term usually refers to an $L^2$-type gradient flow or a closely related relaxational dynamics for $Q$; in broader continuum settings it also appears as the no-flow limit of Beris–Edwards models on fixed or evolving surfaces (Majumdar et al., 2015, Nitschke et al., 2023, Elafandi et al., 2024).

1. Energetic formulation and admissible states

The classical Landau–de Gennes framework combines an elastic energy with a bulk potential. In the one-constant setting,

$$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$

with

$$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$

At the isotropic–nematic transition temperature, $A=B^2/(27C)$, and the bulk potential is minimized both by $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$0 and by the continuum of uniaxial states $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$1 with $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$2 (Majumdar et al., 2015).

More general Landau–de Gennes flows arise by replacing the isotropic elastic density with anisotropic quadratic or quartic elastic terms. One anisotropic quadratic elastic energy is

$$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$3

with the structural condition $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$4, which yields strong Legendre ellipticity and a coercive lower bound. A quartic elastic variant introduces tensorial factors $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$5, $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$6 and the additional term $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$7; when $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$8, it reduces to the Golovaty–Novack–Sternberg quartic elastic model (Liu et al., 2020, Elafandi et al., 2024).

A distinct class of models replaces the quartic bulk polynomial by the Ball–Majumdar singular potential. In that case the physically admissible set is characterized by eigenvalue constraints

$$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$9

and the bulk potential is finite only on this set, convex and lower semicontinuous on $$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$0, smooth in the interior, and divergent at the physical boundary. This enforces physical admissibility directly at the continuum level rather than through a posteriori estimates (Feireisl et al., 2012, Feireisl et al., 2013).

This variety of energies implies that “Landau–de Gennes flow” is not a single canonical PDE. It is a variational class of $$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$1-tensor evolutions whose analytical and physical properties depend strongly on the elastic sector, the bulk potential, the boundary conditions, and whether hydrodynamic or thermal couplings are retained.

2. Relaxational dynamics and dissipation structure

The basic gradient-flow structure is

$$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$2

up to mobility, viscosity, or projection factors. For the classical one-constant energy, the $$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$3 gradient flow is

$$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$4

which becomes

$$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$5

This is a coupled system of five nonlinear parabolic PDEs for the independent components of $$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$6, and it satisfies the dissipation identity

$$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$7

(Majumdar et al., 2015).

For quartic elastic energies, the symmetric-traceless constraint can be enforced explicitly by projection. One formulation is

$$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$8

with a variational derivative containing nonlinear $$\mathcal S=\{Q\in \mathbb{R}^{3\times 3}:Q^\top=Q,\ \operatorname{tr}Q=0\}$$9, $Q=s(n\otimes n-I/3)$0, and $Q=s(n\otimes n-I/3)$1 contributions. The associated continuous energy law is

$Q=s(n\otimes n-I/3)$2

so the dynamics are purely relaxational and converge to local minimizers of the chosen energy (Elafandi et al., 2024).

With anisotropic quadratic elasticity and singular bulk potential, the flow remains gradient-driven but the parabolic operator becomes non-diagonal because of the $Q=s(n\otimes n-I/3)$3 and $Q=s(n\otimes n-I/3)$4 terms. The full PDE contains

$Q=s(n\otimes n-I/3)$5

together with the projected singular bulk force and the linear destabilizing term $Q=s(n\otimes n-I/3)$6. The convexity of the singular potential is central in the dissipation and comparison theory (Liu et al., 2020).

On surfaces, suppressing hydrodynamics and fixing the geometry yields the pure Landau–de Gennes flow limit

$Q=s(n\otimes n-I/3)$7

or, in conforming surface variables,

$Q=s(n\otimes n-I/3)$8

where curvature enters explicitly through the surface mean curvature $Q=s(n\otimes n-I/3)$9, Gaussian curvature $Q=0$0, and shape operator $Q=0$1 (Nitschke et al., 2023).

3. Phase fronts, mean-curvature motion, and geometric limits

At the isotropic–nematic transition, the coexistence of the isotropic well $Q=0$2 and the uniaxial nematic well creates a distinguished front-propagation regime. In a three-dimensional droplet with radial hedgehog boundary data, radially symmetric uniaxial dynamics reduce to a scalar equation for the radial order parameter $Q=0$3. For “efficient interface” initial data and sufficiently small elastic constant, the isotropic–nematic interface is well defined and, for small times, its radius $Q=0$4 satisfies

$Q=0$5

which is motion by mean curvature for spheres. Numerical experiments also show rapid front formation for broader uniaxial, biaxial, and non-radially symmetric initial data (Majumdar et al., 2015).

A separate sharp-interface scaling uses

$Q=0$6

As $Q=0$7, the energy concentrates near a codimension-one interface $Q=0$8. On the isotropic side, $Q=0$9; on the nematic side, $L^2$0, where the director satisfies

$L^2$1

with homogeneous Neumann boundary condition on the evolving interface, while the interface itself satisfies

$L^2$2

The rigorous convergence is proved under well-prepared initial data and as long as the limiting mean curvature flow remains smooth (Laux et al., 2020).

On a disc, the long-time behavior depends strongly on planarity and boundary forcing. Planar initial conditions remain planar for all time and preserve an isotropic core, whereas non-planar perturbations can trigger escape into the third dimension and convergence to a smooth globally ordered uniaxial state. Under non-minimal biaxial Dirichlet data, boundary layers form and the interior may become largely ordered or almost entirely isotropic depending on the initial interface location (Majumdar et al., 2015).

Surface formulations introduce an additional geometric layer. In the conforming decomposition $L^2$3, the elastic molecular field contains terms such as

$L^2$4

so curvature can directly bias alignment and defect structure. This identifies curvature-driven reorientation as an intrinsic part of surface Landau–de Gennes flow rather than an external correction (Nitschke et al., 2023).

4. Existence, regularity, and physical admissibility

For the classical quartic-bulk gradient flow on bounded smooth domains with smooth Dirichlet data, standard parabolic theory yields a unique smooth solution for all $L^2$5. In that setting the flow admits an $L^2$6 bound

$L^2$7

which is used repeatedly in the analysis of fronts and long-time dynamics (Majumdar et al., 2015).

With anisotropic elasticity and singular potential, the analytical framework is more delicate. For periodic domains $L^2$8, $L^2$9, and initial data $Q$0 with $Q$1, the energy is proper, bounded from below, lower semicontinuous, and $Q$2-convex with $Q$3. This yields a unique global EVI gradient-flow solution. For every $Q$4, the solution is strong, belongs to $Q$5, and satisfies an exact energy identity. Under the stronger coercivity condition

$Q$6

the solution eventually detaches from the physical boundary: after some $Q$7, all eigenvalues satisfy

$Q$8

Before that time, the contact set $Q$9 satisfies $$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$0 in three dimensions and $$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$1 in two dimensions (Liu et al., 2020).

Non-isothermal extensions couple $$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$2 to incompressible flow and temperature. In one periodic three-dimensional model with singular bulk potential, the unknowns are $$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$3, the molecular field is

$$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$4

and the total energy and entropy satisfy thermodynamically consistent balances. Global-in-time weak solutions exist for arbitrary physically relevant initial data, and the temperature remains strictly positive: $$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$5 (Feireisl et al., 2012).

A related nonisothermal Beris–Edwards-type model on the torus employs the free energy

$$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$6

and proves global weak solutions under integrated entropy inequalities. A key estimate is a Hessian coercivity inequality for the Ball–Majumdar potential, used to control $$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$7 and $$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$8 in $$F[Q]=\int_\Omega \left(\frac{L}{2}|\nabla Q|^2+f_B(Q)\right)\,dx,$$9 (Feireisl et al., 2013).

5. Finite element discretization and computational analysis

A recent finite element treatment of quartic elastic Landau–de Gennes flow uses quasi-uniform triangulations and the discrete spaces

$$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$0

consisting of continuous, piecewise linear, symmetric traceless $$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$1 fields. A basis is built from five symmetric-traceless generators $$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$2 and nodal hat functions $$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$3, so that

$$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$4

is a basis of $$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$5 (Elafandi et al., 2024).

Time discretization is nonlinearly implicit and midpoint-like. Given $$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$6, one solves for $$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$7 through

$$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$8

with $$f_B(Q)=\frac{A}{2}\operatorname{tr}(Q^2)-\frac{B}{3}\operatorname{tr}(Q^3)+\frac{C}{4}\big(\operatorname{tr}(Q^2)\big)^2.$$9 representing the quartic elastic and bulk contributions evaluated at temporal midpoints. The scheme satisfies the discrete energy dissipation law

$A=B^2/(27C)$0

which gives unconditional energy stability. Solvability at each time step is obtained by a fixed-point iteration starting from $A=B^2/(27C)$1; for sufficiently small $A=B^2/(27C)$2, the fixed-point map is contractive and the solution is unique (Elafandi et al., 2024).

The same analysis proves coercivity and $A=B^2/(27C)$3-convergence of the discrete energies when $A=B^2/(27C)$4. In weak $A=B^2/(27C)$5, cluster points of discrete global minimizers converge to global minimizers of the continuum energy, and isolated local minimizers are approximated by discrete minimizers. The $A=B^2/(27C)$6-convergence argument uses truncation, mollification, and piecewise linear interpolation. For $A=B^2/(27C)$7, the numerical scheme remains energy stable and exhibits similar dynamics, but the $A=B^2/(27C)$8-convergence proof is not established (Elafandi et al., 2024).

The numerical experiments reproduce isotropic-to-nematic phase transitions and tactoid dynamics. On $A=B^2/(27C)$9, the reported convergence is approximately second order in $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$00 for $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$01 and the energy, and second order in $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$02. A practical CFL-like restriction is observed: the maximum stable $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$03 scales like $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$04. On the unit disk, degree $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$05, degree $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$06, and degree $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$07 tactoids show, respectively, splitting into $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$08 or $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$09 defect pairs, or collapse to a uniform nematic state; in all cases the discrete energy decays monotonically (Elafandi et al., 2024).

6. Relation to Oseen–Frank theory, hydrodynamics, and other uses of “flow”

A central structural property is the reduction to Oseen–Frank elasticity on the uniaxial manifold. For

$$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$10

the quartic elastic model of Golovaty et al. reduces to the Oseen–Frank energy for suitable parameter identifications. The additional quartic term in the modified model is motivated by the identity

$$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$11

which justifies including $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$12 and links its coefficient directly to $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$13 and $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$14 (Elafandi et al., 2024).

Landau–de Gennes flow is not synonymous with the Beris–Edwards system. Pure gradient-flow models suppress advection, corotation, velocity-pressure coupling, and Ericksen stresses. By contrast, bulk non-isothermal Beris–Edwards models use

$$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$15

or closely related forms, and surface Beris–Edwards models use objective derivatives $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$16 or $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$17, inextensible surface flow, and coupled momentum balances. The pure Landau–de Gennes flow is recovered in the no-flow regime, for example by setting $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$18, $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$19, or imposing the NF constraint on surfaces (Feireisl et al., 2012, Nitschke et al., 2023).

A separate use of the word “flow” appears in functional renormalization-group studies of the Landau–de Gennes model. There the flow variable is the coarse-graining scale $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$20, with $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$21, and the effective average action satisfies the Wetterich equation rather than a time-dependent relaxational PDE. In that framework, the numerical solution of the RG flow indicates a first-order nematic–isotropic transition and yields the estimate

$$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$22

for the transition-temperature difference in the studied parametrization (Qin et al., 2018).

Taken together, these developments establish Landau–de Gennes flow as a broad variational framework for nematic evolution. It accommodates isotropic and nematic phases, biaxiality, defect cores, front propagation, geometric coupling to curvature, singular physical constraints on eigenvalues, and structure-preserving numerical discretization within a common $$Q \in S_0^{3\times 3}=\{Q\in \mathbb{R}^{3\times 3}:Q^T=Q,\ \operatorname{tr}Q=0\},$$23-tensor formalism.