Parabolic Ericksen–Leslie System

• The parabolic Ericksen–Leslie system is a dissipative PDE model for nematic liquid crystals, coupling fluid flow with molecular orientation dynamics.
• It employs an energy–dissipation variational framework with stress decomposition and Parodi’s relation to ensure stability and smooth evolution.
• Rigorous mathematical analysis confirms global regularity and well-posedness under sufficient fluid dissipation and controlled singularity formation.

The parabolic Ericksen–Leslie system describes the macroscopic evolution of nematic liquid crystals, coupling incompressible viscous fluid flow with the evolution of a molecular orientation field (“director”). The system encodes complex interactions between velocity, elastic and viscous stresses, and the order parameter, and its “parabolic” character refers to the dissipative and smoothing effects provided by viscous and relaxation terms. The model has crystallized into an analytically tractable PDE system studied for both its rigorous mathematical properties and its fidelity to the physics of liquid crystalline flows.

1. Physical Derivation and Mathematical Structure

The derivation of the general Ericksen–Leslie system proceeds from a variational energy–dissipation principle (EnVarA). The total energy, combining kinetic energy and elastic (Oseen–Frank or Landau–de Gennes type) energy, is coupled to a quadratic dissipation potential in terms of the strain rate and director derivative rates.

Systematically, the action functional is

$$A(x) = \int_0^T \left( E_{\text{kinetic}} - E_{\text{int}} \right)\,dt,$$

where the flow map encodes the Lagrangian structure. By the least action principle (Hamiltonian component) and the maximum dissipation principle (dissipative component), one obtains a coupled PDE for the velocity $v(x,t)$ and director $d(x,t)$: $$v_t + v\cdot \nabla v = -\nabla P - \nabla\cdot(\nabla d\odot \nabla d) + \nabla\cdot\tilde\sigma,$$ and an evolution (transport-relaxation) equation for $d$.

The induced elastic stress $\tilde{\sigma}$ splits as

$$\tilde{\sigma} = \tilde{\sigma}_{\text{cons}} + \tilde{\sigma}_{\text{diss}},$$

with the Hamiltonian (conservative) component

$$\tilde{\sigma}_{\text{cons}} = -\frac{1}{1-\lambda_2}(A d - f(d))\otimes d,$$

and the dissipative component expressed in terms of Leslie coefficients (see (Wu et al., 2011), eqn. (1.5)-(1.6)) and kinematic terms $A = \frac{1}{2}(\nabla v+\nabla v^T)$, $N = d_t + v\cdot \nabla d - \Omega d$.

Critical in this decomposition is Parodi’s relation,

$\lambda_2 = - (\mu_2 + \mu_3),$

which yields cancellations between stress components, making possible a clean separation between energy-conserving and dissipative dynamics.

2. Well-Posedness: Existence, Uniqueness, and Regularity

The global well-posedness of the parabolic Ericksen–Leslie system is achieved under physical assumptions that include a sufficiently strong dissipation in the fluid (i.e., large enough fluid viscosity $\mu_4$). In particular, global regularity in three dimensions is established under the condition

$\mu_4 \geq \mu_4^*,$

where the threshold $\mu_4^*$ is determined by the configuration and the other Leslie coefficients.

Energy inequalities form the analytical backbone: $$\frac{d}{dt} E(t) \le -\int_Q \left[ c_1 |\nabla v|^2 + c_2\|A d-f(d)\|^2 \right]\,dx.$$ For higher-regularity, evolution of the higher-order energy

$$\mathcal{A}(t) = \| \nabla v(t) \|_{L^2}^2 + \|A d(t)- f(d(t))\|_{L^2}^2$$

is controlled by differential inequalities of the form

$$\frac{d}{dt}\mathcal{A}(t) + \alpha \mathcal{A}_1(t) \le C[\mathcal{A}(t)^3 + \mathcal{A}(t)],$$

enabling global control via a bootstrap and Grönwall-type arguments (see Theorem 4.1, (Wu et al., 2011)).

Well-posedness (existence, uniqueness, and regularity) holds for sufficiently smooth initial data (e.g., $v_0 \in V$, $d_0 \in H^3$), and the solution lies in

$$v\in L^\infty(0,T;V)\cap L^2(0,T;H^3),\quad d\in L^\infty(0,T;H^2)\cap L^2(0,T;H^3).$$

For small initial data, global existence for arbitrary times follows. The system can become ill-posed or develop singularities if dissipation falls below thresholds or physical constraints on coefficients are violated.

Constraints on Leslie coefficients, such as in condition (2.6) of (Wang et al., 2012), ensure the dissipative character and are typically necessary and sufficient for well-posedness (e.g., $B_2 \geq 0$, $2B_1+B_3 \geq 0$, $B_1+B_2+B_3 \geq 0$ for appropriate $B_j$ linear combinations of Leslie coefficients).

3. Energy Dissipation and Stability Near Equilibrium

Dissipativity is encoded in energy laws: $$\frac{d}{dt} \int (|v|^2 + E_F)\, dx = -\int \left( \gamma |\nabla v|^2 + (\text{dissipative terms}) + |n \times h|^2 \right) dx,$$ with $E_F$ the Oseen–Frank energy and $h$ the molecular field ((Wang et al., 2012), eqn. (2.1)).

Lyapunov stability is established near local minimizers $d^*$ of the elastic energy: $$E(d) = \int_Q \left[ \frac{1}{2} |\nabla d|^2 + F(d) \right] dx, \quad F(d) = \frac{1}{4} (|d|^2 - 1)^2.$$ If initial data are sufficiently close to $(0,d^*)$, the solution decays to $(0, d^*)$ in norm: $$\lim_{t\to\infty} \|v(t)\|_{V} + \|d(t) - d^*\|_{H^2} = 0,$$ with the energy converging to $E(d^*)$ (see Theorem 5.2, (Wu et al., 2011)).

The convergence rate is quantified using the Lojasiewicz–Simon inequality,

$$\|-\Delta d + f(d)\|_{H^{-1}} \geq C |E(d)-E(d^*)|^{1-\theta},$$

with $0<\theta<1/2$, providing a gradient structure and rate for relaxation toward equilibrium.

4. Role and Consequence of Parodi’s Relation

Parodi’s relation is essential for several structural, analytical, and stability properties:

• In the formal derivation, it ensures exact cancellation between certain conservative and dissipative stress components, allowing the system to be written with a split of energy-conserving and dissipative dynamics ((Wu et al., 2011), Section 1.2).
• In nonlinear and linear stability analyses, Parodi’s relation enables higher-order energy inequalities (e.g., Lemma 5.1) that are otherwise inaccessible.
• Linearized plane-wave analysis shows that violation of Parodi’s relation leads to the existence of unstable modes. Explicitly, for disturbances of the form $v(x,t) = b \exp(-i(m\cdot x - \omega t))$, $d(x,t) = a \exp(-i(m\cdot x - \omega t))$, the dispersion relation produces instability (positive $\operatorname{Im} \omega$) unless Parodi’s relation holds ((Wu et al., 2011), Theorem 6.1).
• Mathematical necessity is further confirmed: if certain explicit combinations of the Leslie coefficients vanish for angles parameterizing the director-wave vector alignment, Parodi’s relation must hold (Lemma 6.1).

The necessity and sufficiency of Parodi’s relation thus ties directly to Onsager reciprocity and the guarantee of stable, dissipative evolution.

5. Singularities, Weak Solutions, and Partial Regularity

While strong parabolic regularity is present under sufficient energy dissipation, singularity scenarios arise when this is violated or when the system is in a mixed parabolic–hyperbolic regime. For one-dimensional reductions (such as Poiseuille flows), cusp singularities (blow-up of derivatives while energy and solution remain H\"older continuous) occur due to nonlinear wave steepening (Chen et al., 2019).

In such scenarios, weak solutions can still be constructed and are globally defined owing to cancellation mechanisms. A key object is the "flux density" $J = u_x + \theta_t$, which remains bounded even as $u_x$ and $\theta_t$ separately blow up. The evolution of $J$ absorbs singularities and enables the construction of energy-bounded global weak solutions (Chen et al., 2019, Chen et al., 2023).

Partial regularity results, such as those for kinematic transport models, confirm that singular sets are small (with parabolic Hausdorff dimension at most $15/7$ in $\mathbb{R}^3$), and outside these sets, solutions are smooth (Du et al., 2020).

6. Extensions: Regularized, Thermodynamic, and Stochastic Variants

A family of regularized Ericksen–Leslie models (including Leray–$\alpha$, Bardina, and Navier–Stokes–Voigt models) augments classical equations with smoothing operators while retaining the essential parabolic-dissipative structure (Gal et al., 2014). For such models, global attractors can be constructed, and convergence to equilibrium is established using the Lojasiewicz–Simon inequality.

Thermodynamically consistent non-isothermal extensions include variable temperature, energy, and entropy equations, with the first and second laws rigorously enforced. Dissipation inequalities constrain the generalized Leslie coefficients and guarantee energy decay or entropy production (Anna et al., 2017).

Stochastic versions (Du et al., 8 Jan 2024) incorporate external white noise and rigorously demonstrate the existence of martingale suitable weak solutions. Even in the presence of stochastic forcing, the set of singular points in space-time paths has one-dimensional parabolic Hausdorff measure zero.

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The synthesis above clarifies that the parabolic Ericksen–Leslie system is a variationally structured, energy-dissipative PDE model for nematic liquid crystals whose analytical and physical properties are governed by precise relationships between viscosity, elasticity, and coupling coefficients. The theory encompasses global existence and uniqueness in strong norms under sufficient dissipation, nonlinear stability near equilibria, partial regularity for weak solutions, and broad applicability under generalized boundary and constitutive conditions. The energetic and dissipative structure—especially the energy law, the splitting of stress components, and Parodi’s symmetry—remains the unifying theme in both the rigorous analysis and the modeling of liquid crystalline flows.