Free-Boundary Lin-Liu Equations

• Free-Boundary Lin-Liu Equations extend the classical Ericksen-Leslie model by coupling inviscid fluid velocity with director fields in domains with moving free surfaces.
• The analysis employs both Eulerian and Lagrangian formulations, leveraging Sobolev space techniques and κ-regularization to ensure local well-posedness.
• Novel cancellation mechanisms between hyperbolic fluid dynamics and parabolic director heat-flow enable high-order energy estimates and pave the way for multiphysics extensions.

The free-boundary Lin-Liu equations describe the dynamics of inviscid nematic liquid crystals with a moving free surface under the influence of surface tension, constituting a fundamental model for the coupled evolution of incompressible fluid velocity and director fields in mobile domains. This system extends the classical simplified Ericksen-Leslie model by accounting for geometric boundary regularity, nonlinear boundary laws, and intricate fluid-director coupling effects. The central mathematical challenge lies in establishing local well-posedness for the associated initial-boundary value problem, due to loss of symmetry in the linearized operator and coupling of hyperbolic (fluid) and parabolic (director heat-flow) dynamics, all interacting with a moving interface. Recent advances provide a rigorous framework for analyzing local solutions and their regularity, forming a basis for future research into singularities and multiphysics extensions (Luo et al., 13 Dec 2025).

1. Eulerian and Lagrangian Formulations

The free-boundary Lin-Liu system is naturally described in both Eulerian and Lagrangian coordinates. In the Eulerian setting, the unknowns are the incompressible velocity $u(t,x)\in\mathbb{R}^3$, pressure $P(t,x)\in\mathbb{R}$, and the director field $d(t,x)\in S^2$, defined inside a moving domain $D_t\subset\mathbb{R}^3$ with free boundary $\Gamma_t$ and fixed bottom $\Gamma_0$. The equations are: $$\begin{aligned} & u_t + (u\cdot\nabla)u + \nabla P = -\nabla\cdot(\nabla d \otimes \nabla d), \ & \nabla\cdot u = 0, \ & d_t + (u\cdot\nabla)d - \Delta d = |\nabla d|^2 d, \end{aligned}$$ supplemented by kinematic (moving boundary), dynamic (surface tension law), Neumann (director), and bottom slip boundary conditions.

In the Lagrangian frame, the domain is fixed as $\Omega$ and the unknowns $(\eta, v, \phi, q)$ describe the mapping to the moving fluid domain, velocities, director fields, and pressure, with transformed differential operators defined through $A = (D\eta)^{-1}$ and $J=\det D\eta = 1$ by incompressibility. The system is geometrically expressed as: $$\begin{aligned} & \eta_t = v, \ & v_t + \nabla_A q = -\nabla_A \cdot (\nabla \phi \otimes \nabla \phi), \ & \text{div}_A v = 0, \ & \phi_t - \Delta_g \phi = |\nabla \phi|^2 \phi, \end{aligned}$$ with precise geometric boundary conditions using normal projections and surface tension laws.

2. Boundary Conditions and Compatibility Structures

The system imposes several boundary conditions to ensure physical and mathematical well-posedness. The moving boundary $\Gamma_t$ follows the kinematic law $$(\partial_t + u\cdot\nabla)|_{x\in\Gamma_t}f(x) = 0$$, reflecting advective transport. At the free surface, the dynamic boundary relates the pressure to surface tension via $P|_{\Gamma_t} = \sigma H(\Gamma_t)$, with $\sigma$ as a constant surface tension and $H$ twice the mean curvature. The director field satisfies a Neumann boundary condition $\partial_n d = 0$ on moving boundaries, and the velocity obeys slip conditions on the fixed bottom.

The compatibility of initial data is required up to fourth-order time derivatives within Sobolev spaces of high regularity, specifically $u_0\in H^{5.5}(\Omega)$ with divergence-free constraint, and $d_0\in H^{5.5}(\Omega)$ with boundary compatibility. This ensures the absence of singularities at initiation and enables closure of all corresponding a priori estimates at the boundary.

3. Functional Analytic Setting and Energy Functionals

Analysis employs Sobolev spaces $H^s(\Omega)$ and $H^s(\Gamma)$, capturing spatial regularity up to fractional order (notably $5.5$ on $\Omega$, $6.5$ on $\Gamma$). The high-order energy functional aggregates derivatives of $\eta$, $v$, and $\phi$ weighted by surface tension and boundary norms. Specifically,

$$E(t) = \sum_{k=0}^4 \|\partial_t^k\eta(t)\|_{H^{5.5-k}}^2 + \|v_{tttt}(t)\|_{L^2}^2 + \sum_{k=0}^3 \|\partial_t^k\phi(t)\|_{H^{4.5-k}}^2 + \|\partial_t^4\phi\|_{L^2_tL^2_x}^2 + \sigma\|\eta(t)\|_{H^{6.5}(\Gamma)}^2 + \cdots$$

This functional is a key tool in controlling the regularity and convergence of solutions under the dynamics of the coupled system.

4. κ-Regularization, Approximate Systems, and Energy Cancellation Mechanisms

A horizontal mollifier $\Lambda_\kappa$ and small interior mollifier $\rho_\kappa$ are used to construct smoothed approximate problems $(\kappa\text{-system})$ with additional parabolic viscosity terms $\kappa(\partial_A(v\cdot n^\kappa))$. This yields a mixed parabolic-hyperbolic PDE system on the fixed reference domain, solvable via Galerkin/Picard methods. Uniform $\kappa$-independent energy estimates are established by

• Testing high-order time and tangential derivatives (up to fourth order)
• Careful use of geometric commutators to treat variable coefficient issues
• Exploitation of director–velocity cancellation, where cross-terms between $\nabla \phi \otimes \nabla \phi$ and gradient velocity are exactly offset in energy identities
• Parabolic regularity gain for the heat-flow (director variable), bridging regularity gaps

Critical cancellation mechanisms enable closure of energy estimates without derivative loss in the inviscid setting where operator symmetry is broken, utilizing Nash–Moser and tame estimates. Elliptic theory for pressure recovery uses Dirichlet-Neumann maps and Hodge decomposition in $H^s(\Omega)$.

5. Local Well-Posedness Theorem and Passage to Limit

Assuming $u_0, d_0\in H^{5.5}(\Omega)$ and compatibility up to fourth order, one obtains existence of a unique solution $(\eta, v, \phi, q)$ to the Lagrangian system on a short time interval $[0,T]$, with

$$\eta - \mathrm{Id} \in C^0_t H^{5.5}, \quad v \in C^0_t H^{5.5}_x, \quad \phi \in C^0_t H^{5.5}_x \cap L^2_t H^{6.5}, \quad q \in C^0_t H^{4.5}_x.$$

The energy functional $E(t)$ remains finite, depending continuously on initial data. The solution is constructed as the strong limit of regularized approximations as $\kappa \to 0$, extracting subsequences with compactness arguments. Uniqueness is achieved via difference energy estimates, fully replacing the Taylor-sign condition with surface tension stabilization (Luo et al., 13 Dec 2025).

6. Main Energy Identities and Parabolic Gain

Fundamental energy identities are derived for both fluid and director subsystems. For the director field, the basic $L^2$ energy and higher-order derivatives yield, for example,

$$E_0(t) = \|\nabla \phi(t)\|^2_{L^2(\Omega)} + \int_0^t \|\phi_t\|^2_{L^2(\Omega)} \quad \Longrightarrow \quad \frac{d}{dt} E_0 = 0,$$

as fluid-director coupling terms cancel. At order four, the parabolic equation,

$$D_4(t) = \int_0^t \|\partial_t^5 \phi\|^2_{L^2} + \frac{1}{2} \|\partial_t^4 \phi\|^2_{H^1},$$

displays “parabolic gain,” with one half-derivative enhancement in time which is crucial for regularity closure.

Fluid energy similarly respects non-increasing norms up to leading order; full high-order energy closure is achieved via Grönwall-type inequalities for short time. Key to the entire analysis is the exact cancellation of high-order coupling terms in the nonlinear energy functional, structured to exploit geometric identities in Lagrangian coordinates.

7. Comparison with Prior Work and Extensions

This framework accomplishes several firsts relative to existing literature:

• Establishes the first local-wellposedness result for the fully coupled inviscid liquid-crystal flow with Euler-harmonic heat flow on a moving free boundary.
• Recognizes and utilizes a novel cancellation between $\nabla d \otimes \nabla d$ and heat flow structure in the high-order energy hierarchies.
• Adapts the $\kappa$-regularization/energy method of Coutand–Shkoller to settings where hyperbolic and parabolic regimes coexist genuinely and interact at the boundary.
• Provides a precise treatment of regularizing effects of surface tension and Neumann boundary conditions for the director.
• Closes the scheme with no loss of derivatives either in fluid or director, leveraging the parabolic gain to bridge regularity mismatches.

This analysis facilitates further investigation into singularity formation, vanishing-viscosity limits, and multiphysical coupling such as magnetohydrodynamics (MHD) in free-boundary nematic flows (Luo et al., 13 Dec 2025). A plausible implication is that the geometric techniques developed herein will be applicable to a broader class of free-boundary problems with similar coupling structures.