Existence of weak solutions of the surface Beris-Edwards model

Abstract: We prove the existence of weak solutions to the surface Beris-Edwards model for nematic liquid crystals posed on a $d$-dimensional ($d \in {2,3}$) closed hypersurface of class $C^{2,1}$. This thermodynamically consistent model, recently introduced by Bouck, Nochetto and Yushutin (2024), couples the incompressible tangent Navier-Stokes equations with a kinematic equation for the Q-tensor field that encodes the orientation of the liquid crystal particles with a general state of orientational order. Extending ideas by Abels, Dolzmann and Liu (2014) and Guillén-González and Rodríguez-Bellido (2015) for the Beris-Edwards model in flat domains, we design a Faedo-Galerkin scheme based upon eigenfunctions of an appropriate tangent Stokes operator and tensor-valued Laplace-Beltrami operator and recover a weak solution via standard compactness arguments.

• The paper establishes the existence of global weak solutions to the surface Beris–Edwards model on closed, smooth hypersurfaces.
• It leverages a Galerkin approximation and discrete energy law to derive uniform bounds in relevant Sobolev spaces.
• The analysis recovers the pressure via a surface Helmholtz–Weyl decomposition, ensuring thermodynamic consistency on curved geometries.

Existence of Weak Solutions to the Surface Beris–Edwards Model: Analytical Framework and Implications

Introduction and Context

The mathematical modeling of nematic liquid crystals (NLCs) on curved hypersurfaces has advanced significantly, driven by applications in soft matter, active materials, and nano/bio-technology. The Beris–Edwards model provides a thermodynamically consistent macroscopic description of NLC hydrodynamics via a coupled system involving the incompressible Navier–Stokes equations and a transport equation for the Q-tensor, which encodes the orientational order. While analytical results for the Beris–Edwards model in flat domains are well established, rigorous existence theory for such systems posed on manifolds—and especially for the new generalized, fully-thermodynamically consistent form recently derived for surfaces [Bouck, Nochetto, Yushutin (2024)]—has remained open.

This work (2607.01638) closes this gap by establishing the existence of global-in-time weak solutions for the surface Beris–Edwards system posed on closed, smooth hypersurfaces in spatial dimensions 2 and 3. This is the first such existence result, providing an analytical template for future study of hydrodynamic Q-tensor models in curved geometries.

Mathematical Setting and Model Description

The study considers a $d$-dimensional closed, compact, $C^{2,1}$ hypersurface $\Gamma \subset \mathbb{R}^{d+1}$, $d=2,3$. The unknowns are a tangential velocity field $u$, a symmetric and traceless Q-tensor field $Q$, and a scalar pressure $\pi$, governed by:

• The incompressible Navier–Stokes equations for $u$, restricted to the tangent bundle of the surface and augmented by elastic and Ericksen stresses,
• The transport–relaxation equation for $Q$, containing both corotational (kinematic) and dissipative (relaxational) terms,
• The incompressibility constraint $div_\Gamma u = 0$.

The model uses the Landau–de Gennes free energy incorporating a double-well potential in $C^{2,1}$0 and a surface Laplacian regularization. Both equations couple via the molecular field (variational derivative of the energy) and by specific geometric differential operators (surface gradient, divergence, Laplace–Beltrami). Essential to the well-posedness analysis is the thermodynamic consistency: the system formally dissipates the sum of kinetic and Landau–de Gennes energies.

Analytical Approach

The core mathematical contribution is a rigorous construction of weak solutions to the above system, exploiting several key advances:

• Galerkin Approximation: Eigenfunction bases of the tangent surface Stokes and tensor Laplace–Beltrami operators (whose regularity and existence are established in [Benavides, Nochetto, Shakipov 2025]) span the discrete finite-dimensional subspaces used for the Faedo–Galerkin construction.
• Discrete Energy Law: The Galerkin scheme admits an exact energy law analog to the continuum model:

$C^{2,1}$1

leading to uniform a priori bounds for $C^{2,1}$2 and $C^{2,1}$3 in $C^{2,1}$4 and $C^{2,1}$5 Sobolev spaces, and for $C^{2,1}$6 in $C^{2,1}$7.

• Compactness and Weak Solution Recovery: Aubin–Lions–Simon lemma and careful interpolation yield subsequential convergence in appropriate Bochner spaces; strong convergence in $C^{2,1}$8 is combined with weak convergence in higher-order Sobolev norms to handle nonlinearities.
• Pressure Recovery: Due to the absence of boundary, the pressure $C^{2,1}$9 can be post-processed by using the surface Helmholtz–Weyl decomposition and a variational Laplace–Beltrami theory, yielding higher regularity for $\Gamma \subset \mathbb{R}^{d+1}$0 than in the Euclidean case.

Main Results

Two main theorems are established:

Existence of Weak Solutions

Given initial data $\Gamma \subset \mathbb{R}^{d+1}$1 (tangential, divergence-free), $\Gamma \subset \mathbb{R}^{d+1}$2 (symmetric, traceless), and for all $\Gamma \subset \mathbb{R}^{d+1}$3, there exists a pair $\Gamma \subset \mathbb{R}^{d+1}$4 with

• $\Gamma \subset \mathbb{R}^{d+1}$5,
• $\Gamma \subset \mathbb{R}^{d+1}$6,

that satisfies the surface Beris–Edwards system (in the sense of variational identities) and the energy dissipation inequality. This result holds for arbitrary co-dimension (2 or 3), and the necessary function spaces are precisely matched to the surface geometry.

Pressure Recovery in High Regularity

Thanks to the closed geometry, the pressure field $\Gamma \subset \mathbb{R}^{d+1}$7 is shown to exist uniquely in $\Gamma \subset \mathbb{R}^{d+1}$8 for all $\Gamma \subset \mathbb{R}^{d+1}$9 slightly greater than the spatial dimension, with the solution obtained from the Helmholtz decomposition and careful elliptic estimates on the manifold. This is superior to analogous statements obtainable in bounded domains with boundary.

Notable Analytical Features and Claims

• The construction and analysis generalize the well-posedness theory for Beris–Edwards-type flows to the case where all geometric operations are surface-based. The structure of the proof closely parallels, but is not reducible to, flat-domain results [Abels, Dolzmann, Liu 2014; Guillén-González, Rodríguez-Bellido 2015].
• Uniform control is achieved without assuming any conformity or alignment of the Q-tensor with the surface normal, distinguishing this model from earlier surface Q-tensor flows.
• The proof leverages recent advances in regularity theory for elliptic and Stokes operators on manifolds of low regularity ([Benavides, Nochetto, Shakipov 2025-a,b]), which is essential since classical Sobolev theory is nontrivially extended to the curved, boundaryless setting.
• The energy dissipation argument holds at the discrete Galerkin level and propagates to the limit, a key step for compactness in strongly coupled systems.

Implications and Future Directions

Theoretical Implications:

This existence result lays the foundation for future analytical investigations, such as uniqueness (weak–strong and strong–strong), higher regularity theory, and long-time behavior for Beris–Edwards flows on manifolds. It also acts as a bridge for passing rigorous asymptotic regimes (e.g., thin-film limits, axisymmetric reductions) on nontrivial geometries.

Practical Implications:

With applications in modeling NLC shells, thin films, biological membranes, and soft active matter, the availability of a solid analytical framework is essential for the convergence and accuracy of structure-preserving numerical schemes—crucial for the reliable simulation of dynamics involving topological defects, pattern formation, and active stresses in complex geometry.

Extensions and Future Work:

Assuming further regularity or restrictions, one may adapt energy methods here to study:

• Evolutionary domains and moving hypersurfaces, as in recent Lagrangian approaches [Nitschke, Voigt 2025].
• Active matter extensions including additional coupling terms or body forces modeled via generalized Onsager principles.
• Defect dynamics and singularities, leveraging the global bounds here to study the evolution and interaction of defects in Q-tensor flows on surfaces.
• Analysis of finite-element or structure-preserving discretizations, building on the continuous-in-time Faedo–Galerkin methods here.

Conclusion

This work delivers the first comprehensive existence analysis for the full, thermodynamically consistent Beris–Edwards model on closed hypersurfaces. The methodology bridges geometric analysis, modern PDE techniques, and variational approaches. It consolidates the mathematical foundation necessary for the study and simulation of nematic liquid crystals and active flows constrained to curved surfaces, opening avenues for both deeper mathematical understanding and advanced numerical experiments in complex geometries.

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Reference:

Gonzalo A. Benavides, Ricardo H. Nochetto, and Mansur Shakipov, "Existence of weak solutions of the surface Beris–Edwards model" (2607.01638).