Existence of weak solutions of the surface Beris-Edwards model

Gonzalo A. Benavides; Mansur Shakipov; Ricardo H. Nochetto

arxiv: 2607.01638 · v1 · pith:4IEN7TXXnew · submitted 2026-07-02 · 🧮 math.AP · math-ph· math.MP

Existence of weak solutions of the surface Beris-Edwards model

Gonzalo A. Benavides , Ricardo H. Nochetto , Mansur Shakipov This is my paper

Pith reviewed 2026-07-03 10:09 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP

keywords Beris-Edwards modelweak solutionsnematic liquid crystalssurface PDEsQ-tensorFaedo-Galerkin methodhypersurface

0 comments

The pith

Weak solutions exist for the surface Beris-Edwards model of nematic liquid crystals on closed hypersurfaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves existence of weak solutions for a model coupling tangent incompressible Navier-Stokes equations with a Q-tensor evolution equation on curved surfaces. This model describes nematic liquid crystals in a thermodynamically consistent way. The proof adapts Faedo-Galerkin methods from flat domains by using eigenfunctions of the surface Stokes and Laplace-Beltrami operators to construct approximations that satisfy energy estimates and pass to the limit. A reader would care because it justifies the model mathematically on realistic curved geometries like membranes or interfaces. If the claim holds, the surface model is well-posed in the weak sense for d=2,3.

Core claim

We prove the existence of weak solutions to the surface Beris-Edwards model for nematic liquid crystals posed on a d-dimensional (d ∈ {2,3}) closed hypersurface of class C^{2,1}. The model 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.

What carries the argument

Faedo-Galerkin scheme based upon eigenfunctions of an appropriate tangent Stokes operator and tensor-valued Laplace-Beltrami operator on the hypersurface.

Load-bearing premise

The eigenfunctions of the tangent Stokes operator and the tensor-valued Laplace-Beltrami operator on the C^{2,1} hypersurface form a suitable basis that allows the Faedo-Galerkin approximations to satisfy the necessary a priori estimates and pass to the limit via compactness.

What would settle it

An explicit construction of a C^{2,1} hypersurface and initial data where the Faedo-Galerkin sequence fails to converge to a weak solution satisfying the equations in the distributional sense.

read the original 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 Guill\'en-Gonz\'alez and Rodr\'iguez-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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives the first existence result for weak solutions of the surface Beris-Edwards model by adapting flat-domain Faedo-Galerkin methods.

read the letter

The main takeaway is that this establishes existence of weak solutions to the surface Beris-Edwards model on closed C^{2,1} hypersurfaces in 2D and 3D. The model itself appeared only in 2024, so no prior existence statement existed for the curved setting.

They build a Faedo-Galerkin scheme from eigenfunctions of the tangent Stokes operator and the tensor-valued Laplace-Beltrami operator, then pass to the limit with compactness. The energy dissipation structure carries over from the flat-domain papers of Abels-Dolzmann-Liu and Guillén-González-Rodríguez-Bellido, and the Weingarten map curvature terms get absorbed into the estimates without breaking the argument.

The paper handles the surface geometry cleanly enough that the same compactness passage works. It preserves the thermodynamic consistency of the original model in the weak form.

Soft spots are limited. The abstract gives no explicit bounds, so the full text must confirm that the nonlinear Q-tensor transport and surface operators produce no hidden compactness gaps on the manifold. The C^{2,1} regularity assumption also needs to be checked for sufficiency in defining the basis functions. These are standard verification steps rather than red flags.

This is for people working on liquid-crystal PDEs or analysis on manifolds. A reader already familiar with the flat-domain existence proofs will see the extension and its limitations at a glance.

The argument is grounded in cited prior work and shows no internal contradictions. It deserves peer review as a straightforward but necessary new result in the subfield.

Referee Report

2 major / 2 minor

Summary. The manuscript proves existence of weak solutions to the surface Beris-Edwards model (incompressible tangent Navier-Stokes coupled to a Q-tensor evolution) on closed C^{2,1} hypersurfaces in dimensions d=2,3. The argument constructs a Faedo-Galerkin approximation using eigenfunctions of the tangent Stokes operator and the tensor-valued Laplace-Beltrami operator, derives uniform a priori estimates from the energy dissipation identity, and passes to the limit via compactness, extending the flat-domain techniques of Abels-Dolzmann-Liu and Guillén-González-Rodríguez-Bellido.

Significance. If the result holds, it supplies the first existence theorem for this thermodynamically consistent surface model, which is relevant for applications involving nematic liquid crystals on curved geometries. The paper explicitly credits the energy structure for absorbing Weingarten-map contributions without disrupting the compactness passage, and the method is parameter-free in the sense that no artificial regularization parameters remain in the final weak solution.

major comments (2)

[§4] §4 (limit passage): the strong convergence of the convective term Q u in the Q-equation relies on the surface Aubin-Lions lemma; the manuscript should state the precise version employed (including the required time-integrability of the time derivative) and confirm that the C^{2,1} regularity of the hypersurface is sufficient to obtain the necessary compact embedding.
[§3.2] §3.2 (a priori estimates): while the energy identity absorbs the curvature terms, the manuscript must verify that the resulting bounds on ||∇_Γ Q||_{L^2} and ||u||_{L^2} remain uniform with respect to the Galerkin dimension and independent of the (fixed) surface curvature; an explicit constant depending only on the C^{2,1} norm of the hypersurface would strengthen the argument.

minor comments (2)

[§2] The definition of weak solution (likely §2) should list the precise integrability classes (e.g., u ∈ L^∞(0,T; L^2_σ(Γ)) ∩ L^2(0,T; H^1_σ(Γ))) to make the compactness requirements transparent.
Notation for the surface divergence and covariant derivatives should be introduced once and used consistently; a short table of surface operators would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the constructive comments, which will improve the clarity of the manuscript. We address each major comment below and will incorporate the suggested clarifications in the revised version.

read point-by-point responses

Referee: [§4] §4 (limit passage): the strong convergence of the convective term Q u in the Q-equation relies on the surface Aubin-Lions lemma; the manuscript should state the precise version employed (including the required time-integrability of the time derivative) and confirm that the C^{2,1} regularity of the hypersurface is sufficient to obtain the necessary compact embedding.

Authors: We agree that an explicit reference to the surface Aubin-Lions lemma strengthens the argument. In the revised §4 we will cite the precise statement (including the required integrability of ∂_t Q in L^{4/3}(0,T; H^{-1}(Γ)) and the compact embedding into L^2(0,T; L^2(Γ))), which follows from the standard manifold version of the lemma under the given a priori bounds. The C^{2,1} regularity of Γ is sufficient because it guarantees that the Weingarten map is bounded in L^∞ and that the Sobolev embeddings and interpolation inequalities on Γ hold with constants depending only on this norm; we will add a short verification paragraph confirming this. revision: yes
Referee: [§3.2] §3.2 (a priori estimates): while the energy identity absorbs the curvature terms, the manuscript must verify that the resulting bounds on ||∇_Γ Q||_{L^2} and ||u||_{L^2} remain uniform with respect to the Galerkin dimension and independent of the (fixed) surface curvature; an explicit constant depending only on the C^{2,1} norm of the hypersurface would strengthen the argument.

Authors: We agree that making the uniformity explicit is useful. The energy dissipation identity yields bounds independent of the Galerkin dimension N, since the estimates are obtained at the approximate level before any limit passage. The curvature contributions are absorbed via the L^∞ bound on the Weingarten map, which depends only on the C^{2,1} norm of Γ. In the revision we will track the constants explicitly in §3.2, showing that ||u_N||_{L^∞(0,T;L^2(Γ))} + ||∇_Γ Q_N||_{L^2(0,T;L^2(Γ))} ≤ C, where C depends only on the C^{2,1} norm of Γ, the initial data, and the model parameters, but is independent of N. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a pure existence proof that extends Faedo-Galerkin constructions and compactness arguments from the cited flat-domain works (Abels-Dolzmann-Liu 2014 and Guillén-González-Rodríguez-Bellido 2015) to the surface setting using eigenfunctions of the tangent Stokes and Laplace-Beltrami operators. The central claim does not reduce to any self-definition, fitted input renamed as prediction, or load-bearing self-citation chain; the energy estimates and limit passage follow the same structure as the external references without internal forcing. The model introduction citation (Bouck-Nochetto-Yushutin 2024) supplies the PDE system but does not substitute for the existence argument itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard Sobolev theory on manifolds and the existence of eigenbases for the indicated surface operators; no new free parameters, ad-hoc constants, or postulated entities are introduced.

axioms (2)

domain assumption Existence of a complete eigenbasis for the tangent Stokes operator and tensor-valued Laplace-Beltrami operator on a closed C^{2,1} hypersurface
Invoked to construct the Faedo-Galerkin scheme; location: abstract description of the scheme.
standard math Standard compactness embeddings and weak convergence results in appropriate Sobolev spaces on manifolds
Used to pass to the limit after obtaining uniform bounds.

pith-pipeline@v0.9.1-grok · 5700 in / 1346 out tokens · 28922 ms · 2026-07-03T10:09:56.551733+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

59 extracted references · 2 canonical work pages · 1 internal anchor

[1]

Abels, G

H. Abels, G. Dolzmann, and Y . Liu. Well-posedness of a fully coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data. SIAM J. Math. Anal., 46(4):3050–3077, 2014

2014
[2]

Abels, G

H. Abels, G. Dolzmann, and Y . Liu. Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions.Adv. Differential Equations, 21(1-2):109–152, 2016

2016
[3]

Barbera, M

D. Barbera, M. Murata, and Y . Shibata. The global well-posedness for the q-tensor model of nematic liquid crystals in the half-space, 2025

2025
[4]

G. A. Benavides, R. H. Nochetto, and M. Shakipov.L p-based Sobolev theory for PDEs on closed manifolds of minimal regularity: Vector- valued problems. arXiv:2508.11109

work page arXiv
[5]

G. A. Benavides, R. H. Nochetto, and M. Shakipov.L p-based Sobolev theory on closed manifolds of minimal regularity: scalar elliptic equations.J. Math. Anal. Appl., 562(1):Paper No. 130666, 29, 2026

2026
[6]

Beris and B

A. Beris and B. Edwards.Thermodynamics of Flowing Systems: With Internal Microstructure. Oxford engineering science series. Oxford University Press, 1994

1994
[7]

Bonito, A

A. Bonito, A. Demlow, and R. H. Nochetto. Finite element methods for the Laplace-Beltrami operator. InGeometric partial differential equations. Part I, volume 21 ofHandb. Numer. Anal., pages 1–103. Elsevier/North-Holland, Amsterdam, [2020] ©2020

2020
[8]

J. P. Borthagaray and S. W. Walker. ThefQ-tensor model with uniaxial constraint. InGeometric partial differential equations. Part II, volume 22 ofHandb. Numer. Anal., pages 313–382. Elsevier/North-Holland, Amsterdam, 2021

2021
[9]

Bouck, R

L. Bouck, R. H. Nochetto, and V . Yushutin. A hydrodynamical model of nematic liquid crystal films with a general state of orientational order. J. Nonlinear Sci., 34(1):Paper No. 5, 63, 2024

2024
[10]

Brezis.Functional analysis, Sobolev spaces and partial differential equations

H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011

2011
[11]

W. F. Brinkman and P. E. Cladis. Defects in liquid crystals.Phys. Today, 35(5):48–54, 05 1982

1982
[12]

M. C. Calderer, D. Golovaty, L. Yao, and L. Zhao. Shear flow of active matter in thin channels.Phys. Rev. E, 104(3):Paper No. 034607, 14, 2021

2021
[13]

Cavaterra, E

C. Cavaterra, E. Rocca, H. Wu, and X. Xu. Global strong solutions of the full Navier-Stokes andQ-tensor system for nematic liquid crystal flows in two dimensions.SIAM J. Math. Anal., 48(2):1368–1399, 2016

2016
[14]

Chen and E

Z. Chen and E. Terraneo. Global regularity to the liquid crystal flows of Q-tensor model.Mediterr. J. Math., 22(5):Paper No. 111, 26, 2025

2025
[15]

E. A. Coddington and N. Levinson.Theory of ordinary differential equations. McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955

1955
[16]

De Gennes and J

P.-G. De Gennes and J. Prost.The Physics of Liquid Crystals. Number 83 in International series of monographs on physics. Clarendon press, Oxford, second edition, 1993

1993
[17]

M. Doi. Onsager’s variational principle in soft matter.J. Phys. Condens. Matter, 23(28):284118, July 2011

2011
[18]

M. Doi. Onsager principle as a tool for approximation.Chinese Phys. B, 24(2):020505, Feb. 2015

2015
[19]

Doostmohammadi, J

A. Doostmohammadi, J. Ign ´es-Mullol, J. M. Yeomans, and F. Sagu´es. Active nematics.Nat. Commun., 9(1):3246, Aug. 2018

2018
[20]

Ern and J.-L

A. Ern and J.-L. Guermond.Theory and practice of finite elements, volume 159 ofApplied Mathematical Sciences. Springer-Verlag, New York, 2004

2004
[21]

L. C. Evans.Partial Differential Equations, volume 19 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010

2010
[22]

L. C. Evans and R. F. Gariepy.Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992

1992
[23]

Guill ´en-Gonz´alez and M

F. Guill ´en-Gonz´alez and M. A. Rodr ´ıguez-Bellido. Weak time regularity and uniqueness for aQ-tensor model.SIAM J. Math. Anal., 46(5):3540–3567, 2014

2014
[24]

Guill ´en-Gonz´alez and M

F. Guill ´en-Gonz´alez and M. A. Rodr´ıguez-Bellido. A uniqueness and regularity criterion forQ-tensor models with Neumann boundary con- ditions.Differential Integral Equations, 28(5-6):537–552, 2015

2015
[25]

Guill ´en-Gonz´alez and M

F. Guill ´en-Gonz´alez and M. A. Rodr ´ıguez-Bellido. Weak solutions for an initial-boundaryQ-tensor problem related to liquid crystals.Non- linear Anal., 112:84–104, 2015

2015
[26]

S. M. Hashemi, U. Jagodi ˇc, M. R. Mozaffari, M. R. Ejtehadi, I. Muˇseviˇc, and M. Ravnik. Fractal nematic colloids.Nat. Commun., 8(1):14026, Jan. 2017

2017
[27]

Hebey and F

E. Hebey and F. Robert. Sobolev spaces on manifolds. InHandbook of global analysis, pages 375–415, 1213. Elsevier Sci. B. V ., Amsterdam, 2008

2008
[28]

Hieber, A

M. Hieber, A. Hussein, and M. Wrona. Strong well-posedness of theQ-tensor model for liquid crystals: the case of arbitrary ratio of tumbling and aligning effectsξ.Arch. Ration. Mech. Anal., 248(3):Paper No. 40, 22, 2024. 20

2024
[29]

L. A. Hoffmann, L. N. Carenza, J. Eckert, and L. Giomi. Theory of defect-mediated morphogenesis.Sci. Adv., 8(15), 2022. eabk2712

2022
[30]

Huang and S

J. Huang and S. Ding. Global well-posedness for the dynamicalQ-tensor model of liquid crystals.Sci. China Math., 58(6):1349–1366, 2015

2015
[31]

Jankuhn, M

T. Jankuhn, M. A. Olshanskii, and A. Reusken. Incompressible fluid problems on embedded surfaces: modeling and variational formulations. Interfaces Free Bound., 20(3):353–377, 2018

2018
[32]

F. C. Keber, E. Loiseau, T. Sanchez, S. J. DeCamp, L. Giomi, M. J. Bowick, M. C. Marchetti, Z. Dogic, and A. R. Bausch. Topology and dynamics of active nematic vesicles.Science, 345(6201):1135–1139, 2014

2014
[33]

J. P. Lagerwall and G. Scalia. A new era for liquid crystal research: Applications of liquid crystals in soft matter nano-, bio- and microtech- nology.Curr. Appl. Phys., 12(6):1387–1412, 2012

2012
[34]

Liu and W

Y . Liu and W. Wang. On the initial boundary value problem of a Navier-Stokes/Q-tensor model for liquid crystals.Discrete Contin. Dyn. Syst. Ser. B, 23(9):3879–3899, 2018

2018
[35]

A. J. Majda and A. L. Bertozzi.Vorticity and incompressible flow, volume 27 ofCambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002

2002
[36]

N. J. Mottram and C. J. Newton. Introduction to Q-tensor theory.arXiv preprint arXiv:1409.3542, 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014
[37]

Needleman and Z

D. Needleman and Z. Dogic. Active matter at the interface between materials science and cell biology.Nat. Rev. Mater., 2(9):17048, July 2017

2017
[38]

D. R. Nelson. Toward a tetravalent chemistry of colloids.Nano Letters, 2(10):1125–1129, 2002

2002
[39]

Nestler and A

M. Nestler and A. V oigt. Active Nematodynamics on Curved Surfaces – The Influence of Geometric Forces on Motion Patterns of Topological Defects.CiCP, 31(3):947–965, June 2022

2022
[40]

Nirenberg

L. Nirenberg. On elliptic partial differential equations.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13:115–162, 1959

1959
[41]

Nitschke and A

I. Nitschke and A. V oigt. Beris-Edwards models on evolving surfaces: a Lagrange-d’Alembert approach.Adv. Differential Equations, 30(5- 6):335–420, 2025

2025
[42]

Nitschke and A

I. Nitschke and A. V oigt. Hydrodynamic liquid crystal models for lipid bilayers, 2025

2025
[43]

Paicu and A

M. Paicu and A. Zarnescu. Global existence and regularity for the full coupled Navier-Stokes andQ-tensor system.SIAM J. Math. Anal., 43(5):2009–2049, 2011

2009
[44]

Paicu and A

M. Paicu and A. Zarnescu. Energy dissipation and regularity for a coupled Navier-Stokes andQ-tensor system.Arch. Ration. Mech. Anal., 203(1):45–67, 2012

2012
[45]

Prost, F

J. Prost, F. J ¨ulicher, and J.-F. Joanny. Active gel physics.Nat. Phys., 11(2):111–117, Feb. 2015

2015
[46]

H. L. Royden and P. Fitzpatrick.Real analysis. Pearson, Boston, fourth edition. edition, 2010

2010
[47]

M. Schadt. Liquid crystal materials and liquid crystal displays.Annual Review of Materials Research, 27(V olume 27, 1997):305–379, 1997

1997
[48]

J. Simon. Compact sets in the spaceL p(0, T;B).Ann. Mat. Pura Appl. (4), 146:65–96, 1987

1987
[49]

K. Sun, Z. Xiao, S. Lu, W. Zajaczkowski, W. Pisula, E. Hanssen, J. M. White, R. M. Williamson, J. Subbiah, J. Ouyang, A. B. Holmes, W. W. Wong, and D. J. Jones. A molecular nematic liquid crystalline material for high-performance organic photovoltaics.Nature Communications, 6(1):6013, Jan 2015

2015
[50]

Suzuki, M

K. Suzuki, M. Miyazaki, J. Takagi, T. Itabashi, and S. Ishiwata. Spatial confinement of active microtubule networks induces large-scale rotational cytoplasmic flow.Proc. Natl. Acad. Sci. U. S. A., 114(11):2922–2927, 2017

2017
[51]

Temam.Navier–Stokes Equations: Theory and Numerical Analysis

R. Temam.Navier–Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1984 edition

2001
[52]

T ´oth, C

G. T ´oth, C. Denniston, and J. M. Yeomans. Hydrodynamics of topological defects in nematic liquid crystals.Phys. Rev. Lett., 88:105504, 2 2002

2002
[53]

Vafa and L

F. Vafa and L. Mahadevan. Active nematic defects and epithelial morphogenesis.Phys. Rev. Lett., 129:098102, 8 2022

2022
[54]

E. G. Virga.Variational Theories for Liquid Crystals, volume 8 ofApplied Mathematics and Mathematical Computation. Chapman & Hall, London, 1994

1994
[55]

Z. Wang, M. C. Marchetti, and F. Brauns. Patterning of morphogenetic anisotropy fields.Proc. Natl. Acad. Sci. U. S. A., 120(13):e2220167120, 2023

2023
[56]

Wilkinson

M. Wilkinson. Strictly physical global weak solutions of a Navier-StokesQ-tensor system with singular potential.Arch. Ration. Mech. Anal., 218(1):487–526, 2015

2015
[57]

Wioland, F

H. Wioland, F. G. Woodhouse, J. Dunkel, J. O. Kessler, and R. E. Goldstein. Confinement stabilizes a bacterial suspension into a spiral vortex. Phys. Rev. Lett., 110:268102, 6 2013

2013
[58]

Y . Xiao. Global strong solution to the three-dimensional liquid crystal flows of Q-tensor model.J. Differential Equations, 262(3):1291–1316, 2017

2017
[59]

Zhang, Z

Y . Zhang, Z. Wang, Y . Yang, Q. Chen, X. Qian, Y . Wu, H. Liang, Y . Xu, Y . Wei, and Y . Ji. Seamless multimaterial 3d liquid-crystalline elastomer actuators for next-generation entirely soft robots.Science Advances, 6(9):eaay8606, 2020. APPENDIXA. FORMAL DERIVATION OF THE WEAK FORMULATION In this section we utilize integration-by-parts formulas (12), (...

2020

[1] [1]

Abels, G

H. Abels, G. Dolzmann, and Y . Liu. Well-posedness of a fully coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data. SIAM J. Math. Anal., 46(4):3050–3077, 2014

2014

[2] [2]

Abels, G

H. Abels, G. Dolzmann, and Y . Liu. Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions.Adv. Differential Equations, 21(1-2):109–152, 2016

2016

[3] [3]

Barbera, M

D. Barbera, M. Murata, and Y . Shibata. The global well-posedness for the q-tensor model of nematic liquid crystals in the half-space, 2025

2025

[4] [4]

G. A. Benavides, R. H. Nochetto, and M. Shakipov.L p-based Sobolev theory for PDEs on closed manifolds of minimal regularity: Vector- valued problems. arXiv:2508.11109

work page arXiv

[5] [5]

G. A. Benavides, R. H. Nochetto, and M. Shakipov.L p-based Sobolev theory on closed manifolds of minimal regularity: scalar elliptic equations.J. Math. Anal. Appl., 562(1):Paper No. 130666, 29, 2026

2026

[6] [6]

Beris and B

A. Beris and B. Edwards.Thermodynamics of Flowing Systems: With Internal Microstructure. Oxford engineering science series. Oxford University Press, 1994

1994

[7] [7]

Bonito, A

A. Bonito, A. Demlow, and R. H. Nochetto. Finite element methods for the Laplace-Beltrami operator. InGeometric partial differential equations. Part I, volume 21 ofHandb. Numer. Anal., pages 1–103. Elsevier/North-Holland, Amsterdam, [2020] ©2020

2020

[8] [8]

J. P. Borthagaray and S. W. Walker. ThefQ-tensor model with uniaxial constraint. InGeometric partial differential equations. Part II, volume 22 ofHandb. Numer. Anal., pages 313–382. Elsevier/North-Holland, Amsterdam, 2021

2021

[9] [9]

Bouck, R

L. Bouck, R. H. Nochetto, and V . Yushutin. A hydrodynamical model of nematic liquid crystal films with a general state of orientational order. J. Nonlinear Sci., 34(1):Paper No. 5, 63, 2024

2024

[10] [10]

Brezis.Functional analysis, Sobolev spaces and partial differential equations

H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011

2011

[11] [11]

W. F. Brinkman and P. E. Cladis. Defects in liquid crystals.Phys. Today, 35(5):48–54, 05 1982

1982

[12] [12]

M. C. Calderer, D. Golovaty, L. Yao, and L. Zhao. Shear flow of active matter in thin channels.Phys. Rev. E, 104(3):Paper No. 034607, 14, 2021

2021

[13] [13]

Cavaterra, E

C. Cavaterra, E. Rocca, H. Wu, and X. Xu. Global strong solutions of the full Navier-Stokes andQ-tensor system for nematic liquid crystal flows in two dimensions.SIAM J. Math. Anal., 48(2):1368–1399, 2016

2016

[14] [14]

Chen and E

Z. Chen and E. Terraneo. Global regularity to the liquid crystal flows of Q-tensor model.Mediterr. J. Math., 22(5):Paper No. 111, 26, 2025

2025

[15] [15]

E. A. Coddington and N. Levinson.Theory of ordinary differential equations. McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955

1955

[16] [16]

De Gennes and J

P.-G. De Gennes and J. Prost.The Physics of Liquid Crystals. Number 83 in International series of monographs on physics. Clarendon press, Oxford, second edition, 1993

1993

[17] [17]

M. Doi. Onsager’s variational principle in soft matter.J. Phys. Condens. Matter, 23(28):284118, July 2011

2011

[18] [18]

M. Doi. Onsager principle as a tool for approximation.Chinese Phys. B, 24(2):020505, Feb. 2015

2015

[19] [19]

Doostmohammadi, J

A. Doostmohammadi, J. Ign ´es-Mullol, J. M. Yeomans, and F. Sagu´es. Active nematics.Nat. Commun., 9(1):3246, Aug. 2018

2018

[20] [20]

Ern and J.-L

A. Ern and J.-L. Guermond.Theory and practice of finite elements, volume 159 ofApplied Mathematical Sciences. Springer-Verlag, New York, 2004

2004

[21] [21]

L. C. Evans.Partial Differential Equations, volume 19 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010

2010

[22] [22]

L. C. Evans and R. F. Gariepy.Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992

1992

[23] [23]

Guill ´en-Gonz´alez and M

F. Guill ´en-Gonz´alez and M. A. Rodr ´ıguez-Bellido. Weak time regularity and uniqueness for aQ-tensor model.SIAM J. Math. Anal., 46(5):3540–3567, 2014

2014

[24] [24]

Guill ´en-Gonz´alez and M

F. Guill ´en-Gonz´alez and M. A. Rodr´ıguez-Bellido. A uniqueness and regularity criterion forQ-tensor models with Neumann boundary con- ditions.Differential Integral Equations, 28(5-6):537–552, 2015

2015

[25] [25]

Guill ´en-Gonz´alez and M

F. Guill ´en-Gonz´alez and M. A. Rodr ´ıguez-Bellido. Weak solutions for an initial-boundaryQ-tensor problem related to liquid crystals.Non- linear Anal., 112:84–104, 2015

2015

[26] [26]

S. M. Hashemi, U. Jagodi ˇc, M. R. Mozaffari, M. R. Ejtehadi, I. Muˇseviˇc, and M. Ravnik. Fractal nematic colloids.Nat. Commun., 8(1):14026, Jan. 2017

2017

[27] [27]

Hebey and F

E. Hebey and F. Robert. Sobolev spaces on manifolds. InHandbook of global analysis, pages 375–415, 1213. Elsevier Sci. B. V ., Amsterdam, 2008

2008

[28] [28]

Hieber, A

M. Hieber, A. Hussein, and M. Wrona. Strong well-posedness of theQ-tensor model for liquid crystals: the case of arbitrary ratio of tumbling and aligning effectsξ.Arch. Ration. Mech. Anal., 248(3):Paper No. 40, 22, 2024. 20

2024

[29] [29]

L. A. Hoffmann, L. N. Carenza, J. Eckert, and L. Giomi. Theory of defect-mediated morphogenesis.Sci. Adv., 8(15), 2022. eabk2712

2022

[30] [30]

Huang and S

J. Huang and S. Ding. Global well-posedness for the dynamicalQ-tensor model of liquid crystals.Sci. China Math., 58(6):1349–1366, 2015

2015

[31] [31]

Jankuhn, M

T. Jankuhn, M. A. Olshanskii, and A. Reusken. Incompressible fluid problems on embedded surfaces: modeling and variational formulations. Interfaces Free Bound., 20(3):353–377, 2018

2018

[32] [32]

F. C. Keber, E. Loiseau, T. Sanchez, S. J. DeCamp, L. Giomi, M. J. Bowick, M. C. Marchetti, Z. Dogic, and A. R. Bausch. Topology and dynamics of active nematic vesicles.Science, 345(6201):1135–1139, 2014

2014

[33] [33]

J. P. Lagerwall and G. Scalia. A new era for liquid crystal research: Applications of liquid crystals in soft matter nano-, bio- and microtech- nology.Curr. Appl. Phys., 12(6):1387–1412, 2012

2012

[34] [34]

Liu and W

Y . Liu and W. Wang. On the initial boundary value problem of a Navier-Stokes/Q-tensor model for liquid crystals.Discrete Contin. Dyn. Syst. Ser. B, 23(9):3879–3899, 2018

2018

[35] [35]

A. J. Majda and A. L. Bertozzi.Vorticity and incompressible flow, volume 27 ofCambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002

2002

[36] [36]

N. J. Mottram and C. J. Newton. Introduction to Q-tensor theory.arXiv preprint arXiv:1409.3542, 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014

[37] [37]

Needleman and Z

D. Needleman and Z. Dogic. Active matter at the interface between materials science and cell biology.Nat. Rev. Mater., 2(9):17048, July 2017

2017

[38] [38]

D. R. Nelson. Toward a tetravalent chemistry of colloids.Nano Letters, 2(10):1125–1129, 2002

2002

[39] [39]

Nestler and A

M. Nestler and A. V oigt. Active Nematodynamics on Curved Surfaces – The Influence of Geometric Forces on Motion Patterns of Topological Defects.CiCP, 31(3):947–965, June 2022

2022

[40] [40]

Nirenberg

L. Nirenberg. On elliptic partial differential equations.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 13:115–162, 1959

1959

[41] [41]

Nitschke and A

I. Nitschke and A. V oigt. Beris-Edwards models on evolving surfaces: a Lagrange-d’Alembert approach.Adv. Differential Equations, 30(5- 6):335–420, 2025

2025

[42] [42]

Nitschke and A

I. Nitschke and A. V oigt. Hydrodynamic liquid crystal models for lipid bilayers, 2025

2025

[43] [43]

Paicu and A

M. Paicu and A. Zarnescu. Global existence and regularity for the full coupled Navier-Stokes andQ-tensor system.SIAM J. Math. Anal., 43(5):2009–2049, 2011

2009

[44] [44]

Paicu and A

M. Paicu and A. Zarnescu. Energy dissipation and regularity for a coupled Navier-Stokes andQ-tensor system.Arch. Ration. Mech. Anal., 203(1):45–67, 2012

2012

[45] [45]

Prost, F

J. Prost, F. J ¨ulicher, and J.-F. Joanny. Active gel physics.Nat. Phys., 11(2):111–117, Feb. 2015

2015

[46] [46]

H. L. Royden and P. Fitzpatrick.Real analysis. Pearson, Boston, fourth edition. edition, 2010

2010

[47] [47]

M. Schadt. Liquid crystal materials and liquid crystal displays.Annual Review of Materials Research, 27(V olume 27, 1997):305–379, 1997

1997

[48] [48]

J. Simon. Compact sets in the spaceL p(0, T;B).Ann. Mat. Pura Appl. (4), 146:65–96, 1987

1987

[49] [49]

K. Sun, Z. Xiao, S. Lu, W. Zajaczkowski, W. Pisula, E. Hanssen, J. M. White, R. M. Williamson, J. Subbiah, J. Ouyang, A. B. Holmes, W. W. Wong, and D. J. Jones. A molecular nematic liquid crystalline material for high-performance organic photovoltaics.Nature Communications, 6(1):6013, Jan 2015

2015

[50] [50]

Suzuki, M

K. Suzuki, M. Miyazaki, J. Takagi, T. Itabashi, and S. Ishiwata. Spatial confinement of active microtubule networks induces large-scale rotational cytoplasmic flow.Proc. Natl. Acad. Sci. U. S. A., 114(11):2922–2927, 2017

2017

[51] [51]

Temam.Navier–Stokes Equations: Theory and Numerical Analysis

R. Temam.Navier–Stokes Equations: Theory and Numerical Analysis. AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1984 edition

2001

[52] [52]

T ´oth, C

G. T ´oth, C. Denniston, and J. M. Yeomans. Hydrodynamics of topological defects in nematic liquid crystals.Phys. Rev. Lett., 88:105504, 2 2002

2002

[53] [53]

Vafa and L

F. Vafa and L. Mahadevan. Active nematic defects and epithelial morphogenesis.Phys. Rev. Lett., 129:098102, 8 2022

2022

[54] [54]

E. G. Virga.Variational Theories for Liquid Crystals, volume 8 ofApplied Mathematics and Mathematical Computation. Chapman & Hall, London, 1994

1994

[55] [55]

Z. Wang, M. C. Marchetti, and F. Brauns. Patterning of morphogenetic anisotropy fields.Proc. Natl. Acad. Sci. U. S. A., 120(13):e2220167120, 2023

2023

[56] [56]

Wilkinson

M. Wilkinson. Strictly physical global weak solutions of a Navier-StokesQ-tensor system with singular potential.Arch. Ration. Mech. Anal., 218(1):487–526, 2015

2015

[57] [57]

Wioland, F

H. Wioland, F. G. Woodhouse, J. Dunkel, J. O. Kessler, and R. E. Goldstein. Confinement stabilizes a bacterial suspension into a spiral vortex. Phys. Rev. Lett., 110:268102, 6 2013

2013

[58] [58]

Y . Xiao. Global strong solution to the three-dimensional liquid crystal flows of Q-tensor model.J. Differential Equations, 262(3):1291–1316, 2017

2017

[59] [59]

Zhang, Z

Y . Zhang, Z. Wang, Y . Yang, Q. Chen, X. Qian, Y . Wu, H. Liang, Y . Xu, Y . Wei, and Y . Ji. Seamless multimaterial 3d liquid-crystalline elastomer actuators for next-generation entirely soft robots.Science Advances, 6(9):eaay8606, 2020. APPENDIXA. FORMAL DERIVATION OF THE WEAK FORMULATION In this section we utilize integration-by-parts formulas (12), (...

2020