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arxiv: 2605.05139 · v1 · submitted 2026-05-06 · 🧮 math.AP

On a Partial Voigt Regularization of the 3D Magnetohydrodynamic Equations in Velocity-Vorticity Form

Adam Larios , Yuan Pei This is my paper

Pith reviewed 2026-05-08 16:21 UTC · model grok-4.3

classification 🧮 math.AP
keywords magnetohydrodynamicsVoigt regularizationvelocity-vorticity formulationglobal well-posedness3D MHDblow-up criterion
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The pith

Voigt regularization applied only to the momentum equation yields global well-posedness for the 3D MHD equations in velocity-vorticity form.

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

The paper establishes that the three-dimensional magnetohydrodynamic equations written in velocity-vorticity form remain globally well-posed when a Voigt regularization is added solely to the momentum equation. This single modification suffices even though a symmetric split on the magnetic equation might appear necessary at first glance. The vorticity equation and the magnetic induction equation keep their original structure. The result supplies convergence of the regularized solutions to the original system up to any possible singularity time and supplies a blow-up criterion stated in terms of the vorticity and magnetic field.

Core claim

The authors prove that the 3D MHD system in velocity-vorticity formulation admits global smooth solutions when a Voigt-type regularization is introduced only into the momentum equation. Solutions of this partially regularized system converge strongly to solutions of the unregularized MHD system on any time interval short of a possible blow-up time. A blow-up criterion for the original system is obtained that involves the L^infty norms of the vorticity and the magnetic field.

What carries the argument

Partial Voigt regularization inserted exclusively into the momentum equation, which adds a term that damps high-frequency velocity components while leaving the vorticity transport equation and the magnetic induction equation unchanged in form.

If this is right

  • The partially regularized system possesses global-in-time smooth solutions for arbitrary smooth initial data.
  • Solutions converge to those of the original MHD system on the maximal interval of existence of the latter.
  • A blow-up criterion for the unregularized 3D MHD equations follows directly from the convergence result.
  • The structural advantages of the velocity-vorticity formulation for the magnetic field are preserved.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may allow numerical schemes that stabilize the velocity field without introducing artificial diffusion into the magnetic evolution.
  • Similar partial regularization might be tested on other coupled systems such as rotating fluids or viscoelastic flows where only one equation needs stabilization.
  • The result indicates that the magnetic stretching term already supplies enough control to close estimates once velocity is regularized.

Load-bearing premise

Initial data must be sufficiently regular and the domain must allow energy estimates together with compactness arguments to close without extra modifications.

What would settle it

Construction of a smooth initial datum for which the partially regularized system develops a singularity in finite time or for which the regularized solutions fail to converge to a weak solution of the original MHD system before that time would disprove the global well-posedness statement.

read the original abstract

The Velocity-Vorticity (VV) formulation of the incompressible Navier-Stokes equations has become popular in recent years, especially in numerical studies, due to its structural advantages. Recently, with L. Rebholz, we introduced a Voigt regularization to the momentum equation in this formulation, establishing global well-posedness of the regularized system in 3D, along with convergence results and a blow-up criterion. In the present work, we extend these ideas to the 3D magnetohydrodynamics (MHD) equations. While it may seem that a ``VV-type'' split on the magnetic equation is required, we show that no such modification is necessary, and global well-posedness holds with a Voigt regularization only on the momentum equation, preserving the structure of both the vorticity and magnetic equations. We also prove that the regularized system converges to the original system, up to a possible blow-up time, and we establish a blow-up criterion for solutions to the original 3D MHD system.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript extends prior work on Voigt-regularized Navier-Stokes equations in velocity-vorticity form to the 3D incompressible MHD system. It claims that a Voigt term applied only to the momentum equation yields global well-posedness for the regularized MHD system while leaving the vorticity and magnetic equations structurally unmodified. The authors further establish convergence of regularized solutions to the original MHD system up to a possible blow-up time and derive a blow-up criterion for the unregularized 3D MHD equations.

Significance. If the a priori estimates and passage to the limit are valid, the result is significant because it identifies a minimal regularization that preserves the structural advantages of the velocity-vorticity formulation for both the vorticity and magnetic equations. This partial regularization approach may inform numerical schemes for MHD and clarify the role of regularization in controlling nonlinear couplings, consistent with analogous results for Navier-Stokes.

major comments (1)
  1. [§4] §4 (a priori estimates): the energy estimate for the magnetic field must be shown to close without a Voigt term; the bounding of the stretching term (u·∇)b and the coupling with the regularized velocity requires explicit constants and absorption arguments that are load-bearing for the central claim that no magnetic regularization is needed.
minor comments (2)
  1. [§2] The functional setting (periodic domain or bounded domain with boundary conditions) and the precise Sobolev spaces for initial data should be stated explicitly in the statement of the main theorems rather than only in the preliminaries.
  2. [Introduction] Notation for the Voigt parameter α and the regularized velocity should be introduced uniformly from the outset to avoid redefinition across sections.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation for minor revision. The comment on the a priori estimates in §4 is well-taken and will be addressed by expanding the details in the revised manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (a priori estimates): the energy estimate for the magnetic field must be shown to close without a Voigt term; the bounding of the stretching term (u·∇)b and the coupling with the regularized velocity requires explicit constants and absorption arguments that are load-bearing for the central claim that no magnetic regularization is needed.

    Authors: We agree that additional explicit details will strengthen the presentation. In the revised §4 we will derive the L² energy equality for b by taking the inner product of the magnetic transport equation with b. After integration by parts (using div b = 0), the stretching term becomes −∫ b · (b · ∇)u dx. This is estimated via |∫ b · (b · ∇)u dx| ≤ ||b||₂ ||b||_∞ ||∇u||₂. The regularized velocity equation supplies the necessary control: the Voigt term yields a uniform bound on ||∇u||₂ (and higher norms) that is independent of the magnetic field. We will insert the precise absorption argument using Young’s inequality with a small constant ε = 1/2, absorbing the resulting term into the dissipation already present from the momentum equation while leaving a remainder controlled by the Voigt regularization. The same estimate closes the coupling term arising from the Lorentz force in the momentum equation. These steps confirm that the magnetic energy estimate closes globally without any regularization on the magnetic equation itself. revision: yes

Circularity Check

0 steps flagged

No significant circularity; MHD estimates derived independently

full rationale

The paper's central derivation establishes global well-posedness for the 3D MHD system under partial Voigt regularization applied only to the momentum equation, using standard energy estimates, compactness, and passage to the limit in the velocity-vorticity formulation. These steps are presented as new adaptations to the MHD equations, preserving the structure of the vorticity and magnetic equations without modification. The citation to the authors' prior Navier-Stokes work supplies background but does not serve as a load-bearing premise; the MHD-specific a priori bounds and convergence results are developed separately and do not reduce by construction to the NS case or to any fitted parameter. The functional-setting assumptions are the conventional ones for such PDE analyses and introduce no self-definitional or renaming circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The proofs rest on standard functional-analytic tools for 3D PDEs and the authors' earlier Navier-Stokes regularization; no new physical entities are introduced.

free parameters (1)
  • Voigt regularization parameter alpha
    Positive parameter controlling the strength of the added smoothing term; the limit alpha to 0 is taken to recover the original system.
axioms (2)
  • standard math Standard Sobolev embeddings and interpolation inequalities in 3D
    Invoked to close the a priori estimates for the regularized system.
  • domain assumption Basic existence theory and energy inequalities for the unregularized 3D MHD equations
    Used as background for the convergence and blow-up criterion statements.

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Reference graph

Works this paper leans on

78 extracted references · 78 canonical work pages

  1. [1]

    Benmoussa, J

    K. Benmoussa, J. Deteix, and D. Yakoubi. On the well-posedness of the unsteady velocity-vorticity-helicity formulation of the Navier– Stokes equations.J. Math. Anal. Appl., page Paper No. 130421, 2026

    work page 2026

  2. [2]

    L. C. Berselli and L. Bisconti. On the structural stability of the Euler- Voigt and Navier–Stokes–Voigt models.Nonlinear Anal., 75(1):117– 130, 2012

    work page 2012

  3. [4]

    L. C. Berselli, T.-Y. Kim, and L. G. Rebholz. Analysis of a reduced- order approximate deconvolution model and its interpretation as a Navier–Stokes–Voigt regularization.Discrete Contin. Dyn. Syst. Ser. B, 21(4):1027–1050, 2016

    work page 2016

  4. [5]

    L. C. Berselli and S. Spirito. Suitable weak solutions to the 3D Navier– Stokes equations are constructed with the Voigt approximation.J. Differential Equations, 262(5):3285–3316, 2017

    work page 2017

  5. [6]

    Bisconti and D

    L. Bisconti and D. Catania. On the convergence rates for the three- dimensional filtered Boussinesq equations.Math. Nachr., 294(6):1099– 1114, 2021

    work page 2021

  6. [7]

    M. B¨ ohm. On Navier–Stokes and Kelvin-Voigt equations in three di- mensions in interpolation spaces.Math. Nachr., 155:151–165, 1992

    work page 1992

  7. [8]

    Borges and F

    R. Borges and F. Ramos. Sub-grid effects of the Voigt viscoelastic regularization of a singular dyadic model of turbulence.J. Phys. A, 46(19):195501, 19, 2013

    work page 2013

  8. [9]

    D. W. Boutros, X. Liu, M. Thomas, and E. S. Titi. Global well- posedness of the elastic-viscous-plastic sea-ice model with the inviscid Voigt-regularisation.Math. Models Methods Appl. Sci., 2026

    work page 2026

  9. [10]

    Brze´ zniak, A

    Z. Brze´ zniak, A. Larios, and I. Safarik. Fractional Voigt-regularization of the 3D Navier–Stokes and Euler equations: Global well-posedness and limiting behavior.J. Math. Fluid Mech., 27(45):25 pp., 2025

    work page 2025

  10. [11]

    Y. Cao, E. Lunasin, and E. S. Titi. Global well-posedness of the three- dimensional viscous and inviscid simplified Bardina turbulence models. Commun. Math. Sci., 4(4):823–848, 2006

    work page 2006

  11. [12]

    D. Catania. Global existence for a regularized magnetohydrodynamic-α model.Ann. Univ Ferrara, 56:1–20, 2010. 10.1007/s11565-009-0069-1

    work page doi:10.1007/s11565-009-0069-1 2010

  12. [13]

    Catania and P

    D. Catania and P. Secchi. Global existence for two regularized MHD models in three space-dimension.Portugaliae Math., 68(1):41–52, 2011

    work page 2011

  13. [14]

    Constantin and C

    P. Constantin and C. Foias.Navier–Stokes Equations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988

    work page 1988

  14. [15]

    Coti Zelati and C

    M. Coti Zelati and C. G. Gal. Singular limits of Voigt models in fluid dynamics.J. Math. Fluid Mech., 17(2):233–259, 2015. 32

    work page 2015

  15. [16]

    V. M. Cuff, A. A. Dunca, C. C. Manica, and L. G. Rebholz. The reduced order NS-αmodel for incompressible flow: theory, numerical analysis and benchmark testing.ESAIM Math. Model. Numer. Anal., 49(3):641–662, 2015

    work page 2015

  16. [17]

    H. B. de Oliveira, K. Khompysh, and A. G. Shakir. Strong solutions for the Navier–Stokes–Voigt equations with non-negative density.J. Math. Phys., 66(4):041506, 2025

    work page 2025

  17. [18]

    Di Molfetta, G

    G. Di Molfetta, G. Krstlulovic, and M. Brachet. Self-truncation and scaling in Euler-Voigt-αand related fluid models.Phys. Rev. E, 92:013020, Jul 2015

    work page 2015

  18. [19]

    Di Plinio, A

    F. Di Plinio, A. Giorgini, V. Pata, and R. Temam. Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity. J. Nonlinear Sci., 28(2):653–686, 2018

    work page 2018

  19. [20]

    Duvaut and J.-L

    G. Duvaut and J.-L. Lions. In´ equations en thermo´ elasticit´ e et magn´ eto- hydrodynamique.Arch. Rational Mech. Anal., 46:241–279, 1972

    work page 1972

  20. [21]

    M. A. Ebrahimi, M. Holst, and E. Lunasin. The Navier–Stokes–Voight model for image inpainting.IMA J. App. Math., pages 1–26, 2012. doi:10.1093/imamat/hxr069

    work page doi:10.1093/imamat/hxr069 2012

  21. [22]

    Farhat, E

    A. Farhat, E. Lunasin, and E. S. Titi. Continuous data assimilation for a 2D B´ enard convection system through horizontal velocity measure- ments alone.J. Nonlinear Sci., pages 1–23, 2017

    work page 2017

  22. [23]

    Foias, O

    C. Foias, O. Manley, R. Rosa, and R. Temam.Navier–Stokes Equa- tions and Turbulence, volume 83 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2001

    work page 2001

  23. [24]

    C. G. Gal and T. T. Medjo. A Navier–Stokes–Voight model with mem- ory.Math. Methods Appl. Sci., 36(18):2507–2523, 2013

    work page 2013

  24. [25]

    Gao and C

    H. Gao and C. Sun. Random dynamics of the 3D stochastic Navier–Stokes–Voight equations.Nonlinear Anal. Real World Appl., 13(3):1197–1205, 2012

    work page 2012

  25. [26]

    Garc´ ıa-Luengo, P

    J. Garc´ ıa-Luengo, P. Mar´ ın-Rubio, and J. Real. Pullback attractors for three-dimensional non-autonomous Navier–Stokes–Voigt equations. Nonlinearity, 25(4):905–930, 2012

    work page 2012

  26. [27]

    T. Gatski. Review of incompressible fluid flow computations using the vorticity-velocity formulation.Appl. Numer. Math., 7:227–239, 1991

    work page 1991

  27. [28]

    Guevremont, W

    G. Guevremont, W. G. Habashi, and M. M. Hafez. Finite element solution of the Navier–Stokes equations by a velocity-vorticity method. Int. J. Numer. Methods Fluids, 10:461–475, 1990

    work page 1990

  28. [29]

    Heister, M

    T. Heister, M. A. Olshanskii, and L. G. Rebholz. Unconditional long- time stability of velocity-vorticity method for 2D Navier–Stokes equa- tions.Numer. Math., 135:143–167, 2017

    work page 2017

  29. [30]

    Ilyin, V

    A. Ilyin, V. Kalantarov, and S. Zelik. Attractors and their dimensions for the 3D fractional Navier–Stokes–Voigt equations.arXiv preprint arXiv:2511.04783, 2025

    work page arXiv 2025

  30. [31]

    Ilyin and S

    A. Ilyin and S. Zelik. Attractors for the Navier–Stokes–Voigt equations and their dimension.Lobachevskii J. Math., 46(9):4659–4673, 2025. 33

    work page 2025

  31. [32]

    D. A. Jones and E. S. Titi. Determining finite volume elements for the 2D Navier–Stokes equations.Phys. D, 60(1-4):165–174, 1992. Exper- imental mathematics: computational issues in nonlinear science (Los Alamos, NM, 1991)

    work page 1992

  32. [33]

    V. K. Kalantarov, B. Levant, and E. S. Titi. Gevrey regularity for the attractor of the 3D Navier–Stokes–Voight equations.J. Nonlinear Sci., 19(2):133–152, 2009

    work page 2009

  33. [34]

    V. K. Kalantarov and E. S. Titi. Global attractors and determining modes for the 3D Navier–Stokes–Voight equations.Chinese Ann. Math. B, 30(6):697–714, 2009

    work page 2009

  34. [35]

    Khouider and E

    B. Khouider and E. S. Titi. An inviscid regularization for the surface quasi-geostrophic equation.Comm. Pure Appl. Math., 61(10):1331– 1346, 2008

    work page 2008

  35. [36]

    Krasnoschok, V

    M. Krasnoschok, V. Pata, S. V. Siryk, and N. Vasylyeva. A subdiffusive Navier–Stokes–Voigt system.Phys. D, 409:132503, 2020

    work page 2020

  36. [37]

    Kuberry, A

    P. Kuberry, A. Larios, L. G. Rebholz, and N. E. Wilson. Nu- merical approximation of the Voigt regularization for incompressible Navier–Stokes and magnetohydrodynamic flows.Comput. Math. Appl., 64(8):2647–2662, 2012

    work page 2012

  37. [38]

    Ladyzhenskaya

    O. Ladyzhenskaya. On some nonlinear problems in the mechanics of continuous media.Intern. Congr. Math., Abstracts of Invited Lectures [in Russian], Moscow, page 149, 1966

    work page 1966

  38. [39]

    Larios, E

    A. Larios, E. Lunasin, and E. S. Titi. Global well-posedness for the Boussinesq-Voigt equations.(preprint) arXiv:1010.5024, 2015

    work page arXiv 2015

  39. [40]

    Larios and Y

    A. Larios and Y. Pei. On the local well-posedness and a Prodi-Serrin- type regularity criterion of the three-dimensional MHD-Boussinesq sys- tem without thermal diffusion.J. Differential Equations, 263(2):1419– 1450, 2017

    work page 2017

  40. [41]

    Larios, Y

    A. Larios, Y. Pei, and L. Rebholz. Global well-posedness of the velocity- vorticity-Voigt model of the 3D Navier–Stokes equations.J. Differential Equations, 266(5):2435–2465, 2019

    work page 2019

  41. [42]

    Larios, M

    A. Larios, M. R. Petersen, E. S. Titi, and B. Wingate. A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization.Theor. Comput. Fluid Dyn., 32(1):23–34, 2018

    work page 2018

  42. [43]

    Larios and E

    A. Larios and E. S. Titi. On the higher-order global regularity of the in- viscid Voigt-regularization of three-dimensional hydrodynamic models. Discrete Contin. Dyn. Syst. Ser. B, 14(2):603–627, 2010

    work page 2010

  43. [44]

    Larios and E

    A. Larios and E. S. Titi. Higher-order global regularity of an inviscid Voigt-regularization of the three-dimensional inviscid resistive magne- tohydrodynamic equations.J. Math. Fluid Mech., 16(1):59–76, 2014

    work page 2014

  44. [45]

    W. J. Layton and L. G. Rebholz. On relaxation times in the Navier– Stokes–Voigt model.Int. J. Comput. Fluid Dyn., 27(3):184–187, 2013

    work page 2013

  45. [46]

    H. K. Lee, M. A. Olshanskii, and L. G. Rebholz. On error analysis for the 3D Navier–Stokes equations in velocity-vorticity-helicity form. 34 SIAM J. Numer. Anal., 49(2):711–732, 2011

    work page 2011

  46. [47]

    Levant, F

    B. Levant, F. Ramos, and E. S. Titi. On the statistical properties of the 3D incompressible Navier–Stokes–Voigt model.Commun. Math. Sci., 8(1):277–293, 2010

    work page 2010

  47. [48]

    Li and Y

    H. Li and Y. Qin. Pullback attractors for three-dimensional Navier– Stokes–Voigt equations with delays.Bound. Value Probl., pages 2013:191, 17, 2013

    work page 2013

  48. [49]

    D. C. Lo, D. L. Young, and K. Murugesan. An accurate numerical solution algorithm for 3D velocity-vorticity Navier–Stokes equations by the DQ method.Commun. Numer. Meth. Engng, 22:235–250, 2006

    work page 2006

  49. [50]

    H. L. Meitz and H. F. Fasel. A compact-difference scheme for the Navier–Stokes equations in vorticity-velocity formulation.J. Comput. Phys., 157:371–403, 2000

    work page 2000

  50. [51]

    Nguyen Duong

    T. Nguyen Duong. Existence and long-time behavior of solutions to the velocity–vorticity–Voigt model of the 3D Navier–Stokes equations with damping and memory.Math. Methods Appl. Sci., 45(17):11635–11658, 2022

    work page 2022

  51. [52]

    C. J. Niche. Decay characterization of solutions to Navier–Stokes– Voigt equations in terms of the initial datum.J. Differential Equations, 260(5):4440–4453, 2016

    work page 2016

  52. [53]

    Nirenberg

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

    work page 1959

  53. [54]

    M. A. Olshanskii, T. Heister, L. Rebholz, and K. Galvin. Natural vorticity boundary conditions on solid walls.Comput. Methods Appl. Mech. Engrg., 297:18–37, 2015

    work page 2015

  54. [55]

    M. A. Olshanskii and L. G. Rebholz. Velocity-vorticity-helicity formu- lation and a solver for the Navier–Stokes equations.J. Comput. Phys., 229:4291–4303, 2010

    work page 2010

  55. [56]

    M. A. Olshanskii, L. G. Rebholz, and A. J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier–Stokes equa- tions with no-slip boundary conditions.Discrete Contin. Dyn. Syst. A, 38(7), 2018

    work page 2018

  56. [57]

    A. P. Oskolkov. The uniqueness and solvability in the large of bound- ary value problems for the equations of motion of aqueous solutions of polymers.Zap. Nauˇ cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38:98–136, 1973. Boundary value problems of mathematical physics and related questions in the theory of functions, 7

    work page 1973

  57. [58]

    A. P. Oskolkov. Some quasilinear systems that arise in the study of the motion of viscous fluids.Zap. Nauˇ cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI), 52:128–157, 219, 1975. Boundary value problems of mathematical physics, and related questions in the theory of functions, 8

    work page 1975

  58. [59]

    A. P. Oskolkov. Some nonstationary linear and quasilinear systems that arise in the study of the motion of viscous fluids.Zap. Nauˇ cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 59:133–177, 257, 35

    work page

  59. [60]

    Boundary value problems of mathematical physics and related questions in the theory of functions, 9

    work page

  60. [61]

    A. P. Oskolkov. On the theory of unsteady flows of Kelvin-Voigt flu- ids.Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 115:191–202, 310, 1982. Boundary value problems of mathematical physics and related questions in the theory of functions, 14

    work page 1982

  61. [62]

    Y. Pei. Regularity and convergence results of the velocity-vorticity- Voigt model of the 3D Boussinesq equations.Acta Appl. Math., 176:Pa- per No. 8, 25, 2021

    work page 2021

  62. [63]

    Qin and H

    Y. Qin and H. Jiang. Existence of pullback attractors and invariant measures for 3d Navier-Stokes-Voigt equations with delay.Discrete and Continuous Dynamical Systems - B, 30(1):243–264, 2025

    work page 2025

  63. [64]

    Ramos and E

    F. Ramos and E. S. Titi. Invariant measures for the 3D Navier–Stokes– Voigt equations and their Navier–Stokes limit.Discrete Contin. Dyn. Syst., 28(1):375–403, 2010

    work page 2010

  64. [65]

    Y. Rong, J. A. Fiordilino, F. Shi, and Y. Cao. A modular Voigt regu- larization of the Crank–Nicolson finite element method for the Navier– Stokes equations.J. Sci. Comput., 92(3):Paper No. 101, 35, 2022

    work page 2022

  65. [66]

    Sermange and R

    M. Sermange and R. Temam. Some mathematical questions related to the MHD equations.Comm. Pure Appl. Math., 36(5):635–664, 1983

    work page 1983

  66. [67]

    Q. B. Tang. Dynamics of stochastic three dimensional Navier–Stokes– Voigt equations on unbounded domains.J. Math. Anal. Appl., 419(1):583–605, 2014

    work page 2014

  67. [68]

    Temam.Infinite-Dimensional Dynamical Systems In Mechanics and Physics, volume 68 ofApplied Mathematical Sciences

    R. Temam.Infinite-Dimensional Dynamical Systems In Mechanics and Physics, volume 68 ofApplied Mathematical Sciences. Springer-Verlag, New York, second edition, 1997

    work page 1997

  68. [69]

    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

    work page 2001

  69. [70]

    Thi Ngan and V

    N. Thi Ngan and V. Manh Toi. Feedback control of Navier–Stokes– Voigt equations by finite determining parameters.Acta Math. Viet- nam., 45:917–930, 2020

    work page 2020

  70. [71]

    K. L. Wong and A. J. Baker. A 3D incompressible Navier–Stokes velocity-vorticity weak form finite element algorithm.Int. J. Numer. Meth. Fluids, 38:99–123, 2002

    work page 2002

  71. [72]

    X. H. Wu, J. Z. Wu, and J. M. Wu. Effective vorticity-velocity for- mulations for the three-dimensional incompressible viscous flows.J. Comput. Phys., 122:68–82, 1995

    work page 1995

  72. [73]

    X. Yang, B. Feng, T. M. de Souza, and T. Wang. Long-time dynam- ics for a non-autonomous Navier–Stokes–Voigt equation in Lipschitz domains.Discrete Contin. Dyn. Syst. Ser. B, 24(1):363–386, 2019

    work page 2019

  73. [74]

    X. Yang, P. Huang, and Y. He. A Voigt-regularization of the thermally coupled inviscid, resistive magnetohydrodynamic.Int. J. Numer. Anal. Model., 21(4), 2024. 36

    work page 2024

  74. [75]

    X. Yang, P. Huang, and Y. He. A Voigt regularization for incompress- ible MHD equations in Els¨ asser variables.J. Sci. Comput., 103(1):21, 2025

    work page 2025

  75. [76]

    X.-G. Yang, L. Li, and Y. Lu. Regularity of uniform attractor for 3D non-autonomous Navier–Stokes–Voigt equation.Appl. Math. Comput., 334:11–29, 2018

    work page 2018

  76. [77]

    Yue and J

    G. Yue and J. Wang. Attractors of the velocity–vorticity–Voigt model of the 3D Navier–Stokes equations with damping.Comput. Math. Appl., 80(3):434–452, 2020

    work page 2020

  77. [78]

    A. Zang. Local well-posedness for boundary layer equations of Euler- Voigt equations in analytic setting.J. Differential Equations, 307:1–28, 2022

    work page 2022

  78. [79]

    Zhao and H

    C. Zhao and H. Zhu. Upper bound of decay rate for solutions to the Navier–Stokes–Voigt equations inR 3.Appl. Math. Comput., 256:183– 191, 2015. (Adam Larios)Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE 68588-0130, USA Email address, Adam Larios:alarios@unl.edu (Yuan Pei)Department of Mathematics, Western W ashington University Be...

    work page 2015