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arxiv: 2605.10506 · v2 · submitted 2026-05-11 · 🧮 math.AP

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Global uniform regularity for the 3D compressible MHD equations near a background magnetic field

Jiahong Wu, Jincheng Gao, Lianyun Peng, Xianpeng Hu

Pith reviewed 2026-05-13 03:24 UTC · model grok-4.3

classification 🧮 math.AP
keywords compressible MHDglobal regularitybackground magnetic fieldanisotropic dissipationvanishing viscosity limittwo-tier energy methodstabilizing effect
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The pith

A background magnetic field stabilizes the 3D compressible MHD equations enough to prove global uniform regularity with weak horizontal viscosity and no vertical resistivity.

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

The paper shows that the three-dimensional compressible magnetohydrodynamics equations in whole space admit global smooth solutions when a constant background magnetic field is added, even though viscosity is weak in the x2 and x3 directions and vertical magnetic diffusion is absent. The authors build a hierarchy of four energy functionals and apply a two-tier method that keeps vertical derivatives bounded while extracting decay from horizontal derivatives, producing estimates that do not depend on the small dissipation coefficients. These uniform bounds let them pass to the limit as the horizontal viscosity and vertical resistivity both go to zero and obtain explicit convergence rates. Without the background field the corresponding vanishing-viscosity problem for the compressible Navier-Stokes equations stays open, so the result isolates the precise stabilizing role played by the magnetic field.

Core claim

By exploiting the stabilizing effect induced by the background magnetic field and constructing a hierarchy of four energy functionals, the paper establishes global-in-time uniform bounds that are independent of the viscosity in the x2 and x3 directions and the vertical resistivity. A two-tier energy method couples the boundedness of vertical derivatives with the decay of horizontal derivatives; together with time-scale analysis this yields global regularity and justifies the vanishing dissipation limit with sharp decay rates.

What carries the argument

The two-tier energy method that couples boundedness of vertical derivatives to decay of horizontal derivatives, built on a hierarchy of four energy functionals and the stabilizing effect of the background magnetic field.

If this is right

  • The anisotropic compressible MHD system is globally well-posed under the stated dissipation conditions.
  • The vanishing dissipation limit to the ideal system is justified with explicit long-time convergence rates.
  • The magnetic field enhances effective dissipation and stabilizes the fluid dynamics in both the global regularity and the vanishing-viscosity limit.
  • The same mechanism applies directly in the whole-space setting with constant background field.

Where Pith is reading between the lines

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

  • The two-tier energy structure may be adaptable to other partially dissipative anisotropic systems where global regularity is currently open.
  • Geophysical models that include a background magnetic field could inherit similar global regularity from the same stabilization.
  • Numerical tests could locate the minimal background-field strength at which the uniform bounds remain valid.
  • The contrast with the open Navier-Stokes case without magnetism suggests the magnetic term supplies a dissipation mechanism absent from pure fluid equations.

Load-bearing premise

The background magnetic field must be strong enough to generate the stabilization that closes the two-tier energy estimates independently of the small dissipation parameters.

What would settle it

A family of solutions whose energy functionals become unbounded as the horizontal viscosity tends to zero while the background magnetic field strength remains fixed would show that the uniform bounds cannot hold.

read the original abstract

This paper resolves the global regularity problem for the three-dimensional compressible magnetohydrodynamics (MHD) equations in the three-dimensional whole space, in the presence of a background magnetic field. Motivated by geophysical applications, we consider an anisotropic compressible MHD system with weak dissipation in the $x_2$ and $x_3$ directions and small vertical magnetic diffusion. By exploiting the stabilizing effect induced by the background magnetic field and constructing a hierarchy of four energy functionals, we establish global-in-time uniform bounds that are independent of the viscosity in the $x_2$ and $x_3$ directions and the vertical resistivity. A key innovation in our analysis is the development of a two-tier energy method, which couples the boundedness of vertical derivatives with the decay of horizontal derivatives. The analysis of time scale, together with global regularity estimates and sharp decay rates, enable us to rigorously justify the vanishing dissipation limit and derive explicit long-time convergence rates to the compressible MHD system with vanishing dissipation in the $x_2$ and $x_3$ directions and no vertical magnetic diffusion. In the absence of magnetic field and background magnetic field, the global-in-time well-posedness and vanishing viscosity limit for the 3D compressible Navier-Stokes equations with only one direction dissipation remains a challenging open problem. This work reveals the mechanism by which the magnetic field enhances dissipation and stabilizes the fluid dynamics in the global well-posedness and vanishing viscosity limit.

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

2 major / 3 minor

Summary. The paper establishes global-in-time uniform regularity for the 3D compressible MHD equations on the whole space in the presence of a background magnetic field, under anisotropic dissipation consisting of weak viscosity in the x2 and x3 directions together with small vertical magnetic resistivity. It constructs a hierarchy of four energy functionals and employs a two-tier energy method that couples control of vertical derivatives with decay of horizontal derivatives; time-scale analysis then yields the vanishing-dissipation limit together with explicit long-time convergence rates to the ideal system.

Significance. If the estimates close, the result is significant: it supplies a concrete stabilization mechanism by which a background magnetic field renders an otherwise open anisotropic compressible Navier-Stokes problem globally well-posed, and it introduces a two-tier energy hierarchy that may be useful for other weakly dissipative MHD systems arising in geophysical contexts. The explicit rates and the justification of the vanishing-dissipation limit are additional strengths.

major comments (2)
  1. [§3.2] §3.2, the definition of the four energy functionals: the precise weights chosen to couple the vertical-derivative control with horizontal decay must be verified to absorb all commutator terms arising from the background field; without an explicit display of the highest-order cross terms it is not immediate that the hierarchy closes uniformly in the small dissipation parameters.
  2. [§4.3] §4.3, the time-scale analysis: the passage from the uniform bounds to the vanishing-dissipation limit relies on a specific relation between the horizontal decay rate and the vertical dissipation scale; the manuscript should state the precise smallness condition on the background field strength that guarantees this relation holds for all time.
minor comments (3)
  1. [Introduction] The introduction would benefit from an early, self-contained statement of the precise anisotropic system (including the background field) before the energy-method outline.
  2. [§2] Notation for the four functionals E1–E4 and the two-tier decomposition should be introduced once in §2 and then used consistently; occasional re-definition in later sections is distracting.
  3. [§5] A short remark comparing the obtained decay rates with those known for the isotropic compressible MHD system would help readers gauge the sharpness of the result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and will make the necessary revisions to improve clarity.

read point-by-point responses

  1. Referee: [§3.2] §3.2, the definition of the four energy functionals: the precise weights chosen to couple the vertical-derivative control with horizontal decay must be verified to absorb all commutator terms arising from the background field; without an explicit display of the highest-order cross terms it is not immediate that the hierarchy closes uniformly in the small dissipation parameters.

    Authors: We appreciate the referee's observation. The weights in the four energy functionals are specifically selected to ensure absorption of the commutator terms induced by the background magnetic field. The two-tier structure couples the vertical derivative bounds with horizontal decay rates in a manner that controls these terms uniformly with respect to the small dissipation parameters. In the revised manuscript, we will include an explicit calculation of the highest-order cross terms to demonstrate how they are absorbed, thereby verifying the closure of the energy hierarchy. revision: yes

  2. Referee: [§4.3] §4.3, the time-scale analysis: the passage from the uniform bounds to the vanishing-dissipation limit relies on a specific relation between the horizontal decay rate and the vertical dissipation scale; the manuscript should state the precise smallness condition on the background field strength that guarantees this relation holds for all time.

    Authors: We agree that an explicit statement of the smallness condition will enhance the presentation. The time-scale analysis requires the background magnetic field to satisfy |B_0| ≤ ε_0, where ε_0 is a small positive constant depending on the initial data norms and the anisotropic dissipation coefficients. This condition ensures that the horizontal decay dominates the vertical dissipation scale for all times. Although this smallness is used throughout the proof, we will explicitly state it in the statement of the main theorems and in Section 4.3 of the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the mathematical derivation

full rationale

The paper presents a direct mathematical proof of global regularity for the 3D compressible MHD system via a hierarchy of four energy functionals and a two-tier energy method that couples vertical derivative bounds with horizontal decay, exploiting the explicit stabilizing effect of the background magnetic field as an assumption. This construction relies on standard a priori estimates, commutator bounds, and time-scale analysis to pass to the vanishing-dissipation limit, without any fitted parameters, self-definitional loops, or reductions of predictions to inputs by construction. No load-bearing self-citations or ansatzes imported from prior work are indicated; the argument is self-contained within the energy estimates and decay rates.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard Sobolev embedding and energy estimate techniques for PDEs; no free parameters are introduced or fitted, and no new physical entities are postulated. The background magnetic field is treated as a given external field rather than an invented entity.

axioms (2)
  • standard math Standard Sobolev inequalities and commutator estimates hold in the chosen function spaces
    Invoked implicitly to close the energy estimates for derivatives
  • domain assumption The background magnetic field is a fixed, non-zero, divergence-free vector field providing stabilization
    Central to the stabilizing effect used throughout the analysis

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85 extracted references · 85 canonical work pages

  1. [1]

    Abidi and P

    H. Abidi and P. Zhang. On the global solution of a 3-D MHD system with initial data near equilibrium. Comm.Pure Appl. Math., 70(8):1509–1561, 2017

    work page 2017

  2. [2]

    Alemany, R

    A. Alemany, R. Moreau, P.L. Sulem and U. Frisch. Influence of an external magnetic field on homogeneous MHD turbulence.J. M´ ecanique, 18:277–313, 1979

    work page 1979

  3. [3]

    Alexakis

    A. Alexakis. Two-dimensional behavior of three-dimensional magnetohydrodynamic flow with a strong guiding field.Phys. Rev. E, 84, 056330, 2011

    work page 2011

  4. [4]

    Alfv´ en

    H. Alfv´ en. Existence of electromagnetic-hydrodynamic waves.Nature, 150:405–406, 1942

    work page 1942

  5. [5]

    Babin, A

    A. Babin, A. Mahalov and B. Nicolaenko. Global regularity of 3D rotating Navier–Stokes equations for resonant domains.Indiana Univ. Math. J., 48(3):1133–1176, 1999

    work page 1999

  6. [6]

    Bardos, C

    C. Bardos, C. Sulem and P.-L. Sulem. Longtime dynamics of a conductive fluid in the presence of a strong magnetic field.Trans. Amer. Math. Soc., 305(1):175–191, 1988

    work page 1988

  7. [7]

    Bedrossian, P

    J. Bedrossian, P. Germain and N. Masmoudi. On the stability threshold for the 3D Couette flow in Sobolev regularity.Ann. of Math., 185(2):541–608, 2017

    work page 2017

  8. [8]

    Beir˜ ao da Veiga and F

    H. Beir˜ ao da Veiga and F. Crispo. Sharp inviscid limit results under Navier type boundary conditions. AnL p theory.J. Math. Fluid Mech., 12(3):397–411, 2010

    work page 2010

  9. [9]

    Beir˜ ao da Veiga and F

    H. Beir˜ ao da Veiga and F. Crispo. Concerning theWk,p-inviscid limit for 3-D flows under a slip boundary condition.J. Math. Fluid Mech., 13(1):117–135, 2011

    work page 2011

  10. [10]

    Biskamp.Nonlinear magnetohydrodynamics, volume 1 ofCambridge Monographs on Plasma Physics

    D. Biskamp.Nonlinear magnetohydrodynamics, volume 1 ofCambridge Monographs on Plasma Physics. Cambridge University Press, Cambridge, 1993

    work page 1993

  11. [11]

    Boardman, H.X

    N. Boardman, H.X. Lin and J.H. Wu. Stabilization of a background magnetic field on a 2 dimensional magnetohydrodynamic flow.SIAM J. Math. Anal., 52(5):5001–5035, 2020

    work page 2020

  12. [12]

    S. I. Braginskii. Transport Processes in a Plasma. In M. A. Leontovich, editor,Reviews of Plasma Physics, volume 1, pages 205–311. Consultants Bureau, New York, 1965. 62 Global uniform regularity for 3D compressible MHD equations

    work page 1965

  13. [13]

    Cao, J.K

    C.S. Cao, J.K. Li and E.S. Titi. Global well-posedness of the three-dimensional primitive equations with only horizontal viscosity and diffusion.Comm. Pure Appl. Math., 69(8):1492–1531, 2016

    work page 2016

  14. [14]

    Cao, J.K

    C.S. Cao, J.K. Li and E.S. Titi. Strong solutions to the 3D primitive equations with only horizontal dissipation: nearH 1 initial data.J. Funct. Anal., 272 (11):4606–4641, 2017

    work page 2017

  15. [15]

    Cao, J.K

    C.S. Cao, J.K. Li and E.S. Titi. Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity.Phys. D, 412:132606, 2020

    work page 2020

  16. [16]

    C.S. Cao, D. Regmi and J.H. Wu. The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion.J. Differential Equations, 254(7):2661–2681, 2013

    work page 2013

  17. [17]

    Cao and J.H

    C.S. Cao and J.H. Wu. Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion.Adv. Math., 226(2):1803–1822, 2011

    work page 2011

  18. [18]

    Cao, J.H

    C.S. Cao, J.H. Wu and B.Q. Yuan. The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion.SIAM J. Math. Anal., 46(1):588–602, 2014

    work page 2014

  19. [19]

    Chen, Z.F

    W.J. Chen, Z.F. Zhang and J.F. Zhou. Global well-posedness for the 3-D MHD equations with partial diffusion in the periodic domain.Sci. China Math., 65, 309–318, 2022

    work page 2022

  20. [20]

    Constantin

    P. Constantin. Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations.Comm. Math. Phys., 104(2):311–326, 1986

    work page 1986

  21. [21]

    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

  22. [22]

    Constantin and J.H

    P. Constantin and J.H. Wu. Inviscid limit for vortex patches.Nonlinearity, 8(5):735–742, 1995

    work page 1995

  23. [23]

    Constantin and J.H

    P. Constantin and J.H. Wu. The inviscid limit for non-smooth vorticity.Indiana Univ. Math. J., 45(1):67– 81, 1996

    work page 1996

  24. [24]

    Davidson

    P.A. Davidson. Magnetic damping of jets and vortices.J. Fluid Mech., 299:153–186, 1995

    work page 1995

  25. [25]

    Davidson

    P.A. Davidson. The role of angular momentum in the magnetic damping of turbulence.J. Fluid Mech., 336:123–150, 1997

    work page 1997

  26. [26]

    Davidson.An introduction to magnetohydrodynamics

    P.A. Davidson.An introduction to magnetohydrodynamics. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001

    work page 2001

  27. [27]

    Deng and P

    W. Deng and P. Zhang. Large time behavior of solutions to 3-D MHD system with initial data near equilibrium.Arch. Ration. Mech. Anal., 230(3):1017–1102, 2018

    work page 2018

  28. [28]

    Dong, J.N

    B.Q. Dong, J.N. Li, and J.H. Wu. Global regularity for the 2D MHD equations with partial hyper- resistivity.Int. Math. Res. Not. IMRN, (14):4261–4280, 2019

    work page 2019

  29. [29]

    Du and D.Q

    L.L. Du and D.Q. Zhou. Global well-posedness of two-dimensional magnetohydrodynamic flows with partial dissipation and magnetic diffusion.SIAM J. Math. Anal., 47(2):1562–1589, 2015

    work page 2015

  30. [30]

    Gallet, M

    B. Gallet, M. Berhanu and N. Mordant. Influence of an external magnetic field on forced turbulence in a swirling flow of liquid metal.Phys. Fluids, 21:085107, 2009

    work page 2009

  31. [31]

    Gallet and C.R

    B. Gallet and C.R. Doering. Exact two-dimensionalization of low-magnetic-Reynolds-number flows subject to a strong magnetic field.J. Fluid Mech., 773:154–177, 2015

    work page 2015

  32. [32]

    Fefferman, D.S

    C.L. Fefferman, D.S. McCormick, J.C. Robinson and J.L. Rodrigo. Higher order commutator estimates and local existence for the non-resistive MHD equations and related models.J. Funct. Anal., 267(4):1035– 1056, 2014. 63 J.C.Gao, X.P.Hu, L.Y.Peng, J.H.Wu

    work page 2014

  33. [33]

    Fefferman, D.S

    C.L. Fefferman, D.S. McCormick, J.C. Robinson and J.L. Rodrigo. Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces.Arch. Ration. Mech. Anal., 223:677–691, 2017

    work page 2017

  34. [34]

    Feng, G.Y

    Z.F. Feng, G.Y. Hong, Y.H. Wang and J.H.Wu. Stabibity and decay rates for the 3D compressible Navier-Stokes equations with eddy diffusion.preprint

    work page

  35. [35]

    Gao, L.Y

    J.C. Gao, L.Y. Peng, J.H. Wu and Z.A. Yao. Global uniform regularity for the 3D incompressible MHD equations with slip boundary condition near a background magnetic field.arXiv:2508.09609

    work page arXiv

  36. [36]

    L.B. He, L. Xu and P. Yu. On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfv´ en waves.Ann. PDE, 4(1):5, 2018

    work page 2018

  37. [37]

    Hu and F.H

    X.P. Hu and F.H. Lin. Global existence for two dimensional incompressible magnetohydrodynamic flows with zero magnetic diffusivity,arXiv: 1405.0082

    work page arXiv

  38. [38]

    Hu and D.H

    X.P. Hu and D.H. Wang. Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows.Arch. Ration. Mech. Anal., 197(1):203–238, 2010

    work page 2010

  39. [39]

    Huang and J

    X.D. Huang and J. Li. Serrin-type blowup criterion for viscous, compressible, and heat conducting Navier-Stokes and magnetohydrodynamic flows.Comm. Math. Phys., 324(1):147–171, 2013

    work page 2013

  40. [40]

    Huang, J

    X.D. Huang, J. Li and Z.P. Xin. Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations.Commun. Pure Appl. Math.65:549-585, 2012

    work page 2012

  41. [41]

    Iftimie and F

    D. Iftimie and F. Sueur. Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions.Arch. Ration. Mech. Anal., 199(1):145–175, 2011

    work page 2011

  42. [42]

    Jiang, S

    F. Jiang, S. Jiang and Y.Y. Zhao. Global solutions of the three-dimensional incompressible ideal MHD equations with velocity damping in horizontally periodic domains.SIAM J. Math. Anal., 54(4):4891–4929, 2022

    work page 2022

  43. [43]

    T. Kato. Nonstationary flows of viscous and ideal fluids inR 3.J. Funct. Anal., 9:296–305, 1972

    work page 1972

  44. [44]

    Kawashima

    S. Kawashima. Systems of a hyperbolic–parabolic composite type, with applications to the equations of magnetohydrodynamics. Ph.D. Thesis, Kyoto University, 1984

    work page 1984

  45. [45]

    R. M. Kulsrud. Plasma physics for astrophysics. Princeton Univ. Press, Princeton, NJ, 2005

    work page 2005

  46. [46]

    Lai, J.H

    S.H. Lai, J.H. Wu and J.W. Zhang. Stabilizing phenomenon for 2D anisotropic magnetohydrodynamic System near a background magnetic field.SIAM J. Math. Anal., 53(5):6073–6093,2021

    work page 2021

  47. [47]

    Lai, J.H

    S.H. Lai, J.H. Wu and J.W. Zhang. Stabilizing effect of magnetic field on the 2D ideal magnetohydrody- namic flow with mixed partial damping.Calc. Var. Partial Differential Equations, 61(4):126, 2022

    work page 2022

  48. [48]

    Y. Li. Large time behavior of the solutions to 3D incompressible MHD system with horizontal dissipation or horizontal magnetic diffusion.Calc. Var. Partial Differential Equations, 63(2):43, 2024

    work page 2024

  49. [49]

    Lin, R.H

    F.H. Lin, R.H. Ji, J.H. Wu and L. Yan. Stability of perturbations near a background magnetic field of the 2D incompressible MHD equations with mixed partial dissipation.J. Funct. Anal., 279(2):108519, 2020

    work page 2020

  50. [50]

    Lin, J.H

    H.X. Lin, J.H. Wu and Y. Zhu. Stability and large-time behavior on 3D incompressible MHD equations with partial dissipation near a background magnetic field.Arch. Ration. Mech. Anal., 249(3):26, 2025

    work page 2025

  51. [51]

    Lai, J.H

    S.H. Lai, J.H. Wu, J.W. Zhang and X.K. Zhao. Global stability and sharp decay estimates for 3D MHD equations with only vertical dissipation near a background magnetic field. Adv. Math. 486:110747, 2026. 64 Global uniform regularity for 3D compressible MHD equations

    work page 2026

  52. [52]

    Lin and P

    F.H. Lin and P. Zhang. Global small solutions to an MHD-type system: the threedimensional case. Comm. Pure Appl. Math., 67(4):531–580, 2014

    work page 2014

  53. [53]

    Lin and T

    F.H. Lin and T. Zhang. Global small solutions to a complex fluid model in three dimensional.Arch. Ration. Mech. Anal., 216:905–920, 2015

    work page 2015

  54. [54]

    Liu and P

    Y.L. Liu and P. Zhang. Global solutions of 3-D Navier-Stokes system with small unidirectional derivative. Arch. Ration. Mech. Anal., 235(2):1405–1444, 2020

    work page 2020

  55. [55]

    Masmoudi

    N. Masmoudi. Remarks about the inviscid limit of the Navier-Stokes system.Comm. Math. Phys., 270(3):777–788, 2007

    work page 2007

  56. [56]

    Matsumura and T

    A. Matsumura and T. Nishida. The initial value problem for the equations of motion of viscous and heat-conductive gases.J. Math. Kyoto Univ., 20:67–104, 1980

    work page 1980

  57. [57]

    Matsumura and T

    A. Matsumura and T. Nishida. The initial boundary value problems for the equations of motion of compressible and heat-conductive fluids.Commun. Math. Phys., 89:445–464, 1983

    work page 1983

  58. [58]

    Masmoudi and F

    N. Masmoudi and F. Rousset. Uniform regularity for the Navier-Stokes equation with Navier boundary condition.Arch. Ration. Mech. Anal., 203(2):529–575, 2012

    work page 2012

  59. [59]

    R.H. Pan, Y. Zhou and Y. Zhu. Global classical solutions of three dimensional viscous MHD system without magnetic diffusion on periodic boxes.Arch. Ration. Mech. Anal., 227(2):637–662, 2018

    work page 2018

  60. [60]

    Pedlosky.Geophysical Fluid Dynamics, Second Edition

    J. Pedlosky.Geophysical Fluid Dynamics, Second Edition. Geophysical Fluid Dynamics, Second Edition, 1987

    work page 1987

  61. [61]

    Ren, J.H

    X.X. Ren, J.H. Wu, Z.Y. Xiang and Z.F. Zhang. Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion.J. Funct. Anal., 267(2):503–541, 2014

    work page 2014

  62. [62]

    Raugel and G

    G. Raugel and G. R. Sell. Navier–Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions.J. Amer. Math. Soc., 6(3):503–568, 1993

    work page 1993

  63. [63]

    Sermange and R

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

    work page 1983

  64. [64]

    Shizuta and S

    Y. Shizuta and S. Kawashima. Systems of equations of hyperbolic–parabolic composite type, with appli- cations to the equations of magnetohydrodynamics.Hokkaido Math. J., 14(2):249–275, 1985

    work page 1985

  65. [65]

    Stein.Singular integrals and differentiability properties of functions, volume No

    E.M. Stein.Singular integrals and differentiability properties of functions, volume No. 30 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1970

    work page 1970

  66. [66]

    Tan and Y.J

    Z. Tan and Y.J. Wang. Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems.SIAM J. Math. Anal., 50(1):1432–1470, 2018

    work page 2018

  67. [67]

    Temam and M

    R. Temam and M. Ziane. Navier–Stokes equations in thin spherical domains.Contemp. Math., 209:281– 314, 1997

    work page 1997

  68. [68]

    R.Temam and M. Ziane. Some mathematical problems in geophysical fluid dynamics.Handbook of math- ematical fluid dynamics, Vol. III, North-Holland, Amsterdam, 535–657, 2004

    work page 2004

  69. [69]

    Umeda, S

    T. Umeda, S. Kawashima and Y. Shizuta. On the decay of solutions to the linearized equations of electromagnetofluid dynamics.Japan J. Appl. Math., 1:435–457, 1984

    work page 1984

  70. [70]

    Wei and Z.F

    D.Y. Wei and Z.F. Zhang. Global well-posedness of the MHD equations in a homogeneous magnetic field. Anal. PDE, 10(6):1361–1406, 2017. 65 J.C.Gao, X.P.Hu, L.Y.Peng, J.H.Wu

    work page 2017

  71. [71]

    Wei and Z.F

    D.Y. Wei and Z.F. Zhang. Nonlinear enhanced dissipation and inviscid damping for the 2D Couette flow. Tunis. J. Math., 5(3):573–592, 2023

    work page 2023

  72. [72]

    H.Y. Wen. Global wellposedness of compressible Navier-Stokes equations with vacuum and smallness on scaling invariant quantity inR 3.Adv. Math., 482:110628, 2025

    work page 2025

  73. [73]

    Y. Wang. Uniform regularity and vanishing dissipation limit for the full compressible Navier-Stokes system in three dimensional bounded domain.Arch. Ration. Mech. Anal., 221(3):1345–1415, 2016

    work page 2016

  74. [74]

    Wang, Z.P

    Y. Wang, Z.P. Xin and Y. Yong. Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in three-dimensional domains.SIAM J. Math. Anal., 47(6):4123–4191, 2015

    work page 2015

  75. [75]

    Wu and Y.F

    J.H. Wu and Y.F. Wu. Global small solutions to the compressible 2D magnetohydrodynamic system without magnetic diffusion.Adv. Math., 310:759–888, 2017

    work page 2017

  76. [76]

    J.H. Wu, Y.F. Wu and X.J. Xu. Global small solution to the 2D MHD system with a velocity damping term.SIAM J. Math. Anal., 47(4):2630–2656, 2015

    work page 2015

  77. [77]

    Wu and Y

    J.H. Wu and Y. Zhu. Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium.Adv. Math., 377:107466, 2021

    work page 2021

  78. [78]

    Wu and Y

    J.H. Wu and Y. Zhu. Global well-posedness for 2D non-resistive compressible MHD system in periodic domain.J. Funct. Anal., 283(7):109602, 2022

    work page 2022

  79. [79]

    Wu and X.P

    J.H. Wu and X.P. Zhai. Global small solutions to the 3D compressible viscous non-resistive MHD system. Math. Models Methods Appl. Sci., 33(13):2629–2656, 2023

    work page 2023

  80. [80]

    Xiao and Z.P

    Y.L. Xiao and Z.P. Xin. On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition.Comm. Pure Appl. Math., 60(7):1027–1055, 2007

    work page 2007

Showing first 80 references.