Nonlinear stability of a background magnetic field for the 3D compressible MHD equations with anisotropic dissipation

Jiahong Wu; Jincheng Gao; Lianyun Peng; Xianpeng Hu

REVIEW 1 major objections 1 minor 59 references

Nonlinear stability of a background magnetic field for the 3D compressible MHD equations with anisotropic dissipation

T0 review · 1 major / 1 minor · reviewed 2026-07-09 · glm-5.2

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Background magnetic field stabilizes 3D compressible MHD with minimal dissipation

desk verdict Genuinely new 3D compressible MHD stability result with anisotropic dissipation; the proof hinges on a delicate nonlinear cancellation that needs specialist verification. read the letter →

arxiv 2607.06910 v1 pith:X5S5AGH2 submitted 2026-07-08 math.AP

Jincheng Gao , Xianpeng Hu , Lianyun Peng , Jiahong Wu This is my paper · ORCID

classification math.AP MSC 35Q3576W0535B3576N10

keywords compressibleMHDanisotropicdissipationbackgroundmagneticfieldnonlinearstabilityenhancedcancellationSobolevspacesglobalwell-posedness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that a three-dimensional compressible magnetohydrodynamic (MHD) system, in which the velocity is damped only along two horizontal directions and the magnetic field diffuses along only one direction, is nonetheless globally stable near a quiescent equilibrium carrying a uniform background magnetic field. The equilibrium is a fluid at rest with constant density and a magnetic field pointing in a fixed direction. When the system is perturbed slightly, the natural dissipation built into the equations is far too weak to control the nonlinear interactions: there is no damping of the velocity in the vertical direction, no magnetic diffusion in two of three directions, and no density dissipation at all. The paper identifies two mechanisms that close this gap. First, the background magnetic field, through its linear coupling with the velocity and density, generates hidden enhanced dissipation that effectively damps the magnetic field in a missing direction and the density in horizontal directions. Second, a nonlinear cancellation mechanism absorbs a genuine loss of vertical derivatives that arises from the compressible coupling between density and velocity. Together these yield a global-in-time solution bound for sufficiently small Sobolev-norm initial data.

What carries the argument

The proof rests on three technical pillars. (1) Enhanced dissipation: testing the velocity equation against ∂₂b and against horizontal density gradients ∇ₕϱ extracts hidden damping for ∂₂b and ∇ₕϱ from the linear coupling ∇ϱ + ∇b₂ − ∂₂b, using div b = 0. (2) Nonlinear cancellation: the velocity is weighted by (1+ϱ) in the energy so that terms of the form ∫ b·∇∂ᵢu ∂ᵢb dx and ∫ (1/(1+ϱ)) b·∇∂ᵢb ∂ᵢu dx cancel exactly via div b = 0. (3) Iterative derivative trading: the momentum equation is used to trade vertical derivatives of the density (which has no dissipation) for derivatives of the magnetic field (which has partial dissipation), through four substitution iterations that convert the worst

What would settle it

A specific perturbation of the equilibrium whose nonlinear evolution generates a remainder term from the Lorentz force that does not cancel with the corresponding velocity-equation term, causing the highest-order vertical derivative estimate to grow without bound and breaking the bootstrap closure.

Watch

Extended reading notes

Core claim

The central discovery is that the combination of a background magnetic field and a specific nonlinear cancellation structure is sufficient to compensate for the severe absence of dissipation in a 3D compressible MHD system. The background field generates enhanced dissipation for the magnetic field and density in directions where no direct damping exists, by exploiting the linear coupling between the velocity equation, the pressure gradient, and the divergence-free constraint on the magnetic field. The nonlinear cancellation mechanism resolves what would otherwise be a fatal loss of vertical derivatives: when estimating the highest-order vertical derivatives, the Lorentz force nonlinearity b·

Load-bearing premise

The nonlinear cancellation mechanism requires that a derivative-losing term produced by the Lorentz force nonlinearity is exactly cancelled by an identical term of opposite sign generated when the velocity equation is differentiated and paired against a weighted velocity. If this cancellation is not exact at the highest derivative order, or if the remainder terms cannot be absorbed by the enhanced dissipation estimates, the vertical derivative estimates do not close and the全球

Editorial extensions

If this is right

The enhanced dissipation mechanism should extend to other anisotropic compressible flow models where a background field or background flow provides linear coupling that can be exploited by cross-testing.
The nonlinear cancellation structure may carry over to compressible MHD with different anisotropic dissipation patterns, provided the velocity weighting (1+ϱ) is preserved and div b = 0 holds.
The result suggests that the minimal dissipation threshold for global stability of 3D compressible MHD near a background field is lower than previously expected, potentially guiding the search for sharp conditions.
The iterative derivative-trading technique could be applied to other compressible systems where one variable (density) lacks dissipation but is linearly coupled to a variable with partial dissipation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the background magnetic field were removed (B* = 0), the enhanced dissipation mechanism would fail and the global stability result would likely break down, since the horizontal-only dissipation is insufficient for the compressible Navier-Stokes part alone without the magnetic coupling.
The restriction to P(ρ) = ρ³/3 is likely technical; the authors state the argument extends to general γ-law pressure, but the cancellation mechanism's exactness may depend on the pressure law's compatibility with the (1+ϱ) weighting.
The four-iteration derivative trading suggests that fewer iterations would not suffice, indicating a genuine algebraic obstruction at finite order rather than a convenience of the method.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper proves global nonlinear stability of a background magnetic field equilibrium for 3D compressible MHD where velocity is dissipated only horizontally and the magnetic field diffuses in one direction. That's a real result — the incompressible analog was settled recently (cited [21,36,37]), but the compressible case is harder because the density equation carries no dissipation at all. Nobody has done this combination before, and the result is natural: strip away the magnetic field and you recover the anisotropic compressible Navier-Stokes stability from [20]. The two mechanisms the authors develop are the right ones. The enhanced dissipation extraction — using the background field e₂ to recover ∂₂b and then horizontal density gradients through the pressure coupling — is clean and well-motivated. The nonlinear cancellation mechanism is the load-bearing piece: the Lorentz nonlinearity generates a term that loses a vertical derivative, and the authors show it cancels exactly against a term produced by testing the velocity equation against (1+ϱ)ϱ ∂₃ᵐu₂. The algebra is shown explicitly in equations (2.7)–(2.9) and carried through in Lemma 3.3. The soft spot is the fourth iteration in this chain. The cancellation produces a residual term ∫(1+ϱ)²ϱ ∂₃ᵐ∂₂ϱ ∂₃ᵘu₂ dx, which the authors handle by substituting the momentum equation again (equation 3.13) to trade vertical density derivatives for magnetic field terms. The estimates J₁–J₆ are each stated as ≲ √(E·D), but the verification for J₂ and J₄ leans on the bound ‖∂ₜu_h‖_{Hᵐ⁻¹} ≲ √D (equation 3.14), which depends on the enhanced dissipation being strong enough at the highest derivative order. The anisotropic Sobolev inequalities (Lemma 3.2) are doing real work here — trading vertical regularity for horizontal dissipation — and whether they suffice at order m is the question a specialist needs to check carefully. Many intermediate estimates are deferred to 'similar' arguments, which is standard for this type of paper but adds verification burden. No red flags, no circularity, no invented structures. The proof is long and intricate but internally consistent. This deserves a serious specialist referee who can verify the cancellation closes at the highest order and that the anisotropic inequalities carry enough horizontal dissipation to compensate for the missing vertical direction. I'd recommend sending it out for review.

Referee Report

1 major / 1 minor

Summary. This paper studies the nonlinear stability of the equilibrium $(ρ,u,B)=(1,0,e_2)$ for the 3D compressible MHD equations in $R^3$ under a strongly anisotropic dissipation regime: the velocity is dissipated only in the horizontal directions ($x_1,x_2$) and the magnetic field is diffused only in the $x_1$ direction. The main result (Theorem 1.1) establishes global-in-time existence and quantitative dissipation estimates for sufficiently small initial perturbations in $H^m$ ($m≥3$). The proof relies on two mechanisms: (1) enhanced dissipation extracted from the background magnetic field $e_2$, which provides control of $∂_2 b$ and $∇_h ϱ$; and (2) a nonlinear cancellation mechanism that resolves the loss of vertical derivatives arising from the compressible coupling, particularly from the Lorentz nonlinearity $b·∇b$ in the momentum equation.

Significance. The result is a genuine contribution to the stability theory of compressible MHD with degenerate dissipation. The anisotropic structure studied here—horizontal velocity dissipation plus one-directional magnetic diffusion—is strictly weaker than full dissipation, and the 3D compressible setting is considerably harder than the incompressible analog due to the absence of density dissipation. The two mechanisms developed (enhanced dissipation from the background field and the nonlinear cancellation for vertical derivative loss) are the central technical contributions and appear to be robust. The paper builds on and extends the authors' prior work on incompressible MHD [21,36,37] and the anisotropic compressible Navier-Stokes result [20]. The proof is carried out in substantial detail across Lemmas 3.3–3.9, with explicit energy functionals and dissipation estimates.

major comments (1)

Section 2, Eqs. (2.7)–(2.9) and Lemma 3.3, Eq. (3.13): The nonlinear cancellation mechanism is the structural linchpin of the paper. The algebraic cancellation of the term $∫ ϱ ∂_3^m u_2 b·∂_3^m ∇b_2 dx$ against the identical term from the velocity equation tested against $(1+ϱ)ϱ ∂_3^m u_2$ is verified. However, this cancellation produces the residual term $∫(1+ϱ)^2 ϱ ∂_3^m ∂_2 ϱ ∂_3^m u_2 dx$ (marked 'Bad term' in Eq. (2.9)), whose control is delegated to Eq. (3.13) in Lemma 3.3. In Eq. (3.13), the momentum equation is substituted again to convert $∂_3^m ∂_2 ϱ$ into time derivatives and magnetic field terms, yielding terms $J_1$–$J_6$. The estimates for $J_2$ and $J_4$ rely on the bound $‖∂_t u_h‖_{H^{m-1}} ≲ √{D(t)}$ (Eq. (3.14)), which depends on the enhanced dissipation structure. The concern is whether this fourth iteration genuinely closes: each substitution trades vertical $ϱ$-der

minor comments (1)

The abstract states the pressure law $P(ρ)=ρ^3/3$ for simplicity, with a remark that the general $γ$-law extends with minor modifications. It would help to briefly indicate which step of the proof (if any) uses the specific form of the pressure.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the proof is a self-contained energy argument with standard bootstrap structure

full rationale

The paper proves nonlinear stability of a 3D compressible MHD system with anisotropic dissipation via a standard bootstrap/continuity argument (Proposition 3.1: assume E+∫D ≤ δ, prove E+∫D ≤ δ/2). The three key mechanisms — enhanced dissipation (Lemmas 3.8–3.9, derived by testing the momentum equation against ∂₂b and ∇ₕϱ using div b = 0), the nonlinear cancellation (eqs. 2.7–2.9, verified algebraically as an identity within the paper), and the residual control (eq. 3.13, handled by further substitution of the momentum equation) — are all derived from the PDE system itself, not from definitions or fits. The self-citations are non-load-bearing: [21] is cited as inspiration for the incompressible analog (a different system), [53] provides standard anisotropic Sobolev inequalities (Lemma 3.2), and [42] provides local well-posedness (a classic reference). The closing estimate d/dt ẽE + κ₁κ₂D ≤ √(E·D) is a genuine consequence of the energy estimates, not a tautology. Whether the residual estimates in (3.13) fully close is a correctness concern, not a circularity concern. The score of 1 reflects the minor self-citation to [21] which, while not load-bearing, provides context for the approach.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper is a pure PDE energy-estimate argument. No free parameters are fitted, no new physical entities are postulated, and no ad hoc axioms are introduced. The background magnetic field e₂ and the anisotropic dissipation structure are part of the problem setup, not inventions. All background results are standard and attributed to the literature.

assumptions (5)

standard math Local-in-time existence and uniqueness for system (1.2) in Sobolev spaces H^m, m≥3
Invoked at the start of Section 3 and Section 4, attributed to the framework of Matsumura-Nishida [42]. Standard for compressible Navier-Stokes-type systems.
standard math Anisotropic Sobolev inequalities (Lemma 3.2)
Used throughout all estimates. Attributed to [53, Lemma 1.2]. These are standard 1D Sobolev embedding applications in each direction.
domain assumption The pressure law P(ρ) = ρ³/3 (γ=3)
Stated in Section 1 for simplicity; the authors note the result extends to general γ-law P(ρ)=ρ^γ/γ with minor modifications. This is a modeling assumption, not a mathematical axiom, but it fixes the structure of the pressure term ∇ϱ in the linearized system.
standard math The equilibrium (ρ*,u*,B*) = (1,0,e₂) is a steady state of (1.1)
Directly verifiable by substitution into (1.1). Used to define the perturbation variables.
standard math div b₀ = 0 implies div b = 0 for all time
Standard for the magnetic induction equation; used pervasively in the cancellation arguments (e.g., Step 2, Section 2).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Nonlinear stability of a background magnetic field for the 3D compressible MHD equations with anisotropic dissipation." pith.science (2026). https://pith.science/paper/X5S5AGH2

@misc{pith2026260706910,
  author       = {Pith},
  title        = {Pith review of: Nonlinear stability of a background magnetic field for the 3D compressible MHD equations with anisotropic dissipation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X5S5AGH2}},
  note         = {Machine review of arXiv:2607.06910}
}

abstract

We study the nonlinear stability of an equilibrium with a background magnetic field for the three-dimensional compressible magnetohydrodynamic (MHD) equations in the whole space $\mathbb{R}^3$, in the strongly anisotropic regime where the velocity is dissipated only in the horizontal directions and the magnetic field is diffused in a single direction. We prove that for initial data sufficiently close to the equilibrium in a Sobolev space, the system admits a unique global-in-time solution that remains close to the equilibrium and enjoys quantitative dissipation estimates. The proof overcomes the severe lack of dissipation through two mechanisms: the background magnetic field is shown to generate enhanced dissipation for the magnetic field and the density, while a nonlinear cancellation mechanism is devised to resolve the loss of vertical derivatives caused by the compressible coupling.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages

[20]

Feng, G.Y

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

work page
[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]

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
[3]

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. 42 Stability of the 3D compressible MHD equations with anisotropic dissipation

work page 1993
[4]

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
[5]

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

work page 1965
[6]

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
[7]

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

Show all 59 references

[8]

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

2014
[9]

Cao and J.H

C.S. Cao and J.H. Wu. Stability of the 3D Navier-Stokes equations with anisotropic dissipation.Nonlin- earity, 38(9): No.095012, 2025

2025
[10]

Chemin, B

J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier. Fluids with anisotropic viscosity.Math. Model. Numer. Anal., 34(2):315–335, 2000

2000
[11]

Chemin and P

J.-Y. Chemin and P. Zhang. On the global wellposedness to the 3-D incompressible anisotropic Navier- Stokes equations.Comm. Math. Phys., 272(2):529–566, 2007

2007
[12]

Chen and D.H

G.Q. Chen and D.H. Wang. Global solutions of nonlinear magnetohydrodynamics with large initial data. J. Differential Equations, 182(2):344–376, 2002

2002
[13]

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

2022
[14]

Davidson.An introduction to magnetohydrodynamics

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

2001
[15]

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

2018
[16]

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, 2019(14):4261–4280, 2017

2019
[17]

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

2015
[18]

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

2014
[19]

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

2017
[21]

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

arXiv
[22]

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. 43 J.C.Gao, X.P.Hu, L.Y.Peng, J.H.Wu

2018
[23]

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

arXiv
[24]

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

2010
[25]

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

2013
[26]

D. Iftimie. A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity.SIAM J. Math. Anal., 33(6):1483–1493, 2002

2002
[27]

R.H. Ji, L. Tian and J.H. Wu. 3D anisotropic Navier-Stokes equations inT 2 ×R: stability and large-time behaviour.Nonlinearity, 36(6):3219–3237, 2023

2023
[28]

R.H. Ji, J.H. Wu and W.R. Yang. Stability and optimal decay for the 3D Navier-Stokes equations with horizontal dissipation.J. Differential Equations, 290:57–77, 2021

2021
[29]

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

2022
[30]

Kawashima

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

1984
[31]

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

2005
[32]

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

2021
[33]

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

2022
[34]

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

2024
[35]

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

2020
[36]

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

2025
[37]

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

2026
[38]

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

2014
[39]

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

2015
[40]

Y.L. Liu, M. Paicu and P. Zhang Global well-posedness of 3-D anisotropic Navier-Stokes system with small unidirectional derivative.Arch. Ration. Mech. Anal., 238(2):805–843, 2020

2020
[41]

M. Paicu. ´Equation periodique de Navier-Stokes sans viscosit´ e dans une direction.Comm. Partial Dif- ferential Equations, 30(7-9):1107–1140, 2005. 44 Stability of the 3D compressible MHD equations with anisotropic dissipation

2005
[42]

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

1980
[43]

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

1983
[44]

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

2018
[45]

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

2014
[46]

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

1983
[47]

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

1985
[48]

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

2018
[49]

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

1984
[50]

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

2017
[51]

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

2017
[52]

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

2015
[53]

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

2021
[54]

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

2022
[55]

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

2023
[56]

Xie, Q.S

Y.W. Xie, Q.S. Jiu and J.T. Liu. Sharp decay estimates and asymptotic stability for incompressible MHD equations without viscosity or magnetic diffusion.Calc. Var. Partial Differential Equations, 63(8):191, 2024

2024
[57]

Xu and P

L. Xu and P. Zhang. Global small solutions to three-dimensional incompressible magnetohydrodynamical system.SIAM J. Math. Anal., 47(1):26–65, 2015

2015
[58]

T. Zhang. Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space.Comm. Math. Phys., 287(1):211–224, 2009

2009
[59]

T. Zhang. Global solutions to the 2D viscous, non-resistive MHD system with large background magnetic field.J. Differential Equations, 260(6):5450–5480, 2016. 45

2016

Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.

[1] [20]

Feng, G.Y

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

work page

[2] [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

[3] [2]

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

[4] [3]

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. 42 Stability of the 3D compressible MHD equations with anisotropic dissipation

work page 1993

[5] [4]

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

[6] [5]

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

work page 1965

[7] [6]

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

[8] [7]

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