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Global uniform regularity and large time behavior of solutions to three dimensional compressible MHD equations with vanishing vertical magnetic resistivity in half space

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Pith's one-line read For small smooth data, the 3D compressible MHD system with vanishing vertical magnetic resistivity remains globally regular with ε-independent bounds, and its solutions converge to the horizontal-diffusion limit at an explicit L² rate of ε^

desk verdict Global uniform regularity and an explicit ε^{1/4} convergence rate for vanishing vertical magnetic resistivity in the half-space: a substantial, plausible paper with a genuine gap in ε-uniform local well-posedness that must be fixed before acceptance. read the letter →

arxiv 2607.19710 v1 pith:ZTJ752GS submitted 2026-07-22 math.AP

Song Gao , Jiahong Wu , Feng Xie This is my paper · ORCID
classification math.AP MSC 35Q3535B4035B6576W05
keywords compressibleMHDvanishingresistivityhalfspaceglobalwell-posednessuniformregularityconormalSobolevspacesdecayratessingularlimit
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

This paper tries to prove that the three-dimensional compressible MHD equations in a half space, with a small vertical resistivity coefficient ε, have global-in-time solutions whose regularity bounds do not blow up as ε→0. If true, this justifies taking the limit: as ε goes to zero, the solutions converge, uniformly for all time, to the solutions of the simpler system in which magnetic diffusion acts only horizontally, and the paper pins the L² distance between the two at order ε^{1/4}. The proof works around the boundary layers that inevitably form because only a limited number of normal derivatives can be controlled uniformly: it uses conormal Sobolev spaces (derivatives tangent to the boundary) plus anisotropic inequalities to close a bootstrap energy estimate. As a byproduct, the paper derives large-time decay rates for the rescaled system and shows those rates are slower than for the limit system, which it attributes to the boundary layers.

What carries the argument

The working space is the conormal Sobolev space H^m_co, generated by the vector fields Z_1=∂1, Z_2=∂2, and Z_3=φ(x3)∂3 with φ(x3)=x3/(x3+1), which are tangent to the boundary; this allows normal control only through φ∂3, reflecting that boundary layers prevent uniform control of full normal derivatives. The energy bootstrap combines conormal energy estimates, first-order time and normal derivative estimates, a Stokes-system estimate for dissipation of (a,v), and second-order normal estimates for (v,εB), closed by anisotropic Sobolev and Hardy-type inequalities. Decay is obtained by writing solutions via Duhamel's formula with the linearized semigroup, using the explicit heat-kernel represent

What would settle it

A concrete check: take m=4, choose any smooth compactly supported initial datum satisfying (1.10) and the as-yet-unstated compatibility conditions asserted in Remark 1.2, and test whether the predicted a priori bound (2.40) actually closes. If one can exhibit such data for which no ε-uniform local solution of (1.6) exists—or for which the constant in (2.40) blows up as ε→0—the global uniform regularity claim collapses.

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Extended reading notes

Core claim

The central result is Theorem 1.1: for m≥4, if the initial perturbation is small in the conormal Sobolev norms, the system has a unique global solution satisfying E^ε_m(t)+D^ε_m(t) ≲ δ0, with constants independent of ε. From this uniform bound and a semigroup decay analysis, the paper obtains that (a,m) decays like t^{-3/4} in H^1, B like t^{-1/2}, ∂_h B like t^{-1}, and ∂_3 B like t^{-1/2+δ2}, with the lower bound t^{-3/4} when the mean of a0 is nonzero. Combining regularity, decay, and a carefully crafted energy for the difference between the ε-system and the limit system—using an auxiliary function g(ρ) to handle the pressure term—the paper proves sup_{τ≤t} ||(ρ^ε−ρ^0, v^ε−v^0, B^ε−B^0)(τ

Load-bearing premise

The most load-bearing premise is the unstated ε-uniform local-in-time well-posedness of system (1.6): Section 2.8 refers to prior works for its proof, and Remark 1.2 does not spell out the required compatibility conditions; if that local existence result fails for this boundary-value problem, the bootstrap has no starting point.

Editorial extensions

If this is right

  • If the uniform estimate (1.11) holds, no strong boundary layer forms in the vanishing-resistivity limit; the ε-system stays close to the horizontal-diffusion MHD system for all time.
  • The strong convergence in local conormal space and the ε^{1/4} rate make the limit process quantitative: simulations with tiny vertical resistivity should match the limit system to order ε^{1/4} in L² uniformly in time.
  • The decay rates imply that even with vanishing vertical resistivity the magnetic field's normal derivative decays only like t^{-1/2+δ2}, slower than the limit system's t^{-1/2}; this is a measurable prediction about long-time behavior in the half-space geometry.
  • The lower bound t^{-3/4} for the density-momentum pair shows the decay rates for (a,m) are optimal, not an artifact of the estimates.

Reading between the lines

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

  • Editorial extension: the ε^{1/4} exponent likely comes from the loss in the ∂3B̄ estimate; if future work controls one more normal derivative uniformly, the rate could plausibly improve to ε^{1/2}, matching the natural diffusive scaling.
  • Editorial extension: the same conormal-energy-plus-semigroup template should apply to other singular limits with partial dissipation—for example vanishing horizontal viscosity or resistivity with different boundary conditions—where boundary layers again limit normal regularity.
  • Editorial extension: the auxiliary pressure function g(ρ) trick for handling div v⁰ without L∞ decay is portable: it suggests a general recipe for difference estimates in compressible singular limits when the background velocity has only L²-type decay.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the 3D compressible MHD system in the half-space with vertical magnetic resistivity ε, no-slip velocity, and perfectly conducting magnetic boundary conditions. It claims: (i) global-in-time uniform energy estimates in conormal Sobolev spaces, independent of ε, for small data (Theorem 1.1); (ii) large-time decay rates of solutions in H^1 (Theorem 1.5); and (iii) an explicit time-uniform L^2 convergence rate O(ε^{1/4}) toward the limit system with ε=0 (Theorem 1.11). The proof of Theorem 1.1 is based on a bootstrap argument that chains conormal energy estimates, estimates for time and normal derivatives, and Stokes-type elliptic estimates; §§2.1–2.6 provide the component lemmas, and §2.7 estimates the nonlinear terms L_m and N_m. The decay results in §3 use semigroup estimates for the linearized Navier–Stokes operator from [17,18] and a new heat-kernel representation for the magnetic equation (Lemma 3.5). The convergence rate in §4 uses a difference system, a modified energy with auxiliary functions for the pressure term, and the decay estimates from Theorems 1.5 and 1.9.

Significance. If the results hold, this would be a substantial advance: it provides the first global uniform regularity for the vanishing vertical resistivity MHD limit in a half-space with a boundary, and it gives a quantitative convergence rate. The paper contains genuinely nontrivial elements: the use of conormal spaces to avoid boundary-layer singularities, anisotropic Sobolev inequalities adapted to the partial dissipation, and a careful heat-kernel analysis (Lemma 3.5) that yields decay rates independent of ε. The bootstrap architecture (2.14)–(2.40) is standard in structure but involves many technical estimates. The paper is candid about the slower decay caused by weak boundary layers. However, as detailed below, the foundational local well-posedness step is only invoked by analogy to Navier–Stokes problems and is not proved for the actual MHD system; this is a load-bearing gap that prevents the current version from being fully verified.

major comments (3)
  1. [§2.8] The proof of Theorem 1.1 requires ε-uniform local-in-time well-posedness for the quasilinear MHD system (1.6) with boundary conditions (v, ∂_3 B_h, B_3)=0. The paper states this 'in a similar fashion as in [28] and [32]', but [28] and [32] concern compressible Navier–Stokes equations, not the coupled MHD system with anisotropic resistivity and mixed Neumann/Dirichlet conditions for B. The 'appropriate compatibility conditions' are not stated; Remark 1.2 only says they are analogous to [18,29]. This is not cosmetic: the bootstrap starting time and the initial data for Lemma 2.2 require well-defined initial time-derivatives satisfying the boundary conditions. The uniform-in-ε aspect is also nontrivial because the homogeneous Neumann condition ∂_3 B_h=0 does not appear in the ε→0 limit, so the validity of a uniform Lopatinski condition must be checked. Without an explicit local existence th
  2. [§2.7.3, Eq. (2.33)] The estimate for Σ_{|α|≤m−2} ∫_0^t |B_α| dτ, involving the time-derivative nonlinearities, is asserted by saying 'By a similar argument as before, we derive ... and omit the details for brevity.' This term is essential for the proof of Lemma 2.9(2.16), which in turn is needed to close the bootstrap (2.40). The derivatives counts and the treatment of products with ∂_τ (a^ε, v^ε, B^ε) at the top conormal order are not shown. Since this is a load-bearing nonlinear estimate, the proof should be provided or at least a careful derivative-count argument should be given. A reader cannot currently verify this step from the presented material.
  3. [§4, Eq. (4.13)] The Gronwall argument for the convergence rate relies on the claim ∫_0^t (C(τ)+K(τ)) dτ ≤ C uniformly in ε and t. The proof only checks two representative terms in C(τ) (∥B^ε∥^2_{L∞} and ∥∂B^0∥_{L2} ∥∂_h ∂B^0∥_{L2}) and says the remaining terms are handled similarly. However, C(τ) includes ∥∂_t v^ε∥^2_{H^1}, ∥∂v^ε∥^2_3, ∥∂_a^ε∥^2_2, and ∥B^ε∥^2_{L∞}; each must be uniformly integrable using the Theorems 1.1 and 1.5. The omitted 'similar' terms are not all immediate (e.g., ∥∂_t v^ε∥_{H^1} requires the time-derivative estimates from §2), so the uniform bound (4.13) is not fully established as written.
minor comments (4)
  1. [§2.2, Lemma 2.2] The proof of Lemma 2.2 is omitted with the statement 'The proof is similar as that of Lemma 2.1 and is thus omitted.' A few lines indicating the changes (e.g., the extra boundary terms for ∂_t B^ε) would help the reader.
  2. [§2.7.2, eBα decomposition] In the paragraph following Eq. (2.19), the decomposition eBα = eBα,1 + eBα,2 + eBα,3 and the subsequent estimates appear to mix the roles of v and B: the text says 'For eBα,2' but then estimates contain ε∂_t B^ε terms, while eBα,3 contains ∂_t v^ε terms. Please check the labeling or the displayed definitions to avoid confusion.
  3. [§3.1, Eq. (3.8)] The Minkowski-type inequality (3.7) is stated with 1≤q≤p, and the derivation of (3.8) uses this in the vertical variable for p_3=2 and q=2. This is fine, but the intermediate steps use nested norms and would benefit from a displayed justification of the final exponent.
  4. [Notation, p. 5] The notation ∥·∥_{L^r_{x_3} L^q_{x_2} L^p_{x_1}} appears corrupted in the printed text ('\r\r\r\r\r...'), and the intended order is unclear. Please correct this in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the uniform regularity, decay, and convergence-rate results are derived by energy estimates, Duhamel formulas, and Gronwall arguments; self-citations are supporting only.

full rationale

The paper's derivation chain is not circular. The main uniform regularity estimate (Theorem 1.1) is obtained from a priori energy estimates on the PDE (Lemmas 2.1–2.7), combined with anisotropic Sobolev and commutator estimates (Lemma 2.9). No parameter is fitted to the target quantity: the smallness constants are used only to close standard bootstrap inequalities such as (2.40). The decay results in Section 3 use Duhamel's principle with semigroup estimates from Kagei–Kobayashi [17,18] and with magnetic heat-kernel estimates proved from explicit representations in Lemma 3.5; the bound (3.12) is a usual bootstrap closing M(t) in terms of M(t)^2, which is legitimate under smallness. The convergence-rate proof in Section 4 derives the difference system (4.1) from the original conservative equations, uses an algebraic pressure identity (1.19), and closes with Gronwall's inequality using zero initial difference, so the final ε^{1/4} bound is not a restatement of any input. The self-citations to [10] (commutator properties) and [12] (limit system decay) are supporting references; the commutator properties are stated explicitly, and [12] is a published prior result whose statements are reproduced in Theorems 1.8–1.9. The unproved ε-uniform local well-posedness referenced in §2.8 to [28,32] is an omitted proof / correctness risk, but it is not a circularity: the a priori estimate would still be an independent derivation if local existence were supplied. No fitted input is renamed as a prediction, no uniqueness theorem is imported to force a choice, and no ansatz is smuggled in via self-citation in a way that makes the conclusion equivalent to its assumptions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No numerical parameters are fitted to data; all smallness constants (δ_0, δ_1, δ_*, δ_2∈(0,1/32), δ_4, δ_5) are existential and absorb terms in the estimates. The proof depends on imported semigroup results [17,18], the limit-system results [12], and the unstated compatibility conditions.

assumptions (4)
  • standard math Anisotropic Sobolev inequalities (A.1)–(A.9) hold in R^3_+ with the stated conormal norms.
    Used throughout Sections 2–4 to bound nonlinear terms; cited to [10], [25], [37] without proof in this paper.
  • domain assumption Semigroup decay estimates for the linearized compressible Navier–Stokes operator in the half-space (Lemmas 3.1–3.4) hold.
    Imported without proof from Kagei–Kobayashi [17,18]; used in the Duhamel arguments of Section 3.
  • domain assumption The limit system (1.4)–(1.5) has global solutions and decay rates as stated in Theorems 1.8–1.9, proven in Gao–Xie [12].
    Section 4 uses the limit-system decay ∥∂_3B^0∥=O(t^{-1/2}) and the global well-posedness as black boxes in the convergence-rate proof.
  • ad hoc to paper The initial data satisfy the 'appropriate compatibility conditions' from [18,29] that guarantee local well-posedness for system (1.6) with boundary (1.2).
    Remark 1.2 defers to [18,29] without stating the conditions; Section 2.8 uses them to start the bootstrap. This is a load-bearing but unverified premise.

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Pith. "Pith review of Global uniform regularity and large time behavior of solutions to three dimensional compressible MHD equations with vanishing vertical magnetic resistivity in half space." pith.science (2026). https://pith.science/paper/ZTJ752GS

@misc{pith2026260719710,
  author       = {Pith},
  title        = {Pith review of: Global uniform regularity and large time behavior of solutions to three dimensional compressible MHD equations with vanishing vertical magnetic resistivity in half space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZTJ752GS}},
  note         = {Machine review of arXiv:2607.19710}
}
abstract

This paper aims to establish the global regularity and large time behavior of solutions to the three-dimensional (3D) compressible magnetohydrodynamics (MHD) equations with vanishing vertical magnetic resistivity in the upper half-space with no-slip boundary condition on velocity and perfectly conducting boundary condition on magnetic field. By exploiting anisotropic Sobolev inequalities and elaborated estimates, we are able to achieve global-in-time uniform regularity estimates of solutions, which are independent of small vertical resistivity coefficient $\varepsilon$. These uniform global regularity estimates allow us to pass to the limit as \(\varepsilon \to 0\) and obtain the convergence to the corresponding MHD system without vertical magnetic resistivity globally in time. Moreover, the $H^{1}\left(\mathbb{R}_{+}^{3}\right)$ decay rates of solutions to the original system are also derived based on the detailed analysis on semigroup of the related linearized operator in half space and the uniform energy estimates achieved. In contrast to the decay estimates for the limit system obtained by Gao and Xie \cite{JDE}, the present decay estimate exhibits a slower rate, attributable to the absence of higher-order normal derivative estimates, which results from the occurrence of boundary layers. Finally, we combine both sets of regularity and decay results to rigorously prove an explicit time-uniform $L^2$ convergence rate of order $\varepsilon^{\frac{1}{4}}$ for the vanishing vertical magnetic resistivity limit process.

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Works this paper leans on

41 extracted references · 3 linked inside Pith

  1. [12]

    Gao and F

    S. Gao and F. Xie, Global well-posedness and decay rates of solutions to compressible viscous MHD equations with horizontal magnetic diffusion inR 3 +, J. Differ. Equ. 475 (2026) 114511

    2026

  2. [28]

    Matsumura and T

    A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of general fluids, in Computing Methods in Applied Sciences and Engineering, V (R. Glowinski, F. L. Lions, eds.), North-Holland Publ. Comp., Amsterdam, 1982, pp. 389–406

    1982

  3. [32]

    Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ

    A. Tani, On the first initial-boundary value problem of compressible viscous fluid motion, Publ. RIMS Kyoto Univ. 13 (1977) 193–253

    1977

  4. [1]

    H. W. Alt, Lineare Funktionalanalysis, Berlin Heidelberg New York, Springer, 1985

    1985

  5. [2]

    Biskamp, Nonlinear Magnetohydrodynamics, vol

    D. Biskamp, Nonlinear Magnetohydrodynamics, vol. 1 of Cambridge Monographs on Plasma Physics, Cambridge, Cambridge University Press, 1993

    1993

  6. [3]

    S. I. Braginskii, Transport processes in a plasma, in M. A. Leontovich (ed.), Reviews of Plasma Physics, vol. 1, pp. 205-311, New York, Consultants Bureau, 1965

    1965

  7. [4]

    Chen and D

    G. Chen and D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys. 54 (2003) 608-632

    2003

  8. [5]

    X. Cui, S. Li and F. Xie, Uniform regularity estimates and inviscid limit for the compressible non-resistive magneto- hydrodynamics system, Nonlinearity 36 (2023) 354–400

    2023

Show all 41 references
  1. [6]

    X. Cui, S. Li and F. Xie, Vanishing viscosity limit for compressible magnetohydrodynamic equations with transverse background magnetic field, Commun. Math. Sci. 21 (2023) 1363–1392

    2023

  2. [7]

    P. A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge Texts in Applied Mathematics, Cambridge, Cambridge University Press, 2001

    2001

  3. [8]

    B. Dong, J. Wu and X. Zhai, Global small solutions to a special 2 1 2 -D compressible viscous non-resistive MHD system, J. Nonlinear Sci. 33 (2023) 21

    2023

  4. [9]

    G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, 2nd ed., New York, Springer-Verlag, 2011

    2011

  5. [10]

    J. Gao, J. Wu, Z. Yao and X. Yin, Global uniform regularity for the 3D incompressible MHD equations with slip boundary condition near an equilibrium, (2024), arXiv:2408.14123

    2024 arXiv

  6. [11]

    J. Gao, L. Peng, J. Wu and Z. Yao, Global uniform regularity for the 3D incompressible MHD equations with slip boundary condition near a background magnetic field, (2025), arXiv:2508.09609

    2025 arXiv

  7. [13]

    X. Gu, D. Wang and F. Xie, Vanishing viscosity limit of compressible viscoelastic equations in half space, J. Differ. Equ. 398 (2024) 319–343. 64

    2024

  8. [14]

    G. Hong, X. Hou, H. Peng and C. Zhu, Global existence for a class of large solutions to three-dimensional compressible magnetohydrodynamic equations with vacuum, SIAM J. Math. Anal. 49 (2017) 2409-2441

    2017

  9. [15]

    Hu and D

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

    2010

  10. [16]

    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 (2011) 145–175

    2011

  11. [17]

    Kagei and T

    Y. Kagei and T. Kobayashi, On large time behavior of solutions to the compressible Navier-Stokes equations in the half space inR 3, Arch. Ration. Mech. Anal. 165 (2002) 89-159

    2002

  12. [18]

    Kagei and T

    Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal. 177 (2005) 231-330

    2005

  13. [19]

    Kato, Nonstationary flows of viscous and ideal fluids inR 3, J

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

    1972

  14. [20]

    R. M. Kulsrud, Plasma Physics for Astrophysics, Princeton, NJ, Princeton University Press, 2005

    2005

  15. [21]

    H. Li, X. Xu and J. Zhang, Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum, SIAM J. Math. Anal. 45 (2013) 1356-1387

    2013

  16. [22]

    Li, Global well-posedness for the three-dimensional full compressible viscous non-resistive MHD system, J

    Y. Li, Global well-posedness for the three-dimensional full compressible viscous non-resistive MHD system, J. Math. Fluid Mech. 24 (2022) 1-24

    2022

  17. [23]

    Y. Li, H. Xu and X. Zhai, Global smooth solutions to the 3D compressible viscous non-isentropic magnetohydrodynamic flows without magnetic diffusion, J. Geom. Anal. 33 (2023) 246

    2023

  18. [24]

    Lieb and M

    E. Lieb and M. Loss, Analysis, American Mathematical Society, Providence, 2001

    2001

  19. [25]

    H. Lin, J. 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 (2025) 26

    2025

  20. [26]

    Masmoudi, Remarks about the inviscid limit of the Navier-Stokes system, Comm

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

    2007

  21. [27]

    Masmoudi and F

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

    2012

  22. [29]

    Matsumura and T

    A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys. 89 (1983) 445-464

    1983

  23. [30]

    Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J

    J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure, SIAM J. Math. Anal. 21 (1990) 1093-1117

    1990

  24. [31]

    Tan and Y

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

    2018

  25. [33]

    Wang and F

    D. Wang and F. Xie, Inviscid limit of compressible viscoelastic equations with the no-slip boundary condition, J. Differ. Equ. 353 (2023) 63–113

    2023

  26. [34]

    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 (2015) 4123–4191

    2015

  27. [35]

    G. Wu, Y. Zhang and W. Zou, Optimal time-decay rates for the 3D compressible magnetohydrodynamic flows with discontinuous initial data and large oscillations, J. London Math. Soc. 103 (2021) 817-845

    2021

  28. [36]

    Wu and Y

    J. Wu and Y. Wu, Global small solutions to the compressible 2D magnetohydrodynamic system without magnetic diffusion, Adv. Math. 310 (2017) 759-888

    2017

  29. [37]

    Wu and Y

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

    2021

  30. [38]

    Wu and Y

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

    2022

  31. [39]

    Wu and X

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

    2023

  32. [40]

    Wu and X

    J. Wu and X. Zhai, Global solutions to 3D compressible MHD equations with partial magnetic diffusion, (2025), arXiv:2505.04351

    2025 arXiv

  33. [41]

    Y. Xiao, Z. Xin and J. Wu, Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition, J. Funct. Anal. 257 (2009) 3375-3394. 65

    2009

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