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Magnetic Stabilization of Compressible Flows: Global Existence in 3D Inviscid Non-Isentropic MHD Equations

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Pith's one-line read A background magnetic field satisfying a Diophantine condition prevents finite-time blowup and forces algebraic decay for small smooth perturbations of 3D inviscid, heat-conductive, compressible MHD equations.

desk verdict A real new mechanism and a plausible first-in-class theorem, but the bootstrap does not close as written: the proof needs a fix before the result is trustworthy. read the letter →

arxiv 2507.00888 v2 pith:IA5MLF3C submitted 2025-07-01 math.AP

Jiahong Wu , Fuyi Xu , Xiaoping Zhai This is my paper · ORCID
classification math.AP MSC 35Q3576N1076W05
keywords inviscidcompressibleMHDmagneticstabilizationDiophantineconditionglobalexistencealgebraicdecaywavestructurefinite-timeblowup
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 establishes that a suitable background magnetic field prevents finite-time singularity formation in the 3D inviscid, heat-conductive, compressible MHD equations on the torus. Near the equilibrium $(1,0,1,n)$ with $n$ satisfying a Diophantine irrationality condition, every sufficiently small smooth perturbation is shown to have a unique global smooth solution that decays algebraically back to equilibrium. The result matters because the same equations without the magnetic coupling—the compressible Euler equations—are known to develop shocks and cusps from smooth data; this work supplies a mathematical mechanism by which the magnetic field stabilizes an otherwise unstable inviscid compressible flow.

What carries the argument

The argument rests on two coupled mechanisms. First, the background field $n$ is required to satisfy the Diophantine condition $|n\cdot k| \ge c/|k|^r$ for all nonzero integer $k$, which yields the Sobolev inequality $\|f\|_{H^s} \le C\|n\cdot\nabla f\|_{H^{s+r}}$ (Lemma 2.4); this converts directional control along $n$ into full Sobolev control of $u$. Second, the linearized perturbation system hides three wave structures: the pair $(a, \operatorname{div}u)$ satisfies an acoustic wave equation, the pair $(u,B)$ satisfies degenerate wave equations whose $-(n\cdot\nabla)^2$ term supplies dissipation along the background field, and the pair $(\operatorname{div}u,\theta)$ satisfies wave equations whose interaction yields dissipation of $\operatorname{div}u$. These estimates are assembled into a Lyapunov functional $\mathcal{E}(t)$ obeying $d\mathcal{E}/dt + c\mathcal{E}^{4/3} \le 0$, which forces the algebraic decay stated in Theorem 1.1.

What would settle it

Run a high-resolution numerical simulation of the perturbation system (1.6) on the 3-torus with a Diophantine background field (for instance, $n=(1,\sqrt{2},\sqrt{3})$ normalized) and smooth initial data satisfying (1.7)-(1.8) with $H^N$ norm below the theorem's $\varepsilon$. Theorem 1.1 predicts the solution stays smooth and the $H^{r+4}$ norm decays like $(1+t)^{-3/2}$; observing finite-time blowup or a halt in decay would refute the claim.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for any $N \ge 4r+7$ with $r>2$, if the initial perturbation $(a_0,u_0,\theta_0,B_0)$ lies in $H^N$, respects the constraints (1.7)-(1.8), and has $H^N$ norm below a small $\varepsilon$, then the perturbation system (1.6) admits a unique global solution in $C([0,\infty);H^N)$. Moreover, for every $\beta$ with $r+4 \le \beta < N$, the $H^\beta$ norm decays at the rate $C(1+t)^{-3(N-\beta)/(2(N-r-4))}$. In the authors' words, this rules out finite-time blowup and confirms the stabilizing phenomenon seen in experiments with electrically conducting fluids.

Load-bearing premise

The load-bearing assumption is that the background magnetic field $n$ is sufficiently irrational (Diophantine), so no Fourier mode is exactly silent along the field; if $n$ has rational components, the key inequality $\|u\|_{H^s} \le C\|n\cdot\nabla u\|_{H^{s+r}}$ fails and the proof gives no stabilization.

Editorial extensions

If this is right

  • Global smooth solutions exist for all time near a Diophantine background field, so the inviscid MHD model does not inherit the finite-time shock formation of the compressible Euler equations for these data.
  • The solution decays algebraically to the equilibrium $(1,0,1,n)$, with the fastest decay in the highest regularity.
  • The proof identifies the stabilizing mechanism: wave coupling between $u$ and $B$ turns the background field into directional smoothing $-(n\cdot\nabla)^2 u$, and coupling between $\operatorname{div}u$ and $\theta$ supplies dissipation on $\operatorname{div}u$, compensating for the lack of viscosity in the velocity equation.
  • Per Remark 1.3, the same approach yields a 2D analogue, and in 2D even non-Diophantine backgrounds can be handled under symmetry conditions.
  • Per Remark 1.2, the isentropic version (no temperature equation) is not covered because the $\operatorname{div}u$ dissipation relies on the $\theta$ equation.

Reading between the lines

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

  • Because the Diophantine condition holds for almost every vector $n$, the theorem covers essentially all background fields; the excluded rational fields are exactly those with exact resonances where $n\cdot k=0$, so a plausible extension of the paper's logic is that rational backgrounds may allow genuine blowup or slower decay—an untested prediction.
  • The same wave-structure mechanism may generalize to other inviscid systems with a background vector field and a dissipative partner equation (e.g., rotating or stratified flows), provided an analogous coupling to a diffusion term exists; the paper does not discuss these cases.
  • A direct numerical check comparing a rational background like $n=(1,0,0)$ with a Diophantine background at the same small-amplitude data would test whether the Diophantine condition is a proof artifact or a physical threshold for stabilization.
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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 inviscid non-isentropic compressible MHD system with heat conduction and magnetic diffusion on the torus, in a small H^N neighborhood of the equilibrium (rho, u, theta, b) = (1,0,1,n), where the background magnetic field n satisfies the Diophantine condition (1.3). The perturbation system (1.6) is analyzed through high-order energy estimates, a generalized Poincare inequality for theta, and wave-structure estimates that produce dissipation for nabla a, n·nabla u, and div u. Theorem 1.1 claims global existence and algebraic decay of the perturbation for N >= 4r+7 whenever the initial data are small in H^N and satisfy the constraints (1.7)-(1.8). The proof is self-contained and does not rely on fitted parameters or numerical inputs.

Significance. If the proof is correct, the result is substantial: it would rigorously confirm the magnetic stabilization phenomenon for a 3D inviscid compressible MHD model, in contrast to the finite-time singularity results for the compressible Euler equations. The paper contains a serious and largely explicit technical apparatus: uniform L^2 bounds, high-order energy estimates, wave-structure propositions, and a generalized Poincare inequality for theta. It also states a precise Diophantine condition and an explicit decay rate, and the authors advertise no free parameters or numerical fitting. The main reservation is the bootstrap closure in Section 8, which contains a coefficient gap that is load-bearing for the global existence claim. The result is therefore plausible but not established as written.

major comments (3)
  1. [Section 8, Eqs. (8.3)-(8.5)] The bootstrap closure does not follow from the displayed inequalities. In (8.3), the term C||div u||^2_{H^{r+3}} has a coefficient that is not small: it comes from the identity ||Lambda^s div u||^2 in (5.4) plus the f1-estimate (5.5), so the coefficient is 1+O(delta^2), not epsilon. Proposition 7.1, even if its proof is fully accepted, produces ||div u||^2_{H^{r+3}} on the left of (8.4) but places no small multiple of that same quantity on the right that could absorb the C||div u||^2 term from (8.3). Adding (8.3) and (8.4) therefore leaves an unabsorbed positive constant multiple of ||div u||^2_{H^{r+3}} in (8.5). A large weight on (8.4) before summation could in principle repair this, but that weight changes the Lyapunov functional and the cross-terms in (8.8), and the subsequent closure (8.9)-(8.15) is not re-verified. As written, the strict bootstrap improvement (8.1) and the decay estimate (8.16) are not established.
  2. [Section 8, Eq. (8.12)] The displayed estimate in (8.12) is unsupported. The first inequality bounds gamma ||Delta theta||_{L^infty} ||u||^2_{H^{r+4}} by C gamma ||nabla a||_{H^{r+3}} ||u||^2_{H^{r+4}}, which is not an admissible control of Delta theta; the second inequality then uses ||u||^4_{H^{r+4}} in place of ||u||^2_{H^{r+4}}. If the intended replacement is the analogous estimate with ||theta||_{H^{r+5}}, that term is not present in the dissipation D(t) defined after (8.12), so the absorption into (8.13) needs a separate argument. Since (8.12) is one of the inequalities used to pass from (8.8) to the differential inequality (8.13), this is not a harmless typo in the closure.
  3. [Section 8, definition of E and (8.8)] The quadratic term in the Lyapunov functional is written with (Lambda^s a)^2 in (8.8) and in the definition of E(t), while the preceding estimate (8.7) uses (Lambda^{r+4} a)^2. If the intent is to use the r+4 norm, the notation should be Lambda^{r+4} a; if the intent is s <= r+4, the relation between E(t) and D(t) in (8.15) needs to account for the missing high-order piece. This is a localized consistency issue, but it sits in the final bootstrap and should be corrected.
minor comments (4)
  1. [Section 1, paragraph on Wang-Xin] The phrase 'they approach does not apply' should read 'their approach does not apply'.
  2. [Section 7, beginning of proof] The proof of Proposition 7.1 starts with the equation partial_t theta - Delta theta + div u = f3, but the original system (1.6) contains kappa Delta theta. Either set kappa = 1 explicitly at the start of the paper or carry kappa through the estimates.
  3. [Section 8, after (8.15)] The phrase 'Laputa-type inequality' appears to be a corruption of 'Gronwall-type inequality' or 'differential inequality'; the displayed inequality dE/dt + c E^{4/3} <= 0 should be named clearly.
  4. [Section 8, Eq. (8.8)] The arguments of (Lambda^s a)^2 in the integral term should be reconciled with the r+4 notation used in (8.7); see the third major comment.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the global-existence theorem is derived from the stated PDE system, the Diophantine condition, and self-contained estimates; self-citations are contextual only.

full rationale

The paper's central claim, Theorem 1.1, is a genuine existence-stability result for the perturbation system (1.6) near a Diophantine background magnetic field. The derivation does not reduce to a fit, a renamed known result, or an assumed conclusion. The Diophantine Sobolev inequality of Lemma 2.4 is proved inline via Plancherel and the lower bound |n·k| ≥ c|k|^{-r}; Lemma 2.5 is an elementary perturbation argument. Propositions 5.1, 6.1 and 7.1 are derived from the linearized wave structures of (1.6) and are not imported from prior work. The self-citations [42], [43] and [44] appear in the introduction as background and comparison, not as inputs to the bootstrap or to the energy estimates. No parameter is fitted to data, and no 'prediction' is definitionally equal to an input. The final bootstrap in Section 8 does contain a possible technical gap noted by the reader: adding (8.3) and (8.4) leaves a fixed-coefficient C∥divu∥²_{H^{r+3}} on the right, and the text's 'choosing ε, δ small enough' does not by itself absorb a constant that is not premultiplied by a small parameter. That is a correctness concern about the closure of the bootstrap, not a circularity: the desired inequality (8.1) is not assumed, and no step in the proof is equivalent to its own input by construction. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The theorem relies on no fitted constants and introduces no new entities. The main inputs from outside the paper are standard PDE estimates, the ideal gas law, positive heat conduction and resistivity, and the Diophantine background-field condition. The most consequential assumption is the Diophantine condition, because it is exactly what converts directional magnetic smoothing into full Sobolev control of the velocity field.

assumptions (6)
  • domain assumption The background magnetic field n satisfies the Diophantine condition (1.3) with r greater than 2.
    This is the key structural hypothesis for Lemmas 2.4 and 2.5, which let the proof replace H^s norms with H^{s+r} norms of n dot grad f. Without it, the directional smoothing estimates in Sections 6 and 8 fail.
  • domain assumption Positive heat conduction coefficient kappa and positive magnetic diffusivity sigma, with no velocity viscosity.
    Dissipation in the theta and B equations drives the wave structures that produce decay of divu and n dot grad u. The absence of velocity viscosity is the main difficulty the paper overcomes.
  • domain assumption The pressure law is the ideal gas law P = R rho theta, or the generalized form pi0(rho) plus theta pi1(rho) under constraints.
    The perturbed system (1.5)-(1.6) is written for this constitutive law, and the cancellation structure in the L2 estimates uses it directly.
  • standard math Standard product, commutator, and composition estimates from Kato [28] and Triebel [39], as quoted in Lemmas 2.1-2.3.
    These are the basic analytic tools used repeatedly in Sections 4-7 to bound nonlinear terms in Sobolev spaces.
  • standard math Weighted Poincare inequalities from Desvillettes-Villani [14] and Feireisl [16], used to prove the generalized Poincare inequality for theta in Lemma 3.1.
    Without these, the L2 bound on theta would require a zero-mean condition that the temperature perturbation does not enjoy.
  • domain assumption The asserted local well-posedness of (1.6) in H^N via a contraction mapping argument, attributed to Majda-Bertozzi [33].
    Section 8 starts the bootstrap from a unique local solution. The paper does not prove this local theory, and the cited reference is not specific to heat-conductive resistive inviscid MHD.

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Pith. "Pith review of Magnetic Stabilization of Compressible Flows: Global Existence in 3D Inviscid Non-Isentropic MHD Equations." pith.science (2026). https://pith.science/paper/IA5MLF3C

@misc{pith2026250700888,
  author       = {Pith},
  title        = {Pith review of: Magnetic Stabilization of Compressible Flows: Global Existence in 3D Inviscid Non-Isentropic MHD Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IA5MLF3C}},
  note         = {Machine review of arXiv:2507.00888}
}
abstract

Solutions to the compressible Euler equations in all dimensions have been shown to develop finite-time singularities from smooth initial data such as shocks and cusps. There is an extraordinary list of results on this subject. When the inviscid compressible flow is coupled with the magnetic field in the 3D inviscid non-isentropic compressible magnetohydrodynamic (MHD) equations in $\mathbb{T}^3$, this paper rules out finite-time blowup and establishes the global existence of smooth and stable solutions near a suitable background magnetic field. This result rigorously confirms the stabilizing phenomenon observed in physical experiments involving electrically conducting fluids.

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Works this paper leans on

46 extracted references · 44 canonical work pages

  1. [1]

    Alemany, R

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

    work page 1979

  2. [2]

    Alexakis, Two-dimensional behavior of three-dimensional magnetohydrodynamic flow with a strong guiding field, Phys

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

    work page 2011

  3. [3]

    Alfv ´en, Existence of electromagnetic-hydrodynamic waves, Nature 150 (1942), 405–406

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

    work page 1942

  4. [4]

    Buckmaster, S

    T. Buckmaster, S. Shkoller and V . Vicol, Formation of shocks for 2D isentropic compressible Euler, Commun. Pure Appl. Math. 75 (2022), 2069–2120

    work page 2022

  5. [5]

    Buckmaster, S

    T. Buckmaster, S. Shkoller and V . Vicol, Shock formation and vorticity creation for 3d Euler, Commun. Pure Appl. Math. 76 (2023), 1965–2072

    work page 2023

  6. [6]

    Buckmaster, S

    T. Buckmaster, S. Shkoller and V . Vicol, Formation and development of singularities for the compressible Euler equations, EMS Press. DOI 10.4171/ICM2022/210. Proceedings of the International Congress of Mathemati- cians 2022

    work page doi:10.4171/icm2022/210

  7. [7]

    J. Cassels. An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge, 1957

    work page 1957

  8. [8]

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

    work page 2022

Show all 46 references
  1. [9]

    Christodoulou, The formation of shocks in 3-dimensional fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zurich, 2007

    D. Christodoulou, The formation of shocks in 3-dimensional fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zurich, 2007

    2007

  2. [10]

    Christodoulou, The shock development problem, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zurich, 2019

    D. Christodoulou, The shock development problem, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zurich, 2019

    2019

  3. [11]

    Davidson, Magnetic damping of jets and vortices, J

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

    1995

  4. [12]

    Davidson, The role of angular momentum in the magnetic damping of turbulence,J

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

    1997

  5. [13]

    Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, England, 2001

    P.A. Davidson, An Introduction to Magnetohydrodynamics, Cambridge University Press, Cambridge, England, 2001

    2001

  6. [14]

    Desvillettes and C

    L. Desvillettes and C. Villani. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159 (2005), 245–316

    2005

  7. [15]

    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 Science 33 (2023), Article number 21

    2023

  8. [16]

    Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004

    E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004

    2004

  9. [17]

    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 (2009), 085107

    2009

  10. [18]

    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 (2015), 154–177

    2015

  11. [19]

    Goldstein, D.A

    M.L. Goldstein, D.A. Roberts and W.H. Matthaeus, Magnetohydrodynamic turbulence in the solar wind, Annu. Rev. Astron. Astrophys.33 (1995), 283–326

    1995

  12. [20]

    Guo, Smooth irrotational flows in the large to the Euler-Poisson system in R3+1, Comm

    Y . Guo, Smooth irrotational flows in the large to the Euler-Poisson system in R3+1, Comm. Math. Phys. 195 (1998), 249–265

    1998

  13. [21]

    Y . Guo, A. Ionescu and B. Pausader, Global solutions of the Euler-Maxwell two-fluid system in 3D, Ann. of Math. 183 (2016), 377–498

    2016

  14. [22]

    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

  15. [23]

    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. 34

    2010

  16. [24]

    Huang, J

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

    2012

  17. [25]

    Jiang and S

    F. Jiang and S. Jiang, Nonlinear stability and instability in the Rayleigh-Taylor problem of stratified compressible MHD fluids, Calc. Var. Partial Differ. Equ.58 (2019), 29

    2019

  18. [26]

    Jiang and S

    F. Jiang and S. Jiang, On magnetic inhibition theory in 3D non-resistive magnetohydrodynamic fluids: global existence of large solutions, Arch. Rational Mech. Anal., (2023) 247:96

    2023

  19. [27]

    Jiang and J

    S. Jiang and J. Zhang, On the non-resistive limit and the magnetic boundary-layer for one-dimensional com- pressible magnetohydrodynamics, Nonlinearity 30 (2017), 3587–3612

    2017

  20. [28]

    Kato, Liapunov Functions and Monotonicity in the Euler and Navier-Stokes Equations, Lecture Notes in Mathematics, vol

    T. Kato, Liapunov Functions and Monotonicity in the Euler and Navier-Stokes Equations, Lecture Notes in Mathematics, vol. 1450. Springer, Berlin (1990)

    1990

  21. [29]

    Koch and S

    H. Koch and S. Kocic, Renormalization of vector fields and Diophantine invariant tori, Ergod. Th. & Dynam. Sys. 28 (2008), 1559–1585

    2008

  22. [30]

    Li and Y

    Y . Li and Y . Sun, Global weak solutions to a two-dimensional compressible MHD equations of viscous non- resistive fluids, J. Differential Equations 267 (2019), 3827–3851

    2019

  23. [31]

    Lopes Dias, Renormalisation scheme for vector fields on T2 with a Diophantine frequency, Nonlinearity 15 (2002), 665–679

    J. Lopes Dias, Renormalisation scheme for vector fields on T2 with a Diophantine frequency, Nonlinearity 15 (2002), 665–679

    2002

  24. [32]

    Luk and J

    J. Luk and J. Speck, Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent. Math. 214 (2018), 1–169

    2018

  25. [33]

    Majda and A

    A. Majda and A. Bertozzi, Vorticity and Incompressible Flow , Cambridge University Press, Cambridge, UK, 2002

    2002

  26. [34]

    Merle, P

    F. Merle, P. Raphael, I. Rodnianski and J. Szeftel, On the implosion of a three dimensional compressible fluid, (2019), arXiv:1912.11009

    2019 arXiv

  27. [35]

    H. K. Moffatt, On the suppression of turbulence by a uniform magnetic field, J. Fluid Mech. 28 (1967), 571–592

    1967

  28. [36]

    T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985), 475-485

    1985

  29. [37]

    Sun and Z

    Y . Sun and Z. Zhang, A blow-up criterion of strong solutions to the 2D compressible Navier-Stokes equations, Sci. China Math. 54 (2011), 105-116

    2011

  30. [38]

    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

  31. [39]

    Triebel, Theory of Function Spaces, Monogr

    H. Triebel, Theory of Function Spaces, Monogr. Math., Birkh ¨auser Verlag, Basel, Boston, 1983

    1983

  32. [40]

    Wang and Z

    Y . Wang and Z. Xin, Global well-posedness of the inviscid heat-conductive resistive compressible MHD in a strip domain, Commun. Math. Res. 38 (2022), 1-27

    2022

  33. [41]

    Wang and Z

    Y . Wang and Z. Xin, Existence of multi-dimensional contact discontinuities for the ideal compressible magneto- hydrodynamics, Comm. Pure Appl. Math. 77 (2024), 583–629

    2024

  34. [42]

    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

  35. [43]

    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

  36. [44]

    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), Article number 109602

    2022

  37. [45]

    Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,Commun

    Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density,Commun. Pure Appl. Math. 51 (1998), 229–240

    1998

  38. [46]

    Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data, Nagoya Math

    H. Yin, Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data, Nagoya Math. J. 175 (2004), 125–164. 1 DEPARTMENT OF MATHEMATICS , U NIVERSITY OF NOTRE DAME , N OTRE DAME , IN 46556, USA Email address: jwu29@nd.edu 2SCHO...

    2004

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