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Uniform Anisotropic Regularity and Low Mach Number Limit of Non-isentropic Ideal MHD Equations with a Perfectly Conducting Boundary

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Pith's one-line read Non-isentropic ideal MHD in a conducting-wall half-space has a low-Mach limit for general initial data.

desk verdict A technically serious paper proving the low Mach limit for non-isentropic ideal MHD with boundary and general data; the uniform estimates look credible, but the convergence proof rests on an unpublished companion result. read the letter →

arxiv 2412.09943 v1 pith:4RD4WH4O submitted 2024-12-13 math.AP

Qiangchang Ju , Jiawei Wang , Junyan Zhang This is my paper · ORCID
classification math.AP MSC 35L6035Q3576M4576W05
keywords lowMachnumberlimitidealmagnetohydrodynamicsperfectlyconductingwallanisotropicSobolevspacesincompressiblegeneralinitialdatamicrolocaldefectmeasureentropyregularity
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 as the Mach number ε tends to zero, solutions of the non-isentropic compressible ideal MHD equations in the half-space, with the slip and perfectly conducting wall boundary conditions and with general (not well-prepared) initial data, converge to the incompressible inhomogeneous MHD system coupled to a transport equation for the entropy. The interest is that this is a singular limit: the pressure and velocity divergence oscillate at frequency 1/ε, so uniform estimates alone do not give convergence. The proof supplies uniform-in-ε estimates in anisotropic Sobolev spaces with Mach-number weights, then uses them to filter the acoustic waves via microlocal defect measures and obtain strong local convergence of pressure and divergence. If correct, it rigorously justifies the incompressible model as the leading-order description of low-Mach confined plasmas without imposing extra boundary constraints beyond the physical ones.

What carries the argument

The machinery is the weighted anisotropic energy functional E(t)=Σ ||(εDt)^{k+2l}(q,u,B,S,$ρ^{{−1}}$B·∇S)||^2_{4−k−l} together with a div-curl reduction scheme. The material derivative Dt is used instead of ∂t so that every tangential derivative is aligned with the flow, and each Dt carries a factor ε matching the singular acoustic terms. The argument also uses two structural observations: the Lorentz force produces a weighted anisotropic term that trades a normal derivative for $ε^{2}$ $Dt^{2}$, and the directional derivative $ρ^{{−1}}$B·∇ commutes with Dt, giving the entropy extra regularity along the magnetic field. At the top order, modified Alinhac good unknowns Q8 and U8 absorb the uncontrollable commutator terms so that the differentiated system keeps its anti-symmetric form.

What would settle it

Take a sequence of initial data satisfying the compatibility conditions and all H8/H9 bounds but with |S0(x)| decaying only like (1+|x|)^{−1} (or not decaying), and compute the microlocal defect measure M(t,τ) of the pressure wave packets; if a nonzero measure survives as ε→0, then q fails to converge strongly to zero, contradicting Theorem 1.2.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.2: under H8/H9 initial data satisfying the compatibility and entropy-decay hypotheses, the scaled solution (q,u,B,S) of system (1.8) converges to (0,u0,B0,S0) weakly-* in L∞([0,T];H4(Ω)) and strongly in L2([0,T];$H^{{4−δ}}$_loc(Ω)) for every δ>0. The limit (u0,B0,S0) solves the inhomogeneous incompressible MHD system (1.14) with a transport equation for S0, and the initial velocity w0 is recovered from the original data by the elliptic problem (1.16). The proof is carried by a uniform energy estimate with an anisotropic norm in which each material derivative Dt carries a factor ε and normal derivatives are counted twice; the strongest part of the statement is that the acoustic components q and ∇·u converge strongly to zero in the local Sobolev sense, which is what makes the limiting system incompressible.

Load-bearing premise

The load-bearing premise is that the initial entropy decays at infinity like |x|^{−1−σ} (and its gradient like |x|^{−2−σ}) in the unbounded half-space; this decay is what lets the proof show that acoustic wave energy does not concentrate at infinity, and without it the strong convergence of the pressure and velocity divergence is not established.

Editorial extensions

If this is right

  • For small Mach number, the compressible ideal MHD flow in a conducting-wall half-space is well approximated on a fixed time interval by the incompressible inhomogeneous MHD system with transported entropy, for general initial data.
  • The acoustic waves generated by O(1) initial pressure gradients and divergence are filtered out: q and ∇·u converge strongly to zero in local Sobolev spaces.
  • The uniform-in-ε energy estimates give a lifespan T independent of the Mach number, so the approximation is not just formal.
  • The result removes the extra boundary constraints required by earlier treatments: only the physical slip boundary condition and the perfectly conducting wall condition are needed.
  • In two space dimensions the same limit holds, with the curl condition in (1.16) replaced by the scalar ∇⊥ condition.

Reading between the lines

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

  • My inference: the same div-curl reduction and modified good-unknown argument may transfer to other characteristic-boundary MHD problems, such as current-vortex sheets, where the anti-symmetric structure of the differentiated system is the main obstacle.
  • My inference: because the proof of strong convergence uses global dispersion of acoustic waves in the unbounded half-space, a bounded-domain analogue would require a different mechanism and may genuinely fail.
  • My inference: the entropy decay condition (1.15) is likely not merely technical; if initial entropy does not decay, acoustic energy can escape to infinity or concentrate, and the strong convergence statement of Theorem 1.2 would need modification.
  • My inference: the commutation [ρ^{−1}B·∇,Dt]=0 suggests that in Lagrangian coordinates the magnetic directional derivative is frozen in time, which could simplify numerical or analytic treatments of entropy-vorticity coupling in MHD.
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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

4 major / 4 minor

Summary. The paper studies the scaled non-isentropic ideal MHD system in the half-space with a perfectly conducting wall. Theorem 1.1 claims uniform-in-Mach-number energy estimates in an anisotropic Sobolev-type norm built from material derivatives, using a special Lorentz-force structure, enhanced entropy regularity along the magnetic field, and modified Alinhac good unknowns. Theorem 1.2 claims that, under H4 convergence of the initial data and the entropy decay condition (1.15), the scaled solutions converge weak-* in L∞([0,T];H4) and strongly in L2([0,T];H4−δ_loc) to a solution of the incompressible inhomogeneous MHD system with a transport equation for the entropy. The proof combines the a priori estimates of Section 3 with the microlocal defect-measure analysis of Section 4.

Significance. If the central claims are correct, this is a substantial advance: it removes the extra boundary constraints of [16] and extends the low-Mach-number limit for ideal MHD with a perfectly conducting wall to non-isentropic general initial data. The energy argument in Section 3 is written out in considerable detail and contains genuinely useful structural observations, especially the weighted anisotropic structure contributed by the Lorentz force and the propagation of the enhanced directional regularity of the entropy. The main weakness is not the plausibility of the claims but their verifiability: the decisive defect-measure step in Section 4.2 is delegated to an unpublished companion preprint, and several high-order commutator estimates in Section 3 and Appendix A are asserted by analogy or omitted.

major comments (4)
  1. [§4.2, Eq. (4.20)] The equality M(t,τ)=0 is the only step that upgrades weak convergence to the strong convergence of q and ∇·u stated in Proposition 4.1. The manuscript says that the details can be found in [38, Corollary 4.4], but [38] is an unpublished preprint by two of the authors on elastodynamics, not on MHD. Since the operator P0 in (4.15) has variable coefficients depending on the limit entropy S̃0 and is posed with a Neumann boundary condition, a referee cannot verify that [38, Corollary 4.4] applies to this operator, nor whether (1.15) is sufficient. This is load-bearing: without (4.20), the deductions (4.22)–(4.27), the projection argument in §4.3, and the limit system (1.14) do not follow. Please include a complete proof of (4.20) or state and prove a dedicated lemma with all hypotheses, including the precise role of the entropy decay condition and the boundary condition.
  2. [§3.4, Lemmas 3.7–3.8 and Lemma A.5] The uniform estimate (3.94) is closed using several high-order commutator bounds that are not fully proved. Lemma 3.8 is justified by saying that the proof is obtained by mimicking Lemmas 3.6 and 3.7, and Lemma A.5 states that the details are skipped. These are not merely cosmetic omissions: they control terms such as [(εDt)^{k+2l}, εB×](∇×B), which involve ε-weighted normal derivatives and are used directly in the final closing argument. To make Theorem 1.1 independently checkable, provide complete proofs or a precise reduction of these cases to the displayed estimates in Lemmas 3.6 and A.4.
  3. [§4.1, Eq. (4.3)] The strong convergence of ∇×(ρ0u) in C([0,T];H^{3−δ}_loc) is asserted from the uniform boundedness of Dt∇×(ρ0u), but no compactness argument is supplied. This convergence is used in (4.28) and in the identification of the limit momentum, so the exact functional-space compactness statement (for instance, an Aubin–Lions argument with the relevant anisotropic norms) should be stated and verified, or a precise reference should be given.
  4. [§4.2, Proposition 4.1] The proposition is introduced as a 'slight variant of [1, Prop. 3.1]' and the proof says that some technical details identical to [1] are skipped. Because Proposition 4.1 is the core acoustic-wave mechanism of the low-Mach-number limit, the modifications needed for the variable-coefficient MHD system, the half-space geometry, and the perfectly conducting boundary condition should be written out or listed explicitly. As it stands, this part of the proof is an outline rather than a complete argument.
minor comments (4)
  1. [§3.3.2, Eq. (3.24)] In the definition of G′4, the expression 'D8_t B · R8_B · D8_t B' is not well-formed as written; it should presumably be a single dot product between two vector-valued terms.
  2. [Abstract and Theorem 1.2] The abstract and introduction repeatedly call the data 'general initial data', but Theorem 1.2 imposes the entropy decay condition (1.15). This is acknowledged in Remark 1.5, but the abstract should carry the same qualifier to avoid overstating the result.
  3. [§4.3] In the definition of the space Hσ, the condition that the integral vanish is missing: one should have ∫Ω u·∇φ = 0 for all φ ∈ H1(Ω). As written, the displayed definition is incomplete, and Gσ as stated is not clearly the orthogonal complement of Hσ.
  4. [Theorem 1.2] The two-dimensional case is justified by a reference to [37, Section 3.5] and is not discussed further. Since Theorem 1.2 explicitly claims the d=2 case, a short paragraph explaining the modifications to the vorticity analysis in Section 3.4.1 would be helpful.

Circularity Check

1 steps flagged · score 4.0 of 10

The uniform-energy theorem is self-contained, but the central strong-convergence theorem's decisive defect-measure step (4.20) is delegated to [38, Cor. 4.4], a companion preprint by two of the authors; that self-citation is load-bearing, so the convergence claim is not fully derivable within this paper.

  1. self citation load bearing [Section 4.2, around Eq. (4.20), in the proof of Proposition 4.1]
    "Let M(t,τ ) be the trace-class operator and µ be the microlocal defect measure obtained in Lemma 4.2 by inserting Θε defined in (4.18). Then we can prove M(t,τ ) = 0 µ-a.e., (4.20) whose details can be found in [38, Corollary 4.4]."

    Theorem 1.2 asserts the low-Mach strong convergence (q,∇·u)→0, and Proposition 4.1 is the proof's engine. Its essential step is to conclude M(t,τ)=0; without this, the subsequent deductions (4.21)–(4.27) that give q→0 and ∇·u→0 do not follow. The paper supplies no derivation of (4.20) for the MHD acoustic operator P0 in (4.15) with variable coefficients a0(S̃),ρ0(S̃) and Neumann boundary condition; it states only that the details are in [38, Corollary 4.4], a preprint by the second and third authors on a different system (elastodynamics). Thus the central limit proof passes through a load-bearing self-citation rather than a proof occurring in this manuscript.

full rationale

Apart from the (4.20) step, the derivation is not circular. Section 3 proves Theorem 1.1 directly from (1.8): the L2 estimate, entropy-directional regularity (Prop. 3.2/Cor. 3.3), tangential estimates with the modified Alinhac good unknowns, and the div-curl reduction are all carried out in the text with commutator bounds in Appendix A. No parameter is fitted and no claimed quantity is defined as the output of the estimate. The limit proof uses external results [1, 23] for wave-packet transforms and the injectivity lemma; those are not by the present authors. The entropy decay condition (1.15) is an explicit hypothesis, not a hidden assumption imported from the conclusion. The only load-bearing self-citation is [38, Corollary 4.4] for M(t,τ)=0; because this is a companion paper by two coauthors and its transfer to the MHD operator is not shown here, it raises the circularity score to 4. The central claim still has substantial independent content, so the score is not higher.

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

The proof introduces no new physical entities or fitted parameters. It relies on standard analytic inequalities, prior well-posedness results, and decay assumptions on the initial entropy. The main external dependencies that are not fully derived in the text are the fixed-epsilon existence theorem and the microlocal defect measure lemmas.

assumptions (6)
  • standard math Sobolev embedding and product laws in R^3_+ and on the boundary are valid for the H^s norms used to bound nonlinear terms and commutators.
    Invoked throughout Sections 3 and A, for example in (3.12)-(3.15) and Lemmas A.4-A.5, to control products such as (epsilon partial (epsilon D_t)^5 u)(partial u). The weighted anisotropic norm is used through these estimates.
  • domain assumption Local well-posedness of (1.8) in H^8_* for each fixed epsilon holds for data satisfying the compatibility conditions (1.5), as stated in [32, Theorem 2.1'] or [37, Theorem 1.1].
    Theorem 1.1 is an a priori estimate; applying it to actual solutions requires existence for epsilon fixed, imported from prior hyperbolic boundary well-posedness theory.
  • standard math The wave operator has the dispersion and limiting absorption properties used in the defect measure argument; in particular, Hörmander's theorem applies under the decay condition (1.15).
    Used in Section 4.2, specifically Lemma 4.3 from [23] and the proof that M(t,tau)=0, to show strong convergence of q and ∇·u; Remark 1.5 explains the entropy decay and unbounded domain requirements.
  • domain assumption The entropy decay condition |S0(x)|≤N0|x|^{-1-sigma}, |∇S0(x)|≤N0|x|^{-2-sigma}, and the unbounded half-space geometry hold.
    Assumed in Theorem 1.2 and Remark 1.5; needed for the dispersion and decay estimates of acoustic waves in the half-space. Not needed for the uniform energy estimates of Theorem 1.1.
  • domain assumption The equation of state satisfies rho(p,S)>0 with rho≥rholower>0 and partial rho/partial p>0, and rho is smooth in (p,S).
    This is condition (1.2), which guarantees hyperbolicity and the positivity of a=rho^{-1}partial rho/partial p, used to define the energy (1.13) and to symmetrize the system.
  • standard math For the limiting initial data, the elliptic problem (1.16) has a unique solution w0 in H^4 satisfying the stated boundary, divergence, and weighted-curl equations.
    Theorem 1.2 asserts such w0 exists and is determined by (1.16), following [23, Theorem 1.5]; this is needed to identify the initial data of the limit system.

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Pith. "Pith review of Uniform Anisotropic Regularity and Low Mach Number Limit of Non-isentropic Ideal MHD Equations with a Perfectly Conducting Boundary." pith.science (2026). https://pith.science/paper/4RD4WH4O

@misc{pith2026241209943,
  author       = {Pith},
  title        = {Pith review of: Uniform Anisotropic Regularity and Low Mach Number Limit of Non-isentropic Ideal MHD Equations with a Perfectly Conducting Boundary},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4RD4WH4O}},
  note         = {Machine review of arXiv:2412.09943}
}
read the original abstract

We prove the low Mach number limit of non-isentropic ideal magnetohydrodynamic (MHD) equations with general initial data in the half-space whose boundary satisfies the perfectly conducting wall condition. By observing a special structure contributed by Lorentz force in vorticity analysis, we establish uniform estimates in suitable anisotropic Sobolev spaces with weights of Mach number determined by the number of material derivatives. We also observe that the entropy has the enhanced regularity in the direction of the magnetic field. These two observations help us get rid of the loss of derivatives and weights of Mach number in vorticity analysis caused by the simultaneous appearance of entropy, general initial data and the magnetic field, which is one of the major difficulties that do not appear in Euler equations or the isentropic problems. By utilizing the technique of Alinhac good unknowns, the anti-symmetric structure is preserved in the tangential estimates for the system differenetiated by high-order material derivatives.

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Reference graph

Works this paper leans on

44 extracted references · 43 canonical work pages

  1. [16]

    Incompressible limit of ideal magnetohydrodynamic equati ons with a perfectly con- ducting wall condition

    Ju, Q., Wang, J. Incompressible limit of ideal magnetohydrodynamic equati ons with a perfectly con- ducting wall condition. SIAM J. Math. Anal., 55(6), 7549-7574, 2023

    work page 2023

  2. [37]

    Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall

    Wang, J., Zhang, J. Incompressible limit of compressible ideal MHD flows inside a perfectly conducting wall. arXiv preprint arXiv:2308.01142v5, 2023

    work page Pith review arXiv 2023

  3. [35]

    The Incompressible Limit of the Equations of Compressible I deal Magneto-Hydrodynamics with perfectly conducting boundary

    Secchi, P . The Incompressible Limit of the Equations of Compressible I deal Magneto-Hydrodynamics with perfectly conducting boundary . Commun. Math. Anal. Appl., 3 (2024), pp. 168-198

    work page 2024

  4. [38]

    Low Mach Number Limit of Non-isentropic Inviscid Elastodyn amics with General Initial Data

    Wang, J., Zhang, J. Low Mach Number Limit of Non-isentropic Inviscid Elastodyn amics with General Initial Data. Preprint, 2024

    work page 2024

  5. [1]

    Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions

    Alazard, T. Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions. Adv. Differ. Equ., 10(1):19–44, 2005

    work page 2005

  6. [2]

    Existence d’ondes de rar´ efaction pour des syst` emes quasi -lin´ eaires hyperboliques multi- dimensionnels.(French

    Alinhac, S. Existence d’ondes de rar´ efaction pour des syst` emes quasi -lin´ eaires hyperboliques multi- dimensionnels.(French. English summary) [Existence of rarefaction waves for multidimensional hype r- bolic quasilinear systems]. Commun. Partial Di ffer. Equ., 14(2), 173-230, 1989

    work page 1989

  7. [3]

    On the incompressible limit of the compressible Euler equat ion

    Asano, K. On the incompressible limit of the compressible Euler equat ion. Japan J. Appl. Math., 4(3):455–488, 1987

    work page 1987

  8. [4]

    Initial boundary value problems for quasilinear symmetric hyperbolic systems with char- acteristic boundary

    Chen, S.-X. Initial boundary value problems for quasilinear symmetric hyperbolic systems with char- acteristic boundary. Translated from Chin. Ann. Math. 3(2), 222–232 (1982). Fron t. Math. China 2(1), 87–102 (2007)

    work page 1982

Show all 44 references
  1. [5]

    Three-Scale Singular Limits of Evolutionary PDEs

    Cheng, B., Ju, Q., Schochet, S. Three-Scale Singular Limits of Evolutionary PDEs. Arch. Rational Mech. Anal., 229(2), 601-625, 2018

    2018

  2. [6]

    Convergence Rate Estimates for the Low Mach and Alfv´ en Numb er Three-Scale Singular Limit of Compressible Ideal Magnetoh ydrodynamics

    Cheng, B., Ju, Q., Schochet, S. Convergence Rate Estimates for the Low Mach and Alfv´ en Numb er Three-Scale Singular Limit of Compressible Ideal Magnetoh ydrodynamics. ESAIM: M2AN, V ol.55, S733-S759, 2021

    2021

  3. [7]

    Ebin, D. G. Motion of slightly compressible fluids in a bounded domain. I . Commun. Pure Appl. Math., 35(4):451–485, 1982

    1982

  4. [8]

    and Poedts, S

    Goedbloed, H., Keppens, R. and Poedts, S. Magnetohydrodynamics of Laboratory and Astrophysical plasmas. Cambridge University Press, 2020

    2020

  5. [9]

    On the construction of solutions to the free-surface incomp ressible ideal magnetohy- drodynamic equations

    Gu, X., Wang, Y . On the construction of solutions to the free-surface incomp ressible ideal magnetohy- drodynamic equations. J. Math. Pures Appl., V ol. 128: 1-41, 2019

    2019

  6. [10]

    The analysis of linear partial di fferential equations

    H¨ ormander, L. The analysis of linear partial di fferential equations. V ol. III, Springer-V erlag, New Y ork, 1985

    1985

  7. [11]

    The incompressible limit and the initial layer of the compressible Euler equation in Rn +

    Iguchi, T. The incompressible limit and the initial layer of the compressible Euler equation in Rn +. Math. Methods Appl. Sci., 20(11):945–958, 1997

    1997

  8. [12]

    Singular limits for the compressible Euler equations in an e xterior domain

    Isozaki, H. Singular limits for the compressible Euler equations in an e xterior domain. J. Reine Angew. Math., 381:1-36, 1987

    1987

  9. [13]

    Incompressible limit of the nonisentropic magnetohydrody namic equations

    Jiang, S., Ju, Q., Li, F. Incompressible limit of the nonisentropic magnetohydrody namic equations. SIAM J. Math. Anal., 48(1), 302-319, 2016

    2016

  10. [14]

    Small Alfv´ en number limit for incompressible magnetohydr odynamics in a domain with boundaries

    Jiang, S., Ju, Q., Xu, X. Small Alfv´ en number limit for incompressible magnetohydr odynamics in a domain with boundaries. Sci. China Math., 62, 2229-2248, 2019

    2019

  11. [15]

    Singular limits of the equations of compressible ideal magn etohydrody- namics in a domain with boundaries

    Ju, Q., Schochet, S., Xu, X. Singular limits of the equations of compressible ideal magn etohydrody- namics in a domain with boundaries. Asymptotic Anal., 113, 137-165, 2019

    2019

  12. [17]

    Singular limits of quasilinear hyperbolic systems with lar ge parameters and the incompressible limit of compressible fluids

    Klainerman, S., Majda, A. Singular limits of quasilinear hyperbolic systems with lar ge parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math., 34(4):481–524, 1981

    1981

  13. [18]

    Compressible and incompressible fluids

    Klainerman, S., Majda, A. Compressible and incompressible fluids. Commun. Pure Appl. Math., 35(5):629–651, 1982

    1982

  14. [19]

    Anisotropic Regularity of the Free-Boundary Problem in Com pressible Ideal Magnetohydrodynamics

    Lindblad, H., Zhang, J. Anisotropic Regularity of the Free-Boundary Problem in Com pressible Ideal Magnetohydrodynamics. Arch. Rational Mech. Anal., 247(5), Paper no. 89, 94 pp, 2023 . 33

    2023

  15. [20]

    On the Motion of a Compressible Gravity W ater W ave with V orticity

    Luo, C. On the Motion of a Compressible Gravity W ater W ave with V orticity. Ann. PDE, 4(2): 2506- 2576, 2018

    2018

  16. [21]

    Compressible Gravity-Capillary W ater W aves: Local W ell-Posedness, Incompress- ible and Zero-Surface-T ension Limits

    Luo, C., Zhang, J. Compressible Gravity-Capillary W ater W aves: Local W ell-Posedness, Incompress- ible and Zero-Surface-T ension Limits. arXiv:2211.03600, preprint

    arXiv

  17. [22]

    Compressible Fluids Flow and Systems of Conservation Laws i n Several Space V ariables

    Majda, A. Compressible Fluids Flow and Systems of Conservation Laws i n Several Space V ariables. Applied Mathematical Sciences, V ol. 53, Springer-V erlag New Y ork, 1984

    1984

  18. [23]

    The incompressible limit of the non-isentropic Euler equat ions

    M´ etivier, G., Schochet, S. The incompressible limit of the non-isentropic Euler equat ions. Arch. Ratio- nal Mech. Anal., 158(1):61–90, 2001

    2001

  19. [24]

    Averaging theorems for conservative systems and the weakly compressible Euler equations

    M´ etivier, G., Schochet, S. Averaging theorems for conservative systems and the weakly compressible Euler equations. J. Differ. Equ., 187(1):106-183, 2003

    2003

  20. [25]

    On the initial-boundary-value problem for the linearized e quations of magneto- hydrodynamics

    Ohno, M., Shirota, T. On the initial-boundary-value problem for the linearized e quations of magneto- hydrodynamics. Arch. Rational Mech. Anal., 144(3), 259-299, 1998

    1998

  21. [26]

    Symmetric Positive Systems with Boundary Characteristic o f Constant Multiplicity

    Rauch, J. Symmetric Positive Systems with Boundary Characteristic o f Constant Multiplicity. Trans. Amer. Math. Soc., 291(1), 167-187, 1985

    1985

  22. [27]

    The compressible Euler equations in a bounded domain: Exist ence of solutions and the incompressible limit

    Schochet, S. The compressible Euler equations in a bounded domain: Exist ence of solutions and the incompressible limit. Commun. Math. Phys., 104(1):49–75, 1986

    1986

  23. [28]

    Singular limits in bounded domains for quasilinear symmetr ic hyperbolic systems having a vorticity equation

    Schochet, S. Singular limits in bounded domains for quasilinear symmetr ic hyperbolic systems having a vorticity equation. J. Differ. Equ., 68:400-428, 1987

    1987

  24. [29]

    Fast Singular Limits of Hyperbolic PDEs

    Schochet, S. Fast Singular Limits of Hyperbolic PDEs. J. Differ. Equ., 114(2):476-512, 1994

    1994

  25. [30]

    W ell-posedness for Mixed Problems for the Equations of Idea l Magneto-hydrodynamics

    Secchi, P . W ell-posedness for Mixed Problems for the Equations of Idea l Magneto-hydrodynamics. Archiv. der. Math. 64(3), 237–245, 1995

    1995

  26. [31]

    On an initial boundary value problem for the equations of ide al magnetohydrodynamics

    Secchi, P . On an initial boundary value problem for the equations of ide al magnetohydrodynamics. Math. Methods Appl. Sci., 18, 841-853, 1995

    1995

  27. [32]

    W ell-posedness of characteristic symmetric hyperbolic systems

    Secchi, P . W ell-posedness of characteristic symmetric hyperbolic systems. Arch. Rational Mech. Anal., 134(2), 155-197, 1996

    1996

  28. [33]

    On the Singular Incompressible Limit of Inviscid Compressi ble Fluids

    Secchi, P . On the Singular Incompressible Limit of Inviscid Compressi ble Fluids. J. Math. Fluid Mech., 2(2), 107-125, 2000

    2000

  29. [34]

    An initial boundary value problem in ideal magneto-hydrody namics

    Secchi, P . An initial boundary value problem in ideal magneto-hydrody namics. NoDEA Nonlinear Differential Eequations Appl., 9(4):441-458, 2002

    2002

  30. [36]

    The existence of current-vortex sheets in ideal compressib le magnetohydrodynamics

    Trakhinin, Y . The existence of current-vortex sheets in ideal compressib le magnetohydrodynamics. Arch. Rational Mech. Anal., 191(2), 245-310, 2009

    2009

  31. [39]

    The incompressible limit and the initial layer of the compre ssible Euler equation

    Ukai, S. The incompressible limit and the initial layer of the compre ssible Euler equation. J. Math. Kyoto Univ., 26(2):323–331, 1986

    1986

  32. [40]

    The Initial Boundary V alue Problem for the Equations of Idea l Magneto- Hydrodynamics

    Y anagisawa, T. The Initial Boundary V alue Problem for the Equations of Idea l Magneto- Hydrodynamics. Hokk. Math. J., V ol.16, 295-314, 1987

    1987

  33. [41]

    The fixed boundary value problems for the equations of ideal m agne- tohydrodynamics with a perfectly conducting wall conditio n

    Y anagisawa, T., Matsumura, A. The fixed boundary value problems for the equations of ideal m agne- tohydrodynamics with a perfectly conducting wall conditio n. Commun. Math. Phys., 136(1), 119-140, 1991

    1991

  34. [42]

    Local W ell-posedness and Incompressible Limit of the Free-Boundary Problem in Compress- ible Elastodynamics

    Zhang, J. Local W ell-posedness and Incompressible Limit of the Free-Boundary Problem in Compress- ible Elastodynamics. Arch. Rational Mech. Anal., 244(3), 599-697, 2022

    2022

  35. [43]

    W ell-posedness and Incompressible Limit of Current-V ortex Sheets with Surface T ension in Compressible Ideal MHD

    Zhang, J. W ell-posedness and Incompressible Limit of Current-V ortex Sheets with Surface T ension in Compressible Ideal MHD. arXiv:2312.11254v3, preprint

    arXiv

  36. [44]

    On the Incompressible Limit of Current-V ortex Sheets with o r without Surface T ension

    Zhang, J. On the Incompressible Limit of Current-V ortex Sheets with o r without Surface T ension. arXiv:2405.00421v2, preprint. 34

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Reviewed August 11, 2026 · model on record in the stance chip above.