Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics

Han Liu

REVIEW 3 major objections 4 minor 49 references

Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The Hall-MHD equations are asymptotically finite-dimensional: matching low modes forces long-time convergence.

desk verdict A promising extension of the CDK determining-wavenumber method to Hall-MHD, undermined by a weak/strong regularity mismatch and a sign error in the closing inequality. read the letter →

arxiv 1908.04891 v2 pith:PVCBTJGA submitted 2019-08-13 math.AP

Han Liu This is my paper · ORCID

classification math.AP MSC 35Q3535Q8537L30

keywords Hall-MHDsystemdeterminingwavenumbersmodesLittlewood-Paleytheoryfinite-dimensionaldynamicsstrongsolutionsenergyestimates

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 asks whether the incompressible Hall-MHD equations — a plasma model whose Hall term is nonlinear in the magnetic field — have finite-dimensional long-time behaviour. It claims yes, in the determining-modes sense: for each solution it defines time-dependent wavenumbers, and if two solutions' low modes agree up to those wavenumbers at every time, the full velocity and magnetic fields converge in $L^2$. This extends a known line of results for Navier-Stokes and MHD to a system with a stronger nonlinearity; the payoff is that the infinite-dimensional detail of high-frequency plasma motion is asymptotically controlled by finitely many low modes. The paper also proves the wavenumbers have finite time averages for strong solutions.

What carries the argument

The central object is the pair of time-dependent determining wavenumbers $\Lambda_u(t)$ and $\Lambda_b(t)$: the smallest dyadic scales $2^q$ such that all higher Littlewood-Paley blocks are small compared with the dissipation coefficients, in $L^r$ for the velocity and $L^\infty$ for the magnetic field. These conditions make the high-frequency part of the difference controllable by the dissipative terms in the energy estimates. The estimates are organised with Bony's paraproduct, splitting every nonlinear term into low-low, high-low and low-high interactions; two cancellations carry real weight: one term vanishes because the advecting velocity is divergence-free, and a pair of terms cancels after summation. The remaining high-mode terms are absorbed using the wavenumber conditions and Bernstein's inequality, leaving a differential inequality whose exponential decay gives the theorem.

What would settle it

A reader could try to construct two strong solutions on the three-torus whose projections onto all modes below the determining wavenumbers coincide for every positive time while the full $L^2$ difference fails to decay; if such a pair exists, Theorem 1.1's mechanism is false. Short of an exact construction, a high-resolution numerical experiment maintaining the low-mode coincidence and observing the $L^2$ difference plateau above zero would undermine the claimed exponential control.

Watch

Extended reading notes

Core claim

The paper's central claim is that two solutions of the incompressible Hall-MHD system on the three-torus whose low Fourier modes coincide up to a dynamically determined cutoff must have their full $L^2$ difference decay to zero. Specifically, Theorem 1.1 defines determining wavenumbers $\Lambda_u(t)$ and $\Lambda_b(t)$ for each solution and takes the larger of the two for velocity and magnetic field. If the projections of the two solutions onto all modes at or below these wavenumbers are equal for every $t>0$, then $\|u(t)-v(t)\|_{L^2}+\|b(t)-h(t)\|_{L^2}\to 0$ as $t\to\infty$. The proof derives a frequency-localized differential inequality for the difference and applies an exponential decay estimate. The energy estimates are carried out in Sobolev spaces, the setting in which the abstract and Section 5 state the wavenumber bounds.

Load-bearing premise

The proof tests the difference of the two solutions with weighted high-frequency energy norms and hence needs both solutions to have enough spatial derivatives; that regularity is available for the strong solutions treated in the abstract and Section 5, but not automatically for the weak solutions named in Theorem 1.1.

Editorial extensions

If this is right

If two strong solutions share the same low modes up to their determining wavenumbers for all $t>0$, their full $L^2$ difference decays to zero, so the infinite-dimensional high-frequency part is asymptotically determined by finitely many modes.
For strong solutions, the time averages of the determining wavenumbers are finite, so the number of modes that need to be tracked is finite in an averaged sense.
The electron MHD reduction, with no fluid velocity, already satisfies the same property, so the Hall term alone is compatible with finite-dimensional asymptotic dynamics.
In the limit $b \equiv 0$, the result recovers the known determining-wavenumber statement for the 3D Navier-Stokes equations, showing that adding the Hall term does not destroy the mechanism.
The hypotheses reduce to zero-mean velocity and equal mean magnetic field, so Galilean invariance leaves only physically natural assumptions.

Reading between the lines

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

I infer that, if the regularity gap for weak solutions is closed, the same argument would give a determining-modes result for Leray-Hopf weak solutions, where uniqueness is not otherwise available.
I infer that the cutoff could be used to design reduced-order data assimilation for Hall-MHD: observing only modes below the wavenumber would determine the far future of the high modes.
I infer that the Section 5 bound, which works through $\|\nabla b\|_{L^\infty}$, may be sharpened or shown sharp by testing whether weaker norms also control the average wavenumber.
I infer that the EMHD result isolates the Hall term as the sole mechanism of enslaving; a testable extension is to check whether the same wavenumber bounds hold with different Hall coefficients in the two equations.

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

Colleague — this paper is a genuine attempt to extend Cheskidov–Dai–Kavlie's time-dependent determining wavenumber technique to the 3D Hall-MHD system. That is the right next target after the Navier–Stokes and MHD results, and the author does real work: the Hall term requires new commutator estimates, the reduced EMHD system gets its own treatment in Section 3, and the structural role of the electron wavenumber Λ_b is clearly laid out. For strong solutions, the estimate scheme looks sound in broad strokes. If the argument can be made rigorous, the result would be a solid addition to the finite-dimensional behavior literature for Hall-MHD.

The problem is that Theorem 1.1 is stated for weak solutions, but the proof in Section 4 (and Section 3 for EMHD) runs an H^s energy estimate: multiply by λ_q^{2s}Δ_q^2 of the difference, integrate by parts, and sum over q. For Leray–Hopf weak solutions, the difference (w,m) is only L∞_t L2_x ∩ L2_t H1_x, not H^s for any s>0, and the Hall term is not a function. No regularization or Galerkin argument is supplied. The abstract and Section 5 explicitly restrict to strong solutions, which suggests the author was aware of the regularity requirement but the theorem statement was not updated. This is not a typo; it is the difference between what is proved and what is claimed. The same issue affects Theorem 3.1.

Two smaller points. The constant c_r in Section 3 is set to 1−(2μ)^{-1}, which depends on μ and can be negative; the estimates need c_r to be a sufficiently small absolute constant depending only on r, so this needs a repair. And the closing inequality in Section 4.9, d/dt E ≲ −Σλ_q^2E ≲ E, cannot deliver exponential decay as written; the second bound should be something like ≤ −λ_0^2E. That looks fixable, but as it stands the conclusion does not follow.

The wavenumber bound in Section 5 also leans on H^s bounds and Theorem 2.3, which is fine for global strong solutions, but the paper only has local strong existence or small-data global existence (Theorem 2.2). That part needs more care about the time horizon on which the average is taken.

Net: the central idea is promising and the technical machinery is mostly standard and honestly presented, but the paper needs a substantial revision, not a light edit. It deserves a serious referee who knows the CDK technique and Hall-MHD well. I would not cite it in its current form, and I would not put it on the reading list until the weak/strong gap and the sign issue are addressed.

Referee Report

3 major / 4 minor

Summary. The paper introduces time-dependent determining wavenumbers for the 3D incompressible Hall-MHD system, defined through Littlewood-Paley conditions on each individual solution. The main result, Theorem 1.1, claims that if two weak solutions agree on all low modes up to these time-dependent wavenumbers at every time, then their full L^2 difference decays to zero as t→∞. The proof in Section 4 uses frequency-localized energy estimates with Bony paraproduct decompositions and commutator estimates for both the convection and Hall terms. A reduced EMHD system is analyzed first in Section 3, and Section 5 gives an average bound on the magnetic determining wavenumber for strong solutions in terms of the magnetic dissipation.

Significance. If the main theorem were established at the stated regularity, it would provide the first time-dependent determining-wavenumber result for Hall-MHD and would extend the Cheskidov-Dai-Kavlie approach from the 3D Navier-Stokes equations to a system with a nontrivial Hall term. The commutator estimates controlling the Hall nonlinearity are the main technical contribution, and the wavenumber bound in Section 5 connects the determining wavenumber to the dissipation range. However, the central claim is currently stated for weak solutions while the proof requires regularity that Leray-Hopf weak solutions do not provide; the significance is therefore conditional on a repaired theorem statement or a genuinely weak-solution argument.

major comments (3)

[Theorem 1.1 and Section 4, Eq. (4.12)-(4.13)]
[Section 5, wavenumber bounds]
[Section 4.9, conclusion]

minor comments (4)

[Abstract and Theorem 1.1]
[Section 3, constant c_r]
[Theorem 3.1]
[Notation throughout]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the determining wavenumbers are solution-dependent by design, but the finite-dimensional enslavement theorem rests on nontrivial energy estimates, not on the definitions alone.

full rationale

The paper's determining wavenumbers (1.4)–(1.5) are defined using each solution's own Littlewood–Paley norms, and Theorem 1.1's hypothesis refers to those wavenumbers. This is self-referential in a superficial sense, but not circular: the threshold conditions do not by themselves imply the conclusion. The proof in Sections 3–4 derives a genuine differential inequality through the estimates of A–J, using the threshold conditions only as smallness inputs to absorb nonlinear terms by dissipation; the final Gronwall step then yields exponential decay of the difference. No fitted parameter is later renamed as a prediction, and no parameter is chosen using the target difference (w,m). The bound on the wavenumbers in Section 5 follows from the contrapositive of the definition plus the external Prodi–Serrin regularity criterion (Theorem 2.3, cited from [6]), so it is not a restatement of the theorem's hypothesis. The citation to Cheskidov–Dai–Kavlie [11] supplies the methodological template, but the Hall/EMHD estimates are carried out in this paper and do not reduce to that citation. One genuine concern is a correctness gap, not circularity: Theorem 1.1 is stated for weak solutions, while the proof in §4 tests with Δ_q^2(w,m) after an H^s-weighted multiplication, which requires more regularity than a Leray–Hopf weak solution is known to possess, and the abstract and §5 explicitly restrict to strong solutions. That gap should be assessed as a proof-validity issue, not as a circular derivation. Overall, the derivation chain is not circular: the main theorem is a nontrivial consequence of energy estimates aided by the defining inequalities.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central proof uses standard harmonic analysis tools and cited well-posedness results. The main hand-chosen number is c_r, whose final choice is problematic. No data fitting is involved.

free parameters (1)

c_r = 1 - (2μ)^{-1} (as chosen at end of Section 3)
Intended as a universal constant in Definitions 1.4-1.5, but in the closing step of Theorem 3.1 it is set to 1 - (2μ)^{-1}, which depends on the magnetic resistivity μ and is negative when μ<1/2; this is a hand-chosen number needed to close the estimates.

assumptions (5)

standard math Littlewood-Paley theory: Bernstein inequalities, Bony paraproduct, commutator estimates (Lemmas 2.4-2.6)
Used throughout the proofs as background technology; standard in harmonic analysis and PDE.
domain assumption Existence of global Leray-Hopf weak solutions and local/global strong solutions for the Hall-MHD system (Theorems 2.1-2.2, cited from [1],[5])
The paper relies on these well-posedness results to justify considering weak and strong solutions.
domain assumption Prodi-Serrin type regularity criterion (Theorem 2.3, cited from [6])
Used in Section 5 to bound the determining wavenumber for strong solutions.
domain assumption The defining sets in (1.4)-(1.5) are nonempty, so Λ_u(t) and Λ_b(t) are finite
The theorem and bound assume the wavenumbers exist; the paper does not prove finiteness for arbitrary weak solutions.
ad hoc to paper The energy method is justified for the difference of the two solutions: testing with λ^{2s}Δ_q^2 and integrating by parts, requiring more regularity than weak solutions
Used in Sections 3 and 4 without justification for weak solutions; this is the main gap between the stated theorem and the proof.

invented entities (1)

time-dependent determining wavenumbers Λ_u(t), Λ_b(t)
purpose: Define the frequency cutoff below which two solutions must agree for the asymptotic determining property to hold
Introduced in Definitions 1.4-1.5; well-defined mathematical objects, but their existence (finiteness) is assumed, not proven externally.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics." pith.science (2026). https://pith.science/paper/PVCBTJGA

@misc{pith2026190804891,
  author       = {Pith},
  title        = {Pith review of: Determining wavenumbers for the incompressible Hall-magneto-hydrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVCBTJGA}},
  note         = {Machine review of arXiv:1908.04891}
}

abstract

Using Littlewood-Paley theory, one formulates the determining wavenumbers for the Hall-MHD system, defined for each individual solution $(u,b)$. It is shown that the long time behaviour of strong solutions is almost finite dimensional as the wavenumbers are bounded in certain average senses.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 47 canonical work pages

[1]

Acheritogaray, P

M. Acheritogaray, P. Degond, A. Frouvelle and J. Liu. Kinetic formulation and global exis- tence for the Hall-Magneto-hydrodynamics system . Kinet. Relat. Models Vol. 4(4), 901-918, 2011

work page 2011
[2]

M. J. Benvenutti and L. C. F. Ferreira. Existence and stability of global large strong solutions for the Hall-MHD system. Diﬀer. Integral Equ. Vol. 29(910), 9771000, 2016

work page 2016
[3]

Bahouri, J

H. Bahouri, J. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Diﬀerential Equations. Grundlehren der mathematischen Wissenschaften, 343. Spr inger, Heidelberg, 2011

work page 2011
[4]

L. M. B. C. Campos. On hydromagnetic waves in atmospheres with application to t he Sun. Theor. Comput. Fluid Dyn. 10 (1-4), 37-70, 1998

work page 1998
[5]

D. Chae, P. Degond and J. Liu. Well-posedness for Hall-magnetohydrodynamics . Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire Vol. 31, No. 3, 555-565, 2014

work page 2014
[6]

Chae and J

D. Chae and J. Lee. On the blow-up criterion and small data global existence for the Hall- magnetohydrodynamics. J. Diﬀ. Eq. Vol. 256(11), 3835-3858, 2014

work page 2014
[7]

Chae and M

D. Chae and M. E. Schonbek. On the temporal decay for the Hall-magnetohydrodynamic equations. J. Diﬀ. Eq. Vol. 255(11), 3971-3982, 2013

work page 2013
[8]

D. Chae, R. W an and J. W u. Local well-posedness for the Hall-MHD equations with frac- tional magnetic diﬀusion. J. Math. Fluid Mech. Vol. 17(4), 627-638, 2015. DETERMINING W A VENUMBERS 35

work page 2015

Show all 49 references

[9]

Chae and S

D. Chae and S. W eng. Singularity formation for the incompressible Hall-MHD equ ations without resistivity. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire Vol. 33, No. 4, 10 09-1022, 2016

2016
[10]

Cheskidov and M

A. Cheskidov and M. Dai. Regularity criteria for the 3D Navier-Stokes and MHD equati ons. arXiv:1507.06611

arXiv
[11]

Cheskidov, M

A. Cheskidov, M. Dai and L. Kavlie. Determining modes for the 3D Navier-Stokes equations. Physica D, 374: 1-9, 2018

2018
[12]

Cheskidov and R

A. Cheskidov and R. Shvydkoy. A uniﬁed approach to regularity problems for the 3D Navier- Stokes and Euler equations: the use of Kolmogorovs dissipat ion range. J. Math. Fluid Mech. 16 (2), 263-273, 2014

2014
[13]

Constantin, C

P. Constantin, C. Foia¸ s, O. Manley, and R. Temam. Determining modes and fractal dimen- sion of turbulent ﬂows. J. Fluid Mech., 150: 427-440, 1985

1985
[14]

Constantin, C

P. Constantin, C. Foia¸ s and R. Temam. On the dimension of the attractors in two- dimen- sional turbulence. Physica D, 30, 284-296, 1988

1988
[15]

M. Dai. Regularity criterion for the 3D Hall-magneto-hydrodynami cs. J. Diﬀ. Eq. Vol. 261(1), 573-591, 2016

2016
[16]

M. Dai. Local well-posedness of the Hall-MHD system in H s(Rn) with s > n/ 2. arXiv: 1709.02347

arXiv
[17]

M. Dai. Local well-posedness for the Hall-MHD system in optimal Sob olev Spaces. arXiv: 1803.09556

arXiv
[18]

M. Dai. Non-uniqueness of Leray-Hopf weak solutions of the 3D Hall- MHD system . arXiv: 1812.11311

arXiv
[19]

M. Dai. Regularity criterion and energy conservation for the super critical quasi-geostrophic equation. Journal of Mathematical Fluid Mechanics, DOI:10.1007/s00 021-017-0320-y, 2017

2017 doi
[20]

Eden and A

A. Eden and A. Libin. Explicit dimension estimates of attractors for the MHD equa tions in three-dimensional space. Physica D. 40: 338-352, 1989

1989
[21]

J. Fan, Y. Fukumoto, G.Nakamura and Y. Zhou. Regularity criteria for the incompressible Hall-MHD system. Z. Angew. Math. Mech. 95(11), 1156-1160, 2015

2015
[22]

J. Fan, F. Li and G. Nakamura. Regularity criteria for the incompressible Hall- magnetohydrodynamic equations. Nonlinear Anal. 109: 173-179, 2014

2014
[23]

Foia¸ s, M

C. Foia¸ s, M. Jolly, R. Kravchenko and E. Titi. A determining form for the 2D Navier-Stokes equations the Fourier modes case. J. Math. Phys., 53(11), 115623, 30 pp, 2012

2012
[24]

Foia¸ s, O

C. Foia¸ s, O. Manley, R. Rosa, and R. Temam. Navier-Stokes equations and turbulence. Vol. 83 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 2001

2001
[25]

Foia¸ s, O

C. Foia¸ s, O. Manley, R Temam, and Y. Tr´ eve. Asymptotic analysis of the Navier- Stokes equations. Phys. D, 9 (1-2), 157-188, 1983

1983
[26]

Foia¸ s and G

C. Foia¸ s and G. Prodi. Sur le comportement global des solutions non-stationnaire s des ´ equations de NavierStokes en dimension 2. Rend. Sem. Mat. Univ. Padova 39:1-34, 1967

1967
[27]

Foia¸ s and R

C. Foia¸ s and R. Temam. Some analytic and geometric properties of the solutions of t he Navier-Stokes equations. J. Math. Pures Appl., 58, 339-368, 1979

1979
[28]

Foia¸ s and R

C. Foia¸ s and R. Temam. Determination of the solutions of the Navier-Stokes equati ons by a set of nodal values. Math. Comput., 43, 117-133, 1984

1984
[29]

Foia¸ s and E

C. Foia¸ s and E. Titi. Determining nodes, ﬁnite diﬀerence schemes and inertial ma nifolds. Nonlinearity, 135153, 1991

1991
[30]

S. Galtier. Introduction to Modern Magnetohydrodynamics. Cambridge University Press, Cambridge, UK, 2016

2016
[31]

S. Galtier. Wave turbulence in incompressible Hall magnetohydrodynam ics . Journal of Plasma Physics 72 (5), 721-769, 2006

2006
[32]

Galtier and E Buchlin

S. Galtier and E Buchlin. Multiscale Hall-magnetohydrodynamic turbulence in the so lar wind . The Astrophysical Journal 656 (1), 560, 2007

2007
[33]

S. Galtier. Exact scaling laws for 3D electron MHD turbulence. Journal of Geophysical Research: Space Physics 113 (A1), 2008

2008
[34]

Grafakos

L. Grafakos. Modern Fourier Analysis . Graduate Texts in Mathematics, Vol. 250, 2nd edition, Springer, New York, 2009

2009
[35]

F. He, B. Ahmad, T. Hayat and Y. Zhou. On regularity criteria for the 3D Hall-MHD equations in terms of the velocity. Nonlinear Anal. R W A 32, 35-51, 2016. 36 HAN LIU

2016
[36]

Jeong and S

I. Jeong and S. Oh. On the Cauchy problem for the Hall and electron magnetohydro dy- namic equations without resistivity I: illposedness near d egenerate stationary solutions. arXiv: 1902.02025

1902 arXiv
[37]

Jones and E

D. Jones and E. Titi. Upper bounds on the number of determining modes, nodes, and v olume elements for the Navier-Stokes equations. Indiana Univ. Math. J. 42(3):875887, 1993

1993
[38]

Kolmogorov

A. Kolmogorov. The local structure of turbulence in incompressible viscou s ﬂuids at very large Reynolds numbers. Dokl. Akad. Nauk. SSSR 30: 301-305, 1941

1941
[39]

Kwak and B

M. Kwak and B. Lkhagvasuren. Global wellposedness for Hall-MHD equations . Nonlinear Anal. 174: 104-117, 2018

2018
[40]

M. J. Lighthill. Studies on magneto-hydrodynamic waves and other anisotrop ic wave mo- tions. Phil. Trans. R. Soc. A 252 (1014), 397-430, 1960

1960
[41]

Meyrand and S

R. Meyrand and S. Galtier. Anomalous Spectrum in Electron Magnetohydrodynamic Tur- bulence. Physical review letters 111 (26), 264501, 2013

2013
[42]

J. M. Polygiannakis and X. Moussas. A review of magneto-vorticity induction in Hall-MHD plasmas. Plasma Phys. Control. Fusion 43 (2), 195, 2001

2001
[43]

Robinson

J. Robinson. Attractors and ﬁnite-dimensional behaviour in the 2D Navie r-Stokes equations. ISRN Mathematical Analysis, vol. 2013, Article ID 291823, 2 9 pages, 2013

2013
[44]

W an and Y

R. W an and Y. Zhou. On global existence, energy decay and blow-up criteria for t he Hall- MHD system. J. Diﬀ. Eq. 259 (11), 5982-6008, 2015

2015
[45]

W ang and H

Y. W ang and H. Li. Beale-Kato-Madja type criteria of smooth solutions to 3D Ha ll-MHD ﬂows. Appl. Math. Comput. 286, 41-48, 2016

2016
[46]

Z. Ye. Regularity criterion for the 3D Hall-magnetohydrodynamic equations involving the vorticity. Nonlinear Anal. 144, 182-193, 2016

2016
[47]

Z. Ye. A logarithmically improved regularity criterion for the 3D Hall-MHD equations in Besov spaces with negative indices. Appl. Anal. 96 (16), 2669-2683, 2017

2017
[48]

Ye and Z

Z. Ye and Z. Zhang. A remark on regularity criterion for the 3D Hall-MHD equatio ns based on the vorticity. Appl. Math. Comput. 301: 7077, 2017

2017
[49]

Z. Zhang. A remark on the blow-up criterion for the 3D Hall-MHD system i n Besov spaces. J. Math. Anal. Appl. 441 (2), 692-701, 2016. Department of Mathematics, Stat. and Comp. Sci., University of Illinois Chicago, Chicago, IL 60607,USA E-mail address : hliu94@uic.edu

2016

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Acheritogaray, P

M. Acheritogaray, P. Degond, A. Frouvelle and J. Liu. Kinetic formulation and global exis- tence for the Hall-Magneto-hydrodynamics system . Kinet. Relat. Models Vol. 4(4), 901-918, 2011

work page 2011

[2] [2]

M. J. Benvenutti and L. C. F. Ferreira. Existence and stability of global large strong solutions for the Hall-MHD system. Diﬀer. Integral Equ. Vol. 29(910), 9771000, 2016

work page 2016

[3] [3]

Bahouri, J

H. Bahouri, J. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Diﬀerential Equations. Grundlehren der mathematischen Wissenschaften, 343. Spr inger, Heidelberg, 2011

work page 2011

[4] [4]

L. M. B. C. Campos. On hydromagnetic waves in atmospheres with application to t he Sun. Theor. Comput. Fluid Dyn. 10 (1-4), 37-70, 1998

work page 1998

[5] [5]

D. Chae, P. Degond and J. Liu. Well-posedness for Hall-magnetohydrodynamics . Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire Vol. 31, No. 3, 555-565, 2014

work page 2014

[6] [6]

Chae and J

D. Chae and J. Lee. On the blow-up criterion and small data global existence for the Hall- magnetohydrodynamics. J. Diﬀ. Eq. Vol. 256(11), 3835-3858, 2014

work page 2014

[7] [7]

Chae and M

D. Chae and M. E. Schonbek. On the temporal decay for the Hall-magnetohydrodynamic equations. J. Diﬀ. Eq. Vol. 255(11), 3971-3982, 2013

work page 2013

[8] [8]

D. Chae, R. W an and J. W u. Local well-posedness for the Hall-MHD equations with frac- tional magnetic diﬀusion. J. Math. Fluid Mech. Vol. 17(4), 627-638, 2015. DETERMINING W A VENUMBERS 35

work page 2015