Optimal decay for the compressible MHD equations in the critical regularity framework

Qiru Wang; Qunyi Bie; Zheng-an Yao

arxiv: 1906.09119 · v1 · pith:5R7X42JVnew · submitted 2019-06-20 · 🧮 math.AP

Optimal decay for the compressible MHD equations in the critical regularity framework

Qunyi Bie , Qiru Wang , Zheng-an Yao This is my paper

Pith reviewed 2026-05-25 19:26 UTC · model grok-4.3

classification 🧮 math.AP

keywords compressible MHD equationscritical Besov spacesoptimal decaylarge time behaviorenergy methodlow frequency regularity

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The pith

Compressible MHD solutions in critical Besov spaces decay at the optimal rate t to the minus N/2 times (1/2 minus 1/p) minus sigma1 over 2 when low-frequency initial data satisfy an extra condition.

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

This paper examines the large-time decay of solutions to the compressible magnetohydrodynamic equations in L^p-type critical Besov spaces. It establishes that when low-frequency initial data additionally lie in dot B_{2,infty}^{-sigma_1} for sigma_1 in the interval (1 minus N/2, 2N/p minus N/2], the dot B_{p,1}^0 norm of the global solutions decays optimally as t to the power of minus N/2 times (1/2 minus 1/p) minus sigma_1 over 2. The argument relies on a pure energy method that avoids spectral analysis and dispenses with the usual smallness requirement on low frequencies. A reader would care because the result supplies sharp decay estimates inside an already-established critical regularity framework for MHD flows.

Core claim

If the initial data in the low frequencies additionally belong to some Besov space dot B_{2,infty}^{-sigma_1} with sigma_1 in (1-N/2, 2N/p-N/2], then the dot B_{p,1}^0 norm of the critical global solutions to the compressible MHD equations presents the optimal decay t to the minus N/2 (1/2 minus 1/p) minus sigma_1/2 for t to plus infinity. The proof is carried out by a pure energy argument without spectral analysis.

What carries the argument

Pure energy estimates applied to the low-frequency components that belong to the additional space dot B_{2,infty}^{-sigma_1}, which close the decay without invoking spectral decomposition or smallness.

If this is right

The optimal decay holds for the full range of admissible sigma_1 without any smallness restriction on low frequencies.
The energy method suffices to recover the sharp rate once the extra low-frequency regularity is assumed.
The result applies uniformly to the family of L^p critical spaces for admissible p.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same energy technique could be tested on the compressible Navier-Stokes system to obtain analogous decay statements.
One could ask whether the decay persists when the extra regularity is measured in other Besov or Triebel-Lizorkin spaces.
The removal of the smallness condition suggests the method may tolerate larger low-frequency perturbations than previously expected.

Load-bearing premise

Global existence of solutions in the critical space dot B_{p,1}^0 is already known from prior work.

What would settle it

An explicit solution or numerical example with initial data satisfying the low-frequency Besov condition whose dot B_{p,1}^0 norm decays slower than the predicted rate would falsify the claim.

read the original abstract

In this paper, we study the large time behavior of solutions to the compressible magnetohydrodynamic equations in the $L^p$-type critical Besov spaces. Precisely, we show that if the initial data in the low frequencies additionally belong to some Besov space $\dot{B}_{2,\infty}^{-\sigma_1}$ with $\sigma_1\in (1-N/2, 2N/p-N/2]$, then the $\dot{B}_{p,1}^0$ norm of the critical global solutions presents the optimal decay $t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{\sigma_1}{2}}$ for $t\rightarrow+\infty$. The pure energy argument without the spectral analysis is performed, which allows us to remove the usual smallness assumption of low frequencies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers a pure-energy proof of optimal decay for critical Besov solutions to compressible MHD that drops the smallness assumption on low frequencies at the decay stage, but global existence is still taken from prior small-data results.

read the letter

The core advance here is showing that, given critical global solutions in dot B_{p,1}^0, an extra low-frequency assumption in dot B_{2,infty}^{-sigma_1} (with sigma_1 in (1-N/2, 2N/p - N/2]) yields the optimal decay rate t^{-N/2(1/2-1/p) - sigma_1/2} via a pure energy method instead of spectral analysis. This removes the smallness condition specifically for the decay estimates. The argument is cleanly stated in the abstract and appears to avoid circular fitting or post-hoc parameter choices. The pure-energy route is a genuine technical shift from the usual Fourier or spectral approaches in this literature, and it is worth crediting for that. The range on sigma_1 is chosen so the extra norm directly controls the energy estimates, which looks deliberate and functional. The main limitation is that global existence of the critical solutions is imported from earlier papers on compressible MHD in Besov spaces; those works generally require small low-frequency data to close the a-priori bounds. The new decay therefore applies to solutions that already satisfy the smallness condition at the existence stage, so the removal is narrower than the abstract phrasing might suggest. No independent verification of existence without smallness is provided here. This is a specialized technical note aimed at researchers already working on critical Besov theory for MHD and related fluid systems. It is a modest but honest improvement on the decay side and deserves a serious referee to check the energy estimates in detail.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that for the compressible MHD equations, if the low-frequency component of the initial data lies in the Besov space dot B_{2,infty}^{-sigma_1} with sigma_1 in (1-N/2, 2N/p - N/2], then the dot B_{p,1}^0 norm of the critical global solutions decays at the optimal rate t^{-N/2 (1/2 - 1/p) - sigma_1/2} as t to +infty. The argument proceeds via a pure energy method that avoids spectral analysis and is asserted to remove the usual smallness assumption on low frequencies.

Significance. If the central decay estimate holds, the result would extend the literature on large-time behavior of MHD systems in critical Besov spaces by supplying an optimal decay rate under an additional low-frequency integrability condition and by employing an energy-based approach. The stress-test concern that global existence may still require smallness from prior references does not land on the decay claim itself, since the paper takes existence as given and focuses on decay rates for those solutions; the removal of smallness is therefore scoped to the decay stage.

minor comments (3)

The precise statement of the range for sigma_1 should be cross-checked against the embedding and interpolation inequalities used to close the low-frequency energy estimates (typically in the section deriving the decay).
Clarify in the introduction or the statement of the main theorem whether the smallness removal applies only to the decay analysis or also affects the global-existence hypothesis inherited from cited works.
Ensure that the definition of the low-frequency projection and the precise meaning of 'critical global solutions' are stated explicitly before the decay estimates begin.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of our work, including the accurate summary of the main result and the recommendation of minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No circularity; decay estimates are independent energy bounds on assumed global solutions.

full rationale

The paper takes global existence of solutions in the critical space dot B_{p,1}^0 as given from prior literature and derives the stated decay rate via a pure energy method that incorporates the additional low-frequency Besov norm. No quoted step defines the decay rate in terms of itself, fits a parameter to the target quantity, or reduces the central claim to a self-citation chain. The sigma_1 interval is chosen to close the a-priori estimates but does not create a definitional or fitted-input loop. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of Besov spaces and on the prior existence of global solutions; no free parameters or invented entities appear in the abstract statement.

axioms (2)

standard math Standard embedding and interpolation properties of Besov spaces
Invoked to relate the low-frequency space dot B_{2,infty}^{-sigma_1} to the critical norm dot B_{p,1}^0.
domain assumption Global existence of solutions in the critical Besov space dot B_{p,1}^0
The decay statement is made for 'critical global solutions', presupposing their existence from earlier results.

pith-pipeline@v0.9.0 · 5670 in / 1503 out tokens · 40004 ms · 2026-05-25T19:26:44.215371+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

[1]

Bahouri, J.-Y

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier analysis and nonlinear partial diﬀerential equa- tions, vol. 343 of Grundlehren der Mathematischen Wissenschaften, Sp ringer, Heidelberg, 2011

work page 2011
[2]

Bian and B

D. Bian and B. Guo, Well-posedness in critical spaces for the full c ompressible MHD equations, Acta Math. Sci. , 33 (2013), 1153–1176

work page 2013
[3]

Bian and B

D. Bian and B. Yuan, Local well-posedness in critical spaces for t he compressible MHD equations, Appl. Anal. , 95 (2016), 239–269

work page 2016
[4]

Bony, Calcul symbolique et propagation des singularit´ es po ur les ´ equations aux d´ eriv´ ees par- tielles non lin´ eaires,Ann

J.-M. Bony, Calcul symbolique et propagation des singularit´ es po ur les ´ equations aux d´ eriv´ ees par- tielles non lin´ eaires,Ann. Sci. ´Ecole Norm. Sup. (4) , 14 (1981), 209–246

work page 1981
[5]

Cabannes, Theoretical magnetoﬂuid-dynamics, Academic Press, NewYork, 1970

H. Cabannes, Theoretical magnetoﬂuid-dynamics, Academic Press, NewYork, 1970

work page 1970
[6]

Cannone, A generalization of a theorem by Kato on Navier-Sto kes equations, Rev

M. Cannone, A generalization of a theorem by Kato on Navier-Sto kes equations, Rev. Mat. Iberoamericana, 13 (1997), 515–542

work page 1997
[7]

Charve and R

F. Charve and R. Danchin, A global existence result for the comp ressible Navier–Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal. , 198 (2010), 233–271

work page 2010
[8]

Chemin and N

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipsch itziens et ´ equations de Navier- Stokes, J. Diﬀerential Equations , 121 (1995), 314–328

work page 1995
[9]

Chemin, I

J.-Y. Chemin, I. Gallagher, D. Iftimie, J. Ball and D. Welsh, Perfect incompressible ﬂuids, Clarendon Press Oxford, 1998

work page 1998
[10]

Chen and D

G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohy drodynamics with large initial data, J. Diﬀerential Equations , 182 (2002), 344–376

work page 2002
[11]

Chen and D

G.-Q. Chen and D. Wang, Existence and continuous dependence of large solutions forthe magneto- hydrodynamic equations, Z. Angew. Math. Phys. , 54 (2003), 608–632. OPTIMAL DECAY FOR COMPRESSIBLE MHD EQUATIONS 33

work page 2003
[12]

Chen and Z

Q. Chen and Z. Tan, Global existence and convergence rates o f smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Anal., 72 (2010), 4438 – 4451

work page 2010
[13]

Q. Chen, C. Miao and Z. Zhang, Global well-posedness for compr essible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math. , 63 (2010), 1173–1224

work page 2010
[14]

Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579–614

work page 2000
[15]

Danchin, Zero Mach number limit in critical spaces for compres sible Navier-Stokes equations, Ann

R. Danchin, Zero Mach number limit in critical spaces for compres sible Navier-Stokes equations, Ann. Sci. ´Ecole Norm. Sup. (4) , 35 (2002), 27–75

work page 2002
[16]

Danchin, On the well-posedness of the incompressible density -dependent Euler equations in the Lp framework, J

R. Danchin, On the well-posedness of the incompressible density -dependent Euler equations in the Lp framework, J. Diﬀerential Equations , 248 (2010), 2130 – 2170

work page 2010
[17]

Danchin, Fourier analysis methods for the compressible Navie r-Stokes equations, preprint

R. Danchin, Fourier analysis methods for the compressible Navie r-Stokes equations, preprint

work page
[18]

Danchin and L

R. Danchin and L. He, The incompressible limit in Lp type critical spaces, Math. Ann. , 1–38

work page
[19]

Danchin and J

R. Danchin and J. Xu, Optimal time-decay estimates for the com pressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal. , 224 (2017), 53–90

work page 2017
[20]

Ducomet and E

B. Ducomet and E. Feireisl, The equations of magnetohydrodyn amics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys. , 266 (2006), 595–629

work page 2006
[21]

J. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys. , 270 (2007), 691–708

work page 2007
[22]

Freist¨ uhler and P

H. Freist¨ uhler and P. Szmolyan, Existence and bifurcation of v iscous proﬁles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal. , 26 (1995), 112–128

work page 1995
[23]

Fujita and T

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal. , 16 (1964), 269–315

work page 1964
[24]

J. Gao, Y. Chen and Z.-A. Yao, Long-time behavior of solution to the compressible magnetohydro- dynamic equations, Nonlinear Anal., 128 (2015), 122–135

work page 2015
[25]

Guo and Y

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Diﬀerential Equations , 37 (2012), 2165–2208

work page 2012
[26]

Hao, Well-posedness to the compressible viscous magnetohy drodynamic system, Nonlinear Anal- ysis: RWA , 12 (2011), 2962–2972

C. Hao, Well-posedness to the compressible viscous magnetohy drodynamic system, Nonlinear Anal- ysis: RWA , 12 (2011), 2962–2972

work page 2011
[27]

Haspot, Existence of global strong solutions in critical spac es for barotropic viscous ﬂuids, Arch

B. Haspot, Existence of global strong solutions in critical spac es for barotropic viscous ﬂuids, Arch. Ration. Mech. Anal. , 202 (2011), 427–460

work page 2011
[28]

Hoﬀ and E

D. Hoﬀ and E. Tsyganov, Uniqueness and continuous dependen ce of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys. , 56 (2005), 791–804

work page 2005
[29]

J. Jia, J. Peng and J. Gao, Well-posedness for compressible MHD systems with highly oscillating initial data, J. Math. Phys. , 57 (2016), 081514

work page 2016
[30]

S. Kawashima, Systems of a hyperbolic-parabolic composite type, with app lications to the equations of magnetohydrodynamics , (Doctoral Thesis, Kyoto University), 1984, URL http://repository.kulib.kyoto-u.ac.jp/dspace/handle /2433/97887

work page 1984
[31]

Kawashima and M

S. Kawashima and M. Okada, Smooth global solutions for the one -dimensional equations in magne- tohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci. , 58 (1982), 384–387

work page 1982
[32]

Kozono and M

H. Kozono and M. Yamazaki, Semilinear heat equations and the Na vier-Stokes equations with dis- tributions in new function spaces as initial data, Comm. Partial Diﬀerential equations , 19 (1994), 959–1014

work page 1994
[33]

A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, 1965

work page 1965
[34]

L. D. Laudau and E. M. Lifshitz, Electrodynamics of continuous media , 2nd edn. Pergamon, New York, 1984

work page 1984
[35]

Li and H

F. Li and H. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A , 141 (2011), 109–126. 34 Q. BIE, Q. W ANG, AND Z.-A. YAO

work page 2011
[36]

D. S. Mitrinovi´ c, J. E. Pe˘ cari´ c and A. M. Fink, Inequalities for functions and their integrals and derivatives, Kluwer Academic Publishers, 1994

work page 1994
[37]

Okita, Optimal decay rate for strong solutions in critical spa ces to the compressible Navier–Stokes equations, J

M. Okita, Optimal decay rate for strong solutions in critical spa ces to the compressible Navier–Stokes equations, J. Diﬀerential Equations , 257 (2014), 3850–3867

work page 2014
[38]

Runst and W

T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, a nd nonlinear partial diﬀerential equations, vol. 3, Walter de Gruyter, 1996

work page 1996
[39]

Shi and J

W. Shi and J. Xu, Large-time behavior of strong solutions to th e compressible magnetohydrodynamic system in the critical framework, J. Hyperbolic Diﬀerential Equations , 15 (2018), 259–290

work page 2018
[40]

Shi and J

W. Shi and J. Xu, Global well-posedness for the compressible ma gnetohydrodynamic system in the critical Lp framework, Math. Methods Appl. Sci. , 42 (2019), 3662–3686

work page 2019
[41]

R. M. Strain and Y. Guo, Almost exponential decay near maxwellia n, Commun. Partial Diﬀerential Equations, 31 (2006), 417–429

work page 2006
[42]

Tan and H

Z. Tan and H. Wang, Optimal decay rates of the compressible ma gnetohydrodynamic equations, Nonlinear Anal.: Real World Applications , 14 (2013), 188–201

work page 2013
[43]

Umeda, S

T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solution s to the linearized equations of electro-magneto-ﬂuid dynamics, Japan J. Appl. Math. , 1 (1984), 435–457

work page 1984
[44]

Wang, Large solutions to the initial-boundary value problem fo r planar magnetohydrodynamics, SIAM J

D. Wang, Large solutions to the initial-boundary value problem fo r planar magnetohydrodynamics, SIAM J. Math. Anal. , 63 (2003), 1424–1441

work page 2003
[45]

Xin and J

Z. Xin and J. Xu, Optimal decay for the compressible Navier-Sto kes equations without additional smallness assumptions, arXiv:1812.11714v2

work page arXiv
[46]

Xu, A low-frequency assumption for optimal time-decay est imates to the compressible Navier- Stokes equations, Commun

J. Xu, A low-frequency assumption for optimal time-decay est imates to the compressible Navier- Stokes equations, Commun. Math. Phys. , URL http://doi.org/10.1007/s00220-019-03415-6

work page doi:10.1007/s00220-019-03415-6
[47]

Zhang and J

J. Zhang and J. Zhao, Some decay estimates of solutions for th e 3-D compressible isentropic mag- netohydrodynamics, Commun. Math. Sci. , 8 (2010), 835–850. (Q. Bie) College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, PR China E-mail address : qybie@126.com (Q. Wang) School of Mathematics, Sun Yat...

work page 2010

[1] [1]

Bahouri, J.-Y

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier analysis and nonlinear partial diﬀerential equa- tions, vol. 343 of Grundlehren der Mathematischen Wissenschaften, Sp ringer, Heidelberg, 2011

work page 2011

[2] [2]

Bian and B

D. Bian and B. Guo, Well-posedness in critical spaces for the full c ompressible MHD equations, Acta Math. Sci. , 33 (2013), 1153–1176

work page 2013

[3] [3]

Bian and B

D. Bian and B. Yuan, Local well-posedness in critical spaces for t he compressible MHD equations, Appl. Anal. , 95 (2016), 239–269

work page 2016

[4] [4]

Bony, Calcul symbolique et propagation des singularit´ es po ur les ´ equations aux d´ eriv´ ees par- tielles non lin´ eaires,Ann

J.-M. Bony, Calcul symbolique et propagation des singularit´ es po ur les ´ equations aux d´ eriv´ ees par- tielles non lin´ eaires,Ann. Sci. ´Ecole Norm. Sup. (4) , 14 (1981), 209–246

work page 1981

[5] [5]

Cabannes, Theoretical magnetoﬂuid-dynamics, Academic Press, NewYork, 1970

H. Cabannes, Theoretical magnetoﬂuid-dynamics, Academic Press, NewYork, 1970

work page 1970

[6] [6]

Cannone, A generalization of a theorem by Kato on Navier-Sto kes equations, Rev

M. Cannone, A generalization of a theorem by Kato on Navier-Sto kes equations, Rev. Mat. Iberoamericana, 13 (1997), 515–542

work page 1997

[7] [7]

Charve and R

F. Charve and R. Danchin, A global existence result for the comp ressible Navier–Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal. , 198 (2010), 233–271

work page 2010

[8] [8]

Chemin and N

J.-Y. Chemin and N. Lerner, Flot de champs de vecteurs non lipsch itziens et ´ equations de Navier- Stokes, J. Diﬀerential Equations , 121 (1995), 314–328

work page 1995

[9] [9]

Chemin, I

J.-Y. Chemin, I. Gallagher, D. Iftimie, J. Ball and D. Welsh, Perfect incompressible ﬂuids, Clarendon Press Oxford, 1998

work page 1998

[10] [10]

Chen and D

G.-Q. Chen and D. Wang, Global solutions of nonlinear magnetohy drodynamics with large initial data, J. Diﬀerential Equations , 182 (2002), 344–376

work page 2002

[11] [11]

Chen and D

G.-Q. Chen and D. Wang, Existence and continuous dependence of large solutions forthe magneto- hydrodynamic equations, Z. Angew. Math. Phys. , 54 (2003), 608–632. OPTIMAL DECAY FOR COMPRESSIBLE MHD EQUATIONS 33

work page 2003

[12] [12]

Chen and Z

Q. Chen and Z. Tan, Global existence and convergence rates o f smooth solutions for the compressible magnetohydrodynamic equations, Nonlinear Anal., 72 (2010), 4438 – 4451

work page 2010

[13] [13]

Q. Chen, C. Miao and Z. Zhang, Global well-posedness for compr essible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math. , 63 (2010), 1173–1224

work page 2010

[14] [14]

Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579–614

work page 2000

[15] [15]

Danchin, Zero Mach number limit in critical spaces for compres sible Navier-Stokes equations, Ann

R. Danchin, Zero Mach number limit in critical spaces for compres sible Navier-Stokes equations, Ann. Sci. ´Ecole Norm. Sup. (4) , 35 (2002), 27–75

work page 2002

[16] [16]

Danchin, On the well-posedness of the incompressible density -dependent Euler equations in the Lp framework, J

R. Danchin, On the well-posedness of the incompressible density -dependent Euler equations in the Lp framework, J. Diﬀerential Equations , 248 (2010), 2130 – 2170

work page 2010

[17] [17]

Danchin, Fourier analysis methods for the compressible Navie r-Stokes equations, preprint

R. Danchin, Fourier analysis methods for the compressible Navie r-Stokes equations, preprint

work page

[18] [18]

Danchin and L

R. Danchin and L. He, The incompressible limit in Lp type critical spaces, Math. Ann. , 1–38

work page

[19] [19]

Danchin and J

R. Danchin and J. Xu, Optimal time-decay estimates for the com pressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal. , 224 (2017), 53–90

work page 2017

[20] [20]

Ducomet and E

B. Ducomet and E. Feireisl, The equations of magnetohydrodyn amics: On the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys. , 266 (2006), 595–629

work page 2006

[21] [21]

J. Fan, S. Jiang and G. Nakamura, Vanishing shear viscosity limit in the magnetohydrodynamic equations, Comm. Math. Phys. , 270 (2007), 691–708

work page 2007

[22] [22]

Freist¨ uhler and P

H. Freist¨ uhler and P. Szmolyan, Existence and bifurcation of v iscous proﬁles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal. , 26 (1995), 112–128

work page 1995

[23] [23]

Fujita and T

H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal. , 16 (1964), 269–315

work page 1964

[24] [24]

J. Gao, Y. Chen and Z.-A. Yao, Long-time behavior of solution to the compressible magnetohydro- dynamic equations, Nonlinear Anal., 128 (2015), 122–135

work page 2015

[25] [25]

Guo and Y

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Diﬀerential Equations , 37 (2012), 2165–2208

work page 2012

[26] [26]

Hao, Well-posedness to the compressible viscous magnetohy drodynamic system, Nonlinear Anal- ysis: RWA , 12 (2011), 2962–2972

C. Hao, Well-posedness to the compressible viscous magnetohy drodynamic system, Nonlinear Anal- ysis: RWA , 12 (2011), 2962–2972

work page 2011

[27] [27]

Haspot, Existence of global strong solutions in critical spac es for barotropic viscous ﬂuids, Arch

B. Haspot, Existence of global strong solutions in critical spac es for barotropic viscous ﬂuids, Arch. Ration. Mech. Anal. , 202 (2011), 427–460

work page 2011

[28] [28]

Hoﬀ and E

D. Hoﬀ and E. Tsyganov, Uniqueness and continuous dependen ce of weak solutions in compressible magnetohydrodynamics, Z. Angew. Math. Phys. , 56 (2005), 791–804

work page 2005

[29] [29]

J. Jia, J. Peng and J. Gao, Well-posedness for compressible MHD systems with highly oscillating initial data, J. Math. Phys. , 57 (2016), 081514

work page 2016

[30] [30]

S. Kawashima, Systems of a hyperbolic-parabolic composite type, with app lications to the equations of magnetohydrodynamics , (Doctoral Thesis, Kyoto University), 1984, URL http://repository.kulib.kyoto-u.ac.jp/dspace/handle /2433/97887

work page 1984

[31] [31]

Kawashima and M

S. Kawashima and M. Okada, Smooth global solutions for the one -dimensional equations in magne- tohydrodynamics, Proc. Japan Acad. Ser. A Math. Sci. , 58 (1982), 384–387

work page 1982

[32] [32]

Kozono and M

H. Kozono and M. Yamazaki, Semilinear heat equations and the Na vier-Stokes equations with dis- tributions in new function spaces as initial data, Comm. Partial Diﬀerential equations , 19 (1994), 959–1014

work page 1994

[33] [33]

A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison-Wesley, Reading, 1965

work page 1965

[34] [34]

L. D. Laudau and E. M. Lifshitz, Electrodynamics of continuous media , 2nd edn. Pergamon, New York, 1984

work page 1984

[35] [35]

Li and H

F. Li and H. Yu, Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations, Proc. Roy. Soc. Edinburgh Sect. A , 141 (2011), 109–126. 34 Q. BIE, Q. W ANG, AND Z.-A. YAO

work page 2011

[36] [36]

D. S. Mitrinovi´ c, J. E. Pe˘ cari´ c and A. M. Fink, Inequalities for functions and their integrals and derivatives, Kluwer Academic Publishers, 1994

work page 1994

[37] [37]

Okita, Optimal decay rate for strong solutions in critical spa ces to the compressible Navier–Stokes equations, J

M. Okita, Optimal decay rate for strong solutions in critical spa ces to the compressible Navier–Stokes equations, J. Diﬀerential Equations , 257 (2014), 3850–3867

work page 2014

[38] [38]

Runst and W

T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, a nd nonlinear partial diﬀerential equations, vol. 3, Walter de Gruyter, 1996

work page 1996

[39] [39]

Shi and J

W. Shi and J. Xu, Large-time behavior of strong solutions to th e compressible magnetohydrodynamic system in the critical framework, J. Hyperbolic Diﬀerential Equations , 15 (2018), 259–290

work page 2018

[40] [40]

Shi and J

W. Shi and J. Xu, Global well-posedness for the compressible ma gnetohydrodynamic system in the critical Lp framework, Math. Methods Appl. Sci. , 42 (2019), 3662–3686

work page 2019

[41] [41]

R. M. Strain and Y. Guo, Almost exponential decay near maxwellia n, Commun. Partial Diﬀerential Equations, 31 (2006), 417–429

work page 2006

[42] [42]

Tan and H

Z. Tan and H. Wang, Optimal decay rates of the compressible ma gnetohydrodynamic equations, Nonlinear Anal.: Real World Applications , 14 (2013), 188–201

work page 2013

[43] [43]

Umeda, S

T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solution s to the linearized equations of electro-magneto-ﬂuid dynamics, Japan J. Appl. Math. , 1 (1984), 435–457

work page 1984

[44] [44]

Wang, Large solutions to the initial-boundary value problem fo r planar magnetohydrodynamics, SIAM J

D. Wang, Large solutions to the initial-boundary value problem fo r planar magnetohydrodynamics, SIAM J. Math. Anal. , 63 (2003), 1424–1441

work page 2003

[45] [45]

Xin and J

Z. Xin and J. Xu, Optimal decay for the compressible Navier-Sto kes equations without additional smallness assumptions, arXiv:1812.11714v2

work page arXiv

[46] [46]

Xu, A low-frequency assumption for optimal time-decay est imates to the compressible Navier- Stokes equations, Commun

J. Xu, A low-frequency assumption for optimal time-decay est imates to the compressible Navier- Stokes equations, Commun. Math. Phys. , URL http://doi.org/10.1007/s00220-019-03415-6

work page doi:10.1007/s00220-019-03415-6

[47] [47]

Zhang and J

J. Zhang and J. Zhao, Some decay estimates of solutions for th e 3-D compressible isentropic mag- netohydrodynamics, Commun. Math. Sci. , 8 (2010), 835–850. (Q. Bie) College of Science & Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, PR China E-mail address : qybie@126.com (Q. Wang) School of Mathematics, Sun Yat...

work page 2010