In this paper, we consider the isentropic compressible magnetohydrodynamic equations on an unbounded three-dimensional domain with a compact Lipschitz boundary and establish the existence of globally defined weak solutions satisfying the energy inequality in differential form.

1.
Beal
,
J. T.
,
Kato
,
T.
, and
Majda
,
A.
, “
Remarks on the breakdown of smooth solutions for the 3D Euler equations
,”
Commun. Math. Phys.
94
,
61
66
(
1984
).
2.
Caflisch
,
R. E.
,
Kapper
,
I.
, and
Steele
,
G.
, “
Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD
,”
Commun. Math. Phys.
184
,
443
455
(
1997
).
3.
Chen
,
Q.
,
Miao
,
C.
, and
Zhang
,
Z.
, “
The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations
,”
Commun. Math. Phys.
275
(
3
),
861
872
(
2007
).
4.
Dell’Oro
,
F.
and
Feireisl
,
E.
, “
On the energy inequality for weak solutions to the Navier-Stokes equations of compressible fluids on unbounded domains
,”
Nonlinear Anal.
128
,
136
148
(
2015
).
5.
Fan
,
J.
,
Li
,
F.
,
Nakamura
,
G.
, and
Tan
,
Z.
, “
Regularity criteria for the three-dimensional magnetohydrodynamic equations
,”
J. Differ. Equations
256
,
2858
2875
(
2014
).
6.
Feireisl
,
E.
,
Novotný
,
A.
, and
Petzeltová
,
H.
, “
On the existence of globally defined weak solutions to the Navier-Stokes equations
,”
J. Math. Fluid Mech.
3
,
358
392
(
2001
).
7.
Feireisl
,
E.
,
Novotný
,
A.
, and
Petzeltová
,
H.
, “
On the domain dependence of solutions to the Navier-Stokes equations of a barotropic fluid
,”
Math. Methods Appl. Sci.
25
,
1045
1073
(
2002
).
8.
He
,
C.
and
Wang
,
Y.
, “
On the regularity criteria for weak solutions to the magnetohydrodynamic equations
,”
J. Differ. Equations
238
,
1
17
(
2007
).
9.
Jiang
,
S.
,
Ju
,
Q.
,
Li
,
F.
, and
Xin
,
Z.
, “
Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data
,”
Adv. Math.
259
,
384
420
(
2014
).
10.
Liu
,
X.
, “
Blow-up criteria for the three-dimensional compressible magnetohydrodynamic equations
Math. Phys. Anal. Geom.
(to be published).
11.
Lu
,
M.
,
Du
,
Y.
, and
Yao
,
Z.
, “
Blow-up criterion for compressible MHD equations
,”
J. Math. Anal. Appl.
379
,
425
438
(
2011
).
12.
Novotný
,
A.
and
Straškraba
,
I.
,
Introduction to the Mathematical Theory of Compressible Flow
(
Oxford University Press
,
New York
,
2004
).
13.
Tong
,
T.
and
Wang
,
Y.
, “
On the ideal compressible magnetohydrodynamic equations
,”
J. Math. Anal. Appl.
430
,
1
19
(
2015
).
14.
Wang
,
S.
and
Xu
,
Z.
, “
Low Mach number limit of non-isentropic magnetohydrodynamic equations in a bounded domain
,”
Nonlinear Anal.
105
,
102
119
(
2014
).
15.
Wang
,
Y.
and
Li
,
S.
, “
Global regularity for the Cauchy problem of three-dimensional compressible magnetohydrodynamics equations
,”
Nonlinear Anal.
18
,
23
33
(
2014
).
16.
Wu
,
J.
, “
Bounds and new approaches for the 3D MHD equations
,”
J. Nonlinear Sci.
12
,
395
413
(
2002
).
17.
Yang
,
X.
, “
Low Mach number limit of the compressible Hall-magnetohydrodynamic system
,”
Nonlinear Anal.
25
,
118
126
(
2015
).
18.
Yang
,
Y.
,
Dou
,
C.
, and
Ju
,
Q.
, “
Weak-strong uniqueness property for the magnetohydrodynamic equations of three-dimensional compressible isentropic flows
,”
Nonlinear Anal.
85
,
23
30
(
2013
).
19.
Zhou
,
Y.
, “
Remarks on regularities for the 3D MHD equations
,”
Discrete Contin. Dyn. Syst.
5
,
881
886
(
2005
).
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