We consider the existence, uniqueness and nonlinear stability of stationary solutions for the three-dimensional compressible micropolar fluids with the external force of general form. More precisely, with the weighted L2 and L estimates on solutions to the linearized problem, we show the existence and uniqueness of stationary solutions in some suitable function space by the contraction mapping principle. The stability of the steady flow is studied by an elementary energy method.

1.
A. C.
Eringen
, “
Theory of micropolar fluids
,”
Indiana Univ. Math. J.
16
,
1
18
(
1966
).
2.
A. C.
Eringen
,
Microcontinuum field theories: I. Foundations and Solids
(
Springer
,
New York
,
1999
).
3.
G.
Lukaszewicz
,
Micropolar Fluids: Theory and Applications
,
Modeling and Simulation in Science, Engineering and Technology
(
Birkhäuser
,
Baston
,
1999
).
4.
N.
Mujaković
, “
Global in time estimates for one-dimensional compressible viscous micropolar fluid model
,”
Glas. Mat.
40
(
1
),
103
120
(
2005
).
5.
N.
Mujaković
, “
Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: A local existence theorem
,”
Ann. Univ. Ferrara, Sez. 7: Sci. Mat.
53
(
2
),
361
379
(
2007
).
6.
N.
Mujaković
, “
The existence of a global solution for one dimensional compressible viscous micropolar fluid with non-homogeneous boundary conditions for temperature
,”
Nonlinear Anal.
19
,
19
30
(
2014
).
7.
H. B.
Cui
and
H. Y.
Yin
, “
Stationary solutions to the one-dimensional micropolar fluid model in a half line: Existence, stability and convergence rate
,”
J. Math. Anal. Appl.
449
(
1
),
464
489
(
2017
).
8.
H. Y.
Yin
, “
Stability of stationary solutions for inflow problem on the micropolar fluid model
,”
Z. Angew. Math. Phys.
68
(
2
),
44
(
2017
).
9.
J. R.
Su
, “
Incompressible limit of a compressible micropolar fluid model with general initial data
,”
Nonlinear Anal.
132
,
1
24
(
2016
).
10.
J.
Jin
and
R.
Duan
, “
Stability of rarefaction waves for 1D compressible viscous micropolar fluid model
,”
J. Math. Anal. Appl.
450
(
2
),
1123
1143
(
2017
).
11.
Y. M.
Qin
,
T. G.
Wang
, and
G. L.
Hu
, “
The Cauchy problem for a 1D compressible viscous micropolar fluid model: Analysis of the stabilization and the regularity
,”
Nonlinear Anal.
13
(
3
),
1010
1029
(
2012
).
12.
I.
Dražić
and
N.
Mujaković
, “
3-D flow of a compressible viscous micropolar fluid with spherical symmetry: A local existence theorem
,”
Bound. Value Probl.
2012
,
69
.
13.
I.
Dražić
and
N.
Mujaković
, “
3-D flow of a compressible viscous micropolar fluid with spherical symmetry: A global existence theorem
,”
Bound. Value Probl.
2015
,
98
.
14.
N.
Mujaković
and
I.
Dražić
, “
3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Uniqueness of a generalized solution
,”
Bound. Value Probl.
2014
,
226
.
15.
I.
Dražić
and
N.
Mujaković
, “
3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Large time behavior of the solution
,”
J. Math. Anal. Appl.
431
,
545
568
(
2015
).
16.
I.
Dražić
,
L.
Simčić
, and
N.
Mujaković
, “
3-D flow of a compressible viscous micropolar fluid with spherical symmetry: Regularity of the solution
,”
J. Math. Anal. Appl.
438
,
162
183
(
2016
).
17.
I.
Dražić
, “
Three-dimensional flow of a compressible viscous micropolar fluid with cylindrical symmetry: A global existence theorem
,”
Math. Methods Appl. Sci.
40
,
4785
4801
(
2017
).
18.
I.
Dražić
,
N.
Mujaković
, and
N.
Črnjarić-Žic
, “
Three-dimensional compressible viscous micropolar fluid with cylindrical symmetry: Derivation of the model and a numerical solution
,”
Math. Comput. Simul.
140
,
107
124
(
2017
).
19.
N.
Mujaković
,
L.
Simčić
, and
I.
Dražić
, “
3-D flow of a compressible viscous micropolar fluid with cylindrical symmetry: Uniqueness of a generalized solution
,”
Math. Methods Appl. Sci.
40
(
7
),
2686
2701
(
2017
).
20.
M. T.
Chen
, “
Blowup criterion for viscous, compressible micropolar fluids with vacuum
,”
Nonlinear Anal.
13
(
2
),
850
859
(
2012
).
21.
M. T.
Chen
,
B.
Huang
, and
J. W.
Zhang
, “
Blowup criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum
,”
Nonlinear Anal.
79
,
1
11
(
2013
).
22.
M. T.
Chen
,
X. Y.
Xu
, and
J. W.
Zhang
, “
Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum
,”
Commun. Math. Sci.
13
,
225
247
(
2015
).
23.
Q. Q.
Liu
and
P. X.
Zhang
, “
Optimal time decay of the compressible micropolar fluids
,”
J. Differ. Equ.
260
(
10
),
7634
7661
(
2016
).
24.
Z. G.
Wu
and
W. K.
Wang
, “
The pointwise estimates of diffusion wave of the compressible micropolar fluids
,”
J. Differ. Equ.
265
,
2544
2576
(
2018
).
25.
X.
Hou
and
H. Y.
Peng
, “
Global existence for a class of large solution to the three-dimensional micropolar fluid equations with vacuum
,”
J. Math. Anal. Appl.
498
,
124931
(
2021
).
26.
L. L.
Tong
,
R. H.
Pan
, and
Z.
Tan
, “
Decay estimates of solutions to the compressible micropolar fluids system in R3
,”
J. Differ. Equ.
293
,
520
552
(
2021
).
27.
J. L.
Boldrini
,
M.
Durán
, and
M. A.
Rojas-Medar
, “
Existence and uniqueness of strong solution for the incompressible micropolar fluid equations in domains of R3
,”
Ann. Univ. Ferrara, Sez. 7: Sci. Mat.
56
(
1
),
37
51
(
2010
).
28.
Q. L.
Chen
and
C. X.
Miao
, “
Global well-posedness for the micropolar fluid system in critical Besov spaces
,”
J. Differ. Equ.
252
,
2698
2724
(
2012
).
29.
G.
Lukaszewicz
, “
On non-stationary flows of incompressible asymmetric fluids
,”
Math. Methods Appl. Sci.
13
(
3
),
219
232
(
1990
).
30.
P.
Braz e Silva
and
E. G.
Santos
, “
Global weak solutions for asymmetric incompressible fluids with variable density
,”
C. R. Math.
346
,
575
578
(
2008
).
31.
Q. Q.
Liu
and
P. X.
Zhang
, “
Long-time behavior of solution to the compressible micropolar fluids with external force
,”
Nonlinear Anal.
40
,
361
376
(
2018
).
32.
A.
Matsumura
and
T.
Nishida
, “
The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids
,”
Proc. Jpn. Acad., Ser. A
55
,
337
342
(
1979
).
33.
A.
Matsumura
and
T.
Nishida
, “
The initial value problem for the equations of motion of viscous and heat-conductive gases
,”
Kyoto J. Math.
20
,
67
104
(
1980
).
34.
A.
Matsumura
and
T.
Nishida
, “
Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids
,”
Commun. Math. Phys.
89
(
4
),
445
464
(
1983
).
35.
J. Z.
Qian
and
H.
Yin
, “
On the stationary solutions of the full compressible Navier-Stokes equations and its stability with respect to initial disturbance
,”
J. Differ. Equ.
237
,
225
256
(
2007
).
36.
Y.
Shibata
and
K.
Tanaka
, “
On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance
,”
J. Math. Soc. Jpn.
55
(
3
),
797
826
(
2003
).
37.
A.
Novotny
and
M.
Padula
, “
Physically reasonable solutions to steady compressible Navier-Stokes equations in 3D-exterior domains (v = 0)
,”
Kyoto J. Math.
36
,
389
422
(
1996
).
38.
A.
Novotny
and
M.
Padula
, “
Physically reasonable solutions to steady compressible Navier-Stokes equations in 3D-exterior domains (v ≠ 0)
,”
Math. Ann.
308
,
439
489
(
1997
).
39.
K.
Tanaka
, “
A stability of steady flow of compressible viscous fluid with respect to initial disturbance (v ≠ 0)
,”
Math. Methods Appl. Sci.
29
(
12
),
1451
1466
(
2006
).
40.
H.
Cai
and
Z.
Tan
, “
Existence and stability of stationary solutions to the compressible Navier-Stokes-Poisson equations
,”
Nonlinear Anal.
32
,
260
293
(
2016
).
41.
Z. Z.
Chen
and
H. J.
Zhao
, “
Existence and nonlinear stability of stationary solutions to the full compressible Navier–Stokes–Korteweg system
,”
J. Math. Pures Appl.
101
,
330
371
(
2014
).
42.
N.
Ju
, “
Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space
,”
Commun. Math. Phys.
251
,
365
376
(
2004
).
43.
S.
Kawashima
, “
Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics
,” Ph.D. Thesis, (
Kyoto University
,
1983
).
You do not currently have access to this content.