In this paper, the steady inviscid flows with radial symmetry for the isothermal Euler system are studied in an annulus. We present a complete classification of transonic radially symmetric flow patterns in term of physical boundary conditions at the inner and outer circle. By solving the one side boundary problem, we obtain that there exist accelerating or decelerating smooth transonic flows in an annulus. Moreover, the structural stability of these smooth symmetric transonic flows with nonzero angular velocity are further investigated. Furthermore, we examine the transonic solutions with shocks as well via prescribing suitable boundary conditions on the inner and outer circle.
REFERENCES
1.
R.
Courant
and K. O.
Friedrichs
, Supersonic Flow and Shock Waves
(Interscience Publishers Inc.
, New York
, 1948
).2.
S.
Weng
, Z.
Xin
, and H.
Yuan
, “Steady compressible radially symmetric flows with nonzero angular velocity in an annulus
,” J. Differ. Equations
286
, 433
–454
(2021
).3.
S.
Weng
, Z.
Xin
, and H.
Yuan
, “On some smooth symmetric transonic flows with nonzero angular velocity and vorticity
,” Math. Models Methods Appl. Sci.
31
(13
), 2773
–2817
(2021
).4.
L.
Bers
, Mathematical Aspects of Subsonic and Transonic Gas Dynamics
(Wiley
, New York
, 1958
).5.
C.
Wang
and Z.
Xin
, “On a degenerate free boundary problem and continuous subsonic-sonic flows in a convergent nozzle
,” Arch. Ration. Mech. Anal.
208
(3
), 911
–975
(2013
).6.
C.
Wang
and Z.
Xin
, “On sonic curves of smooth subsonic-sonic and transonic flows
,” SIAM J. Math. Anal.
48
(4
), 2414
–2453
(2016
).7.
C.
Wang
and Z.
Xin
, “Smooth transonic flows of Meyer type in de Laval nozzles
,” Arch. Ration. Mech. Anal.
232
(3
), 1597
–1647
(2019
).8.
C.
Wang
and Z.
Xin
, “Regular subsonic-sonic flows in general nozzles
,” Adv. Math.
380
, 107578
(2021
).9.
G.
Chen
, F.
Huang
, and T.
Wang
, “Subsonic-sonic limit of approximate solutions to multidimensional steady Euler equations
,” Arch. Ration. Mech. Anal.
219
, 719
–740
(2016
).10.
C.
Xie
and Z.
Xin
, “Global subsonic and subsonic-sonic flows through infinitely long nozzles
,” Indiana Univ. Math. J.
56
, 2991
–3024
(2007
).11.
C.
Xie
and Z.
Xin
, “Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles
,” J. Differ. Equations
248
(11
), 2657
–2683
(2010
).12.
G.
Chen
, C.
Dafermos
, M.
Slemrod
, and D.
Wang
, “On two-dimensional sonic-subsonic flow
,” Commun. Math. Phys.
271
(3
), 635
–647
(2007
).13.
H.
Feimin
, T.
Wang
, and Y.
Wang
, “On multi-dimensional sonic-subsonic flow
,” Acta Math. Sin.
31
(6
), 2131
–2140
(2011
).14.
X.
Zhouping
and W.
Shangkun
, “A deformation-curl decomposition for three dimensional steady Euler equations
,” Sci. Sin. Math.
49
, 307
–320
(2019
).15.
S.
Weng
, “A deformation-curl-Poisson decomposition to the three dimensional steady Euler-Poisson system with applications
,” J. Differ. Equations
267
(11
), 6574
–6603
(2019
).16.
Z.
Xin
and H.
Yin
, “The transonic shock in a nozzle, 2-D and 3-D complete Euler systems
,” J. Differ. Equations
245
(4
), 1014
–1085
(2008
).17.
Z.
Xin
and H.
Yin
, “Transonic shock in a nozzle I: 2D case
,” Commun. Pure Appl. Math.
58
(8
), 999
–1050
(2005
).18.
J.
Li
, Z.
Xin
, and H.
Yin
, “A free boundary value problem for the full Euler system and 2-D transonic shock in a large variable nozzle
,” Math. Res. Lett.
16
, 777
–796
(2009
).19.
J.
Li
, Z.
Xin
, and H.
Yin
, “On transonic shocks in a conic divergent nozzle with axi-symmetric exit pressures
,” J. Differ. Equations
248
, 423
–469
(2010
).20.
S.
Weng
, C.
Xie
, and Z.
Xin
, “Structural stability of the transonic shock problem in a divergent three-dimensional axisymmetric perturbed nozzle
,” SIAM J. Math. Anal.
53
, 279
–308
(2021
).21.
S.
Weng
and Z.
Zhang
, “Two dimensional subsonic and subsonic-sonic spiral flows outside a porous body
,” Acta Math. Sci.
42
(4
), 1569
–1584
(2022
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
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