In this paper, we investigate the global well-posedness and stability of steady supersonic Euler flows in a quasi-one-dimensional convergent nozzle. First, we show that there exists a critical nozzle length L1, the global well-posedness of steady supersonic flows can be obtained and the explicit solution is given as long as the nozzle length L < L1. Then we prove the global stability of steady supersonic flows under small perturbations of initial-boundary values by wave decomposition.

1.
B.
Duan
,
A.
Lan
, and
Z.
Luo
, “
Transonic shock solutions to the Euler system in divergent-convergent nozzles
,”
Acta Math. Sci.
42
,
1536
1546
(
2022
).
2.
T. P.
Liu
, “
Nonlinear stability and instability of transonic flows through a nozzle
,”
Commun. Math. Phys.
83
,
243
260
(
1982
).
3.
C.
Wang
and
Z.
Xin
, “
Global smooth supersonic flows in infinite expanding nozzles
,”
SIAM J. Math. Anal.
47
,
3151
3211
(
2015
).
4.
G.
Xu
and
H. C.
Yin
, “
Three-dimensional global supersonic Euler flows in the infinitely long divergent nozzles
,”
SIAM J. Math. Anal.
53
,
133
180
(
2021
).
5.
F. L.
Wei
,
J. L.
Liu
, and
H. R.
Yuan
, “
Global stability to steady supersonic solutions of the 1-D compressible Euler equations with frictions
,”
J. Math. Anal. Appl.
495
,
124761
(
2021
).
6.
P.
Gao
and
J. L.
Liu
, “
Global stability to steady supersonic flows with mass addition in one-dimensional duct
,”
J. Math. Anal. Appl.
525
,
127128
(
2023
).
7.
G.
Xu
and
H. C.
Yin
, “
On the existence and stability of a global subsonic flow in a 3D infinitely long cylindrical nozzle
,”
Chin. Ann. Math., Ser. B
31
,
163
190
(
2010
).
8.
L.
Liu
,
G.
Xu
, and
H. R.
Yuan
, “
Stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations
,”
Adv. Math.
291
,
696
757
(
2016
).
9.
J. L.
Gao
and
H. R.
Yuan
, “
Stability of stationary subsonic compressible Euler flows with mass-additions in two-dimensional straight ducts
,”
J. Differ. Equ.
334
,
87
156
(
2022
).
10.
T. P.
Liu
, “
Transonic gas flow in a duct of varying area
,”
Arch. Ration. Mech. Anal.
80
,
1
18
(
1982
).
11.
J.
Rauch
,
C. J.
Xie
, and
Z. P.
Xin
, “
Global stability of steady transonic Euler shocks in quasi-one-dimensional nozzles
,”
J. Math. Pures Appl.
99
,
395
408
(
2013
).
12.
J.
Liao
and
Z.
Tan
, “
Transonic shock solutions of the steady Euler flow in quasi-one-dimensional convergent nozzles
,”
J. Differ. Equ.
372
,
657
671
(
2023
).
13.
B.
Duan
,
Z.
Luo
, and
J. J.
Xiao
, “
Transonic shock solutions to the Euler–Poisson system in quasi-one-dimensional nozzles
,”
Commun. Math. Sci.
14
,
1023
1047
(
2016
).
14.
T.
Luo
,
J.
Rauch
,
C. J.
Xie
, and
Z. P.
Xin
, “
Stability of transonic shock solutions for one-dimensional Euler–Poisson equations
,”
Arch. Ration. Mech. Anal.
202
,
787
827
(
2011
).
15.
D. X.
Kong
, “
Cauchy problem for quasilinear hyperbolic systems
,” in
MSJ Memoirs
(
The Math. Society of Japan
,
Tokyo
,
2000
), Vol.
6
.
16.
F.
John
, “
Formation of singularities in one-dimensional nonlinear wave propagation
,”
Commun. Pure Appl. Math.
27
,
377
405
(
1974
).
17.
T. T.
Li
,
Y.
Zhou
, and
D. X.
Kong
, “
Global classical solutions for general quasilinear hyperbolic systems with decay initial data
,”
Nonlinear Anal.
28
,
1299
1332
(
1997
).
18.
T. T.
Li
and
L. B.
Wang
, “
Blow-up mechanism of classical solutions to quasilinear hyperbolic systems in the critical case
,”
Chin. Ann. Math., Ser. B
27
,
53
66
(
2006
).
19.
T. T.
Li
and
W. C.
Yu
, “
Boundary value problems for quasilinear hyperbolic systems
,” Duke University Math. Series V (
1985
).
You do not currently have access to this content.