In this paper, we first give the global stability of the steady flow of the one-dimensional rotating shallow water equations, by using wave decomposition and uniform prior estimates. Under the assumption that the boundary data is temporal periodic, we can conclude that the classical solution is also temporal periodic.
REFERENCES
1.
B.
Galperin
, H.
Nakano
, H.-P.
Huang
, and S.
Sukoriansky
, “The ubiquitous zonal jets in the atmospheres of giant planets and Earth’s oceans
,” Geophys. Res. Lett.
31
, L13303
, (2004
).2.
V.
Zeitlin
, S. B.
Medvedev
, and R.
Plougonven
, “Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 1. Theory
,” J. Fluid Mech.
481
, 269
–290
(2003
).3.
C. G.
Rossby
, “On the mutual adjustment of pressure and velocity distributions in certain simple current systems, II
,” J. Mar. Res.
1
, 239
–263
(1938
).4.
5.
B.
Cheng
and A.
Mahalov
, “Euler equation on a fast rotating sphere–Time-averages and zonal flows
,” Eur. J. Mech. B/Fluids
37
, 48
–58
(2013
).6.
B.
Galperin
and P. L.
Read
, Zonal Jets: Phenomenology, Genesis, and Physics
(Cambridge University Press
, 2019
).7.
B.
Cheng
, P.
Qu
, and C.
Xie
, “Singularity formation and global existence of classical solutions for one-dimensional rotating shallow water system
,” SIAM J. Math. Anal.
50
, 2486
–2508
(2018
).8.
Y.
Huang
and C.
Xie
, “Formation of singularity for the classical solutions of the rotating shallow water system
,” J. Differ. Equations
348
, 45
–65
(2023
).9.
P. D.
Lax
, “Development of singularities of solutions of nonlinear hyperbolic partial differential equations
,” J. Math. Phys.
5
, 611
–613
(1964
).10.
L.
Ta-tsien
, Z.
Yi
, and K.
De-xing
, “Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems
,” Commun. Partial Differ. Equations
19
, 1263
–1317
(1994
).11.
J.
Ye
, H.
Guo
, and Y.
Hu
, “An initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations
,” J. Math. Anal. Appl.
527
, 127422
(2023
).12.
P. A.
Gilman
, “Magnetohydrodynamic ‘shallow water’ equations for the solar tachocline
,” Astrophys. J.
544
, L79
(2000
).13.
J. A.
Rossmanith
, “A wave propagation method with constrained transport for ideal and shallow water magnetohydrodynamics
,” Ph.D. thesis, ProQuest LLC, University of Washington
, Ann Arbor, MI
, 2002
, p. 175
.14.
B.
Cheng
and C.
Xie
, “On the classical solutions of two dimensional inviscid rotating shallow water system
,” J. Differ. Equations
250
, 690
–709
(2011
).15.
Y.
Hu
and Y.
Qian
, “An initial-boundary value problem for the two-dimensional rotating shallow water equations with axisymmetry
,” Differ. Integr. Equations
35
, 611
–640
(2022
).16.
C.
Hao
, L.
Hsiao
, and H.-L.
Li
, “Cauchy problem for viscous rotating shallow water equations
,” J. Differ. Equations
247
, 3234
–3257
(2009
).17.
H.
Liu
and E.
Tadmor
, “Rotation prevents finite-time breakdown
,” Physica D
188
, 262
–276
(2004
).18.
F.
Wei
, J.
Liu
, and H.
Yuan
, “Global stability to steady supersonic solutions of the 1-D compressible Euler equations with frictions
,” J. Math. Anal. Appl.
495
, 124761
(2021
).19.
F.
Wei
and J.
Liu
, “Global stability to steady supersonic Rayleigh flows in one-dimensional duct
,” Chin. Ann. Math., Ser. B
45
, 279
–296
(2024
).20.
P.
Gao
and J.
Liu
, “Global stability to steady supersonic flows with mass addition in one-dimensional duct
,” J. Math. Anal. Appl.
525
, 127128
(2023
).21.
H.
Cai
and Z.
Tan
, “Time periodic solutions to the compressible Navier-Stokes-Poisson system with damping
,” Commun. Math. Sci.
15
, 789
–812
(2017
).22.
N.
Tsuge
, “Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force
,” Nonlinear Anal.: Real World Appl.
53
, 103080
(2020
).23.
H.
Yuan
, “Time-periodic isentropic supersonic Euler flows in one-dimensional ducts driving by periodic boundary conditions
,” Acta Math. Sci.
39
, 403
–412
(2019
).24.
H.
Yu
, X.
Zhang
, and J.
Sun
, “Global existence and stability of time-periodic solution to isentropic compressible Euler equations with source term
,” Commun. Math. Sci.
21
, 1333
–1348
(2023
).25.
F.
John
, “Formation of singularities in one-dimensional nonlinear wave propagation
,” Commun. Pure Appl. Math.
27
, 377
–405
(1974
).26.
D.-x.
Kong
, Cauchy Problem for Quasilinear Hyperbolic Systems
, MSJ Memoirs Vol. 6
(Mathematical Society of Japan
, Tokyo
, 2000
), p. viii+213
.27.
T. T.
Li
, Global Classical Solutions for Quasilinear Hyperbolic Systems
, RAM: Research in Applied Mathematics Vol. 32
(Masson; John Wiley & Sons, Ltd.
, Paris; Chichester
, 1994
), p. x+315
.28.
L.
Ta-Tsien
, Z.
Yi
, and K.
De-Xing
, “Global classical solutions for general quasilinear hyperbolic systems with decay initial data
,” Nonlinear Anal.
28
, 1299
–1332
(1997
).29.
T.
Li
and L.
Wang
, “Blow-up mechanism of classical solutions to quasilinear hyperbolic systems in the critical case
,” Chin. Ann. Math., Ser. B
27
, 53
–66
(2006
).30.
L.
Yu
, “Semi-global C1 solution to the mixed initial-boundary value problem for a kind of quasilinear hyperbolic systems
,” Chin. Ann. Math., Ser. A
25
, 549
–560
(2004
); J. Contemp. Math. 25(2004), 335
–346
(2005
).31.
T. T.
Li
and W. C.
Yu
, Boundary Value Problems for Quasilinear Hyperbolic Systems
, Duke University Mathematics Series Vol. V
(Duke University, Mathematics Department
, Durham, NC
, 1985
), p. viii+325
.32.
T.-T.
Li
and Y.
Jin
, “Semi-global C1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems
,” Chin. Ann. Math.
22
, 325
–336
(2001
).33.
T.-T.
Li
and B.
Rao
, “Local exact boundary controllability for a class of quasilinear hyperbolic systems
,” Chin. Ann. Math.
23
, 209
–218
(2002
), Dedicated to the Memory of Jacques-Louis Lions.34.
Z.
Wang
, “Exact boundary controllability for non-autonomous quasilinear wave equations
,” Math. Methods Appl. Sci.
30
, 1311
–1327
(2007
).© 2025 Author(s). Published under an exclusive license by AIP Publishing.
2025
Author(s)
You do not currently have access to this content.
