This paper deals with a model the equatorial water waves with the Coriolis effect in the rotating fluid, which is called rotation-two-component Camassa–Holm system. The purpose of this work is to utilize the pseudo-parabolic regularization to establish the existence and uniqueness of weak solutions in a lower order Sobolev space with .
REFERENCES
1.
L.
Fan
, H.
Gao
, and Y.
Liu
, “On the rotation-two-component Camassa-Holm system modelling the equatorial water waves
,” Adv. Math.
291
, 59
–89
(2016
).2.
A.
Fokas
and B. S.
Fuchssteiner
, “Symplectic structures, their Bäcklund transformation and hereditary symmetries
,” Physica D
4
, 47
–66
(1981
).3.
R.
Camassa
and D. D.
Holm
, “An integrable shallow water equation with peaked solitons
,” Phys. Rev. Lett.
71
, 1661
–1664
(1993
).4.
A.
Constantin
, “The trajectories of particles in Stokes waves
,” Invent. Math.
166
, 523
–535
(2006
).5.
A.
Constantin
, “Particle trajectories in extreme Stokes waves
,” IMA J. Appl. Math.
77
, 293
–307
(2012
).6.
A.
Constantin
and J.
Escher
, “Analyticity of periodic traveling free surface water waves with vorticity
,” Ann. Math.
173
, 559
–568
(2011
).7.
J. F.
Toland
, “Stokes waves
,” Topol. Methods Nonlinear Anal.
7
, 1
–48
(1996
).8.
A.
Constantin
and W. A.
Strauss
, “Stability of peakons
,” Commun. Pure Appl. Math.
53
, 603
–610
(2000
).9.
A.
Constantin
and W. A.
Strauss
, “Stability of the Camassa-Holm solitons
,” J. Nonlinear Sci.
12
, 415
–422
(2002
).10.
R.
Camassa
, D. D.
Holm
, and J. M.
Hyman
, “A new integrable shallow water equation
,” Adv. Appl. Mech.
31
, 1
–33
(1994
).11.
A.
Constantin
and J.
Escher
, “Wave breaking for nonlinear nonlocal shallow water equations
,” Acta Math.
181
, 229
–243
(1998
).12.
A.
Bressan
and A.
Constantin
, “Global conservative solutions of the Camassa-Holm equation
,” Arch. Ration. Mech. Anal.
183
, 215
–239
(2007
).13.
A.
Bressan
and A.
Constantin
, “Global dissipative solutions of the Camassa-Holm equation
,” Anal. Appl.
05
, 1
–27
(2007
).14.
G.
Rodríguez-Blanco
, “On the Cauchy problem for the Camassa-Holm equation
,” Nonlinear Anal.
46
, 309
–327
(2001
).15.
A.
Constantin
and J.
Escher
, “Well-posedness, global existence, and blow up phenomena for a periodic quasi-linear hyperbolic equation
,” Commun. Pure Appl. Math.
51
, 475
–504
(1998
).16.
A.
Costantin
and J.
Escher
, “Global existence of solutions and blow-up for a shallow water equation
,” Ann. Sc. Norm. Super. Pisa CL. Sci.
26
, 303
–328
(1998
).17.
R.
Danchin
, “A few remarks on the Camassa-Holm equation
,” Differ. Integral Equations
14
, 953
–988
(2001
).18.
L.
Brandolese
, “Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces
,” Int. Math. Res. Not. IMRN
2012
, 5161
–5181
.19.
A. A.
Himonas
, G.
Misiołek
, G.
Ponce
, and Y.
Zhou
, “Persistence properties and unique continuation of solutions of the Camassa-Holm equation
,” Commun. Math. Phys.
271
, 511
–522
(2007
).20.
A.
Constantin
and L.
Molinet
, “Global weak solutions for a shallow water equation
,” Commun. Math. Phys.
211
, 45
–61
(2000
).21.
Z.
Xin
and P.
Zhang
, “On the weak solutions to a shallow water equation
,” Commun. Pure Appl. Math.
53
, 1411
–1433
(2000
).22.
M.
Chen
, S.-Q.
Liu
, and Y.
Zhang
, “A 2-component generalization of the Camassa-Holm equation and its solutions
,” Lett. Math. Phys.
75
, 1
–15
(2006
).23.
G.
Falqui
, “On a Camassa-Holm type equation with two dependent variables
,” J. Phys. A: Math. Gen.
39
, 327
–342
(2006
).24.
A.
Constantin
and R. I.
Ivanov
, “On an integrable two-component Camassa-Holm shallow water system
,” Phys. Lett. A
372
, 7129
–7132
(2008
).25.
R.
Ivanov
, “Two-component integrable systems modelling shallow water waves: The constant vorticity case
,” Wave Motion
46
, 389
–396
(2009
).26.
C.
Guan
and Z.
Yin
, “Global weak solutions for a two-component Camassa-Holm shallow water system
,” J. Funct. Anal.
260
, 1132
–1154
(2011
).27.
C.
Guan
, H.
He
, and Z.
Yin
, “Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system
,” Nonlinear Anal. Real World Appl.
25
, 219
–237
(2015
).28.
C.
Guan
and Z.
Yin
, “Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system
,” J. Differ. Equations
248
, 2003
–2014
(2010
).29.
G.
Gui
and Y.
Liu
, “On the global existence and wave-breaking criteria for the two-component Camassa-Holm system
,” J. Funct. Anal.
258
, 4251
–4278
(2010
).30.
G.
Gui
and Y.
Liu
, “On the Cauchy problem for the two-component Camassa-Holm system
,” Math. Z.
268
, 45
–66
(2011
).31.
J.
Escher
, O.
Lechtenfeld
, and Z.
Yin
, “Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation
,” Discrete Contin. Dyn. Syst.
19
, 493
–513
(2007
).32.
L.
Wei
, Y.
Wang
, and H.
Zhang
, “Breaking waves and persistence property for a two-component Camassa-Holm system
,” J. Math. Anal. Appl.
445
, 1084
–1096
(2017
).33.
R. M.
Chen
and Y.
Liu
, “Wave breaking and global existence for a generalized two-component Camassa-Holm system
,” Int. Math. Res. Not. IMRN.
6
, 1381
–1416
(2011
).34.
Y.
Han
, F.
Guo
, and H.
Gao
, “On solitary waves and wave-breaking phenomena for a generalized two-component integrable Dulin-Gottwald-Holm system
,” J. Nonlinear Sci.
23
, 617
–656
(2013
).35.
F.
Guo
, H.
Gao
, and Y.
Liu
, “On the wave-breaking phenomena for the two-component Dullin-Gottwald-Holm system
,” J. London Math. Soc.
86
, 810
–834
(2012
).36.
Y.
Chen
, H.
Gao
, and Y.
Liu
, “On the Cauchy problem for the two-component Dullin-Gottwald-Holm system
,” Discrete Contin. Dyn. Syst.
33
, 3407
–3441
(2013
).37.
X.
Liu
and Z.
Yin
, “Local well-posedness and stability of solitary waves for the two-component Dullin-Gottwald-Holm system
,” Nonlinear Anal. Real World Appl.
88
, 1
–15
(2013
).38.
F.
Guo
and R.
Wang
, “On the persistence and unique continuation properties for an integrable two-component Dullin-Gottwald-Holm system
,” Nonlinear Anal.
96
, 38
–46
(2014
).39.
R. M.
Chen
, L.
Fan
, H.
Gao
, and Y.
Liu
, “Breaking waves and solitary waves to the rotation-two-component Camassa-Holm system
,” SIAM J. Math. Anal.
49
, 3573
–3602
(2017
).40.
L.
Zhang
and B.
Liu
, “Well-posedness, blow-up criteria and Gevrey regularity for a rotation-two-component Camassa-Holm system
,” Discrete Contin. Dyn. Syst.
38
, 2655
–2685
(2018
).41.
E.
Fana
and M.
Yuen
, “Peakon weak solutions for the rotation-two-component Camassa-Holm system
,” Appl. Math. Lett.
97
, 53
–59
(2019
).42.
Y.
Guo
and Z.
Yin
, “The rotational speed limit and the blow-up phenomena of the rotation 2-component Camassa-Holm system
,” Monatsh. Math.
190
, 301
–332
(2019
).43.
C.
Wang
, R.
Zeng
, S.
Zhou
, B.
Wang
, and C.
Mu
, “Continuity for the rotation-two-component Camassa-Holm system
,” Discrete Contin. Dyn. Syst. Ser. B
24
, 6633
–6652
(2019
).44.
Y. A.
Li
and P. J.
Olver
, “Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation
,” J. Differ. Equations
162
, 27
–63
(2000
).45.
T.
Kato
and G.
Ponce
, “Commutator estimates and Navier-Stokes equations
,” Commun. Pure Appl. Math.
41
, 203
–208
(1988
).46.
J. L.
Bona
and R.
Smith
, “The initial value problem for the Korteweg-de Vries equation
,” Philos. Trans. R. Soc. London, Ser. A
278
, 555
–601
(1975
).47.
S. Y.
Lai
and S. Y.
Wu
, “Existence of weak solutions in lower order Sobolev space for a Camassa-Holm-type equation
,” J. Phys. A: Math. Theor.
43
, 13pp
(2010
).48.
W.
Walter
, Differential and Integral Inequalities
(Springer-Verlag
, New York
, 1970
).49.
J. L.
Lions
, Quelques Méthodes dé Résolution des Problémes aux Limites Non linéaires
(Dunod
, Gauthier-Villars, Paris
, 1969
).© 2020 Author(s).
2020
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