Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect

Runzhang Xu; Yanbing Yang

doi:10.3934/dcds.2020288

Article Contents

2020, Volume 40, Issue 11: 6527-6547. Doi: 10.3934/dcds.2020288

This issue Previous Article On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces Next Article The large diffusion limit for the heat equation in the exterior of the unit ball with a dynamical boundary condition

Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect

Runzhang Xu and
Yanbing Yang^,

College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang 150001, China

^* Corresponding author: Yanbing Yang
^* Corresponding author: Yanbing Yang

Received: April 2020

Revised: June 2020

Published: July 2020

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Abstract

Studied herein is the local well-posedness of solutions with the low regularity in the periodic setting for a class of one-dimensional generalized Rotation-Camassa-Holm equation, which is considered as an asymptotic model to describe the propagation of shallow-water waves in the equatorial region with the weak Coriolis effect due to the Earth's rotation. With the aid of the semigroup approach and a refined viscosity technique, the local existence, uniqueness and continuity of periodic solutions in the spatial space $ C^1 $ is established based on the local structure of the dynamics along the characteristics.

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Mathematics Subject Classification: Primary: 35Q53, 35B65, 35A01.

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References

[1]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.
[2]	A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857.
[3]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[4]	C. Cao, D. D. Holm and E. S. Titi, Traveling wave solutions for a class of one-dimensional nonlinear shallow water wave models, J. Dynam. Differential Equations, 16 (2004), 167-178. doi: 10.1023/B:JODY.0000041284.26400.d0.
[5]	P. Cao, F. Li, S. Liu and J. Pan, A conjecture on cluster automorphisms of cluster algebras, Electron. Res. Arch., 27 (2019), 1-6. doi: 10.3934/era.2019006.
[6]	R. M. Chen, G. Gui and Y. Liu, On a shallow-water approximation to the Green-Naghdi equations with the Coriolis effect, Adv. Math., 340 (2018), 106-137. doi: 10.1016/j.aim.2018.10.003.
[7]	Y. Chen, L. Huang and Y. Liu, On the modelling of shallow-water waves with the Coriolis effect, J. Nonlinear Sci., 30 (2020), 93-135. doi: 10.1007/s00332-019-09569-w.
[8]	A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation, Comm. Pure Appl. Math., 51 (1998), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.
[9]	A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586.
[10]	A. Constantin and J. Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z., 233 (2000), 75-91. doi: 10.1007/PL00004793.
[11]	A. Constantin and R. I. Ivanov, Equatorial wave-current interactions, Comm. Math. Phys., 370 (2019), 1-48. doi: 10.1007/s00220-019-03483-8.
[12]	A. Constantin and R. S. Johnson, On the nonlinear, three-dimensional structure of equatorial oceanic flows, J. Phys. Oceanogr., 49 (2019), 2029-2042. doi: 10.1175/JPO-D-19-0079.1.
[13]	A. Constantin and R. S. Johnson, The dynamics of waves interacting with the Equatorial Undercurrent, Geophys. Astrophys. Fluid Dyn., 109 (2015), 311-358. doi: 10.1080/03091929.2015.1066785.
[14]	A. Constantin and B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv., 78 (2003), 787-804. doi: 10.1007/s00014-003-0785-6.
[15]	A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation, Phys. D, 157 (2001), 75-89. doi: 10.1016/S0167-2789(01)00298-6.
[16]	B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. doi: 10.1016/0167-2789(81)90004-X.
[17]	G. Gui, Y. Liu and T. Luo, Model equations and traveling wave solutions for shallow-water waves with the Coriolis effect, J. Nonlinear Sci., 29 (2019), 993-1039. doi: 10.1007/s00332-018-9510-x.
[18]	G. C. Johnson, M. J. McPhaden and E. Firing, Equatorial Pacific ocean horizontal velocity, divergence, and upwelling, J. Phys. Oceanogr., 31 (2001), 839-849. doi: 10.1175/1520-0485(2001)031<0839:EPOHVD>2.0.CO;2.
[19]	J. Johnsen, Well-posed final value problems and Duhamel's formula for coercive Lax-Milgram operators, Electron. Res. Arch., 27 (2019), 20-36. doi: 10.3934/era.2019008.
[20]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.
[21]	C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620. doi: 10.1002/cpa.3160460405.
[22]	S. Kouranbaeva, The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 40 (1999), 857-868. doi: 10.1063/1.532690.
[23]	S. Lang, Differential Manifolds, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1972.
[24]	A. Magaña, A. Miranville and R. Quintanilla, On the time decay in phase lag thermoelasticity with two temperatures, Electron. Res. Arch., 27 (2019), 7-19. doi: 10.3934/era.2019007.
[25]	S. Masaki and J.-I. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 283-326. doi: 10.1016/j.anihpc.2017.04.003.
[26]	F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc., 14 (2001), 555-578. doi: 10.1090/S0894-0347-01-00369-1.
[27]	G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208. doi: 10.1016/S0393-0440(97)00010-7.
[28]	O. G. Mustafa, Existence and uniqueness of solutions with low regularity for a class of nonlinear dispersive equations, SIAM J. Math. Anal., 37 (2005), 1117-1130. doi: 10.1137/040612397.
[29]	O. G. Mustafa, Existence and uniqueness of low regularity solutions for the Dullin-Gottwald-Holm equation, Comm. Math. Phys., 265 (2006), 189-200. doi: 10.1007/s00220-006-1532-9.
[30]	S. Pan, Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle, Electron. Res. Arch., 27 (2019), 89-99. doi: 10.3934/era.2019011.
[31]	P. L. da Silva and I. L. Freire, Well-posedness, travelling waves and geometrical aspects of generalizations of the Camassa-Holm equation, J. Differential Equations, 267 (2019), 5318-5369. doi: 10.1016/j.jde.2019.05.033.
[32]	X. Tu, Y. Liu and C. Mu, Existence and uniqueness of the global conservative weak solutions to the rotation-Camassa-Holm equation, J. Differential Equations, 266 (2019), 4864-4900. doi: 10.1016/j.jde.2018.10.012.
[33]	L. Yang, C. Mu, S. Zhou and X. Tu, The global conservative solutions for the generalized Camassa-Holm equation, Electron. Res. Arch., 27 (2019), 37-67. doi: 10.3934/era.2019009.
[34]	L. Zhang, Non-uniform dependence and well-posedness for the rotation-Camassa-Holm equation on the torus, J. Differential Equations, 267 (2019), 5049-5083. doi: 10.1016/j.jde.2019.05.023.
[35]	M. Zhu, Y. Liu and Y. Mi, Wave-breaking phenomena and persistence properties for the nonlocal rotation-Camassa-Holm equation, Ann. Mat. Pura Appl. (4), 199 (2020), 355-377. doi: 10.1007/s10231-019-00882-5.