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November  2020, 40(11): 6507-6527. doi: 10.3934/dcds.2020288

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

Received  April 2020 Revised  June 2020 Published  July 2020

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.

Keywords: Coriolis effect, Rotation-Camassa-Holm-type equation, low regularity, local well-posedness, nonlinearity.
Mathematics Subject Classification: Primary: 35Q53, 35B65, 35A01.
Citation: Runzhang Xu, Yanbing Yang. Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6507-6527. doi: 10.3934/dcds.2020288
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.

show all references

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.

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