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On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces
Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect
College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang 150001, China |
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.
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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