American Institute of Mathematical Sciences

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August  2018, 38(8): 4163-4187. doi: 10.3934/dcds.2018181

On the Cauchy problem for a higher-order μ-Camassa-Holm equation

Feng Wang 1, , Fengquan Li 2, and  Zhijun Qiao 3,

1. 

School of Mathematics and Statistics, Xidian University, Xi'an 710071, China
2. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
3. 

School of Mathematical & Statistical Sciences, University of Texas-Rio Grande Valley, Texas 78539, USA

Authors to whom correspondence should be addressed. E-mails: wangfeng@xidian.edu.cn, fqli@dlut.edu.cn, zhijun.qiao@utrgv.edu.

Received  December 2017 Revised  February 2018 Published  May 2018

In this paper, we study the Cauchy problem of a higher-order μ-Camassa-Holm equation. We first establish the Green's function of $(μ-\partial_{x}^{2}+\partial_{x}^{4})^{-1}$ and local well-posedness for the equation in Sobolev spaces $H^{s}(\mathbb{S})$, $s>\frac{7}{2}$. Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in $H^{s}(\mathbb{S})$, $s≥ 4$. Finally, we prove that the equation admits single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.

Keywords: Higher-order μ-Camassa-Holm equation, global existence, weak solutions, non-uniformly continuous, peakon solutions.
Mathematics Subject Classification: 35G25, 35L05, 35B30.
Citation: Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181
References:
[1]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar

[2]

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. Google Scholar

[3]

R. Chen, J. Lenells and Y. Liu, Stability of the μ-Camassa-Holm peakons, J. Nonlinear Sci., 23 (2013), 97-112. doi: 10.1007/s00332-012-9141-6. Google Scholar

[4]

G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711. Google Scholar

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G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Diff. Equ., 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014. Google Scholar

[6]

G. M. Coclite and K. H. Karlsen, A note on the Camassa-Holm equation, J. Diff. Equ., 259 (2015), 2158-2166. doi: 10.1016/j.jde.2015.03.020. Google Scholar

[7]

G. M. Coclite and L. Ruvo, A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law, Discrete Contin. Dyn. Syst., 37 (2017), 1247-1282. doi: 10.3934/dcds.2017052. Google Scholar

[8]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Diff. Equ., 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. Google Scholar

[9]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231. Google Scholar

[10]

A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10 (2000), 391-399. doi: 10.1007/s003329910017. Google Scholar

[11]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar

[12]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328. Google Scholar

[13]

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. Google Scholar

[14]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A., 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201. Google Scholar

[15]

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. Google Scholar

[16]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. Google Scholar

[17]

A. Constantin and L. Molinet, Global weak solution solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. Google Scholar

[18]

A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar

[19]

R. Danchin, A few remarks on the Camassa-Holm equation, Diff. Integ. Equ., 14 (2001), 953-988. Google Scholar

[20]

D. Ding, Traveling solutions and evolution properties of the higher order Camassa-Holm equation, Nonlinear Anal., 152 (2017), 1-11. doi: 10.1016/j.na.2016.12.010. Google Scholar

[21]

D. Ding and P. Lv, Conservative solutions for higher-order Camassa-Holm equations, J. Math.Phys., 51 (2010), 072701, 15pp. doi: 10.1063/1.3462917. Google Scholar

[22]

D. Ding and S. Zhang, Lipschitz metric for the periodic second-order Camassa-Holm equation, J. Math. Anal. Appl., 451 (2017), 990-1025. doi: 10.1016/j.jmaa.2017.02.018. Google Scholar

[23]

J. Escher and B. Kolev, Geodesic completeness for Sobolev H^{s}$-metrics on the diffeomorphism group of the circle, J. Evol. Equ., 14 (2014), 949-968. doi: 10.1007/s00028-014-0245-3. Google Scholar

[24]

J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, J. Geom. Mech., 6 (2014), 335-372. doi: 10.3934/jgm.2014.6.335. Google Scholar

[25]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. doi: 10.1016/0167-2789(81)90004-X. Google Scholar

[26]

A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Diff. Int. Eqs., 22 (2009), 201-224. Google Scholar

[27]

A. Himonas, C. Kenig and G. Misiolek, Non-uniform dependence for the periodic CH equation, Comm. Partial Differential Equations, 35 (2010), 1145-1162. doi: 10.1080/03605300903436746. Google Scholar

[28]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, In: Spectral Theory and Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 448 (1975), 25-70. Google Scholar

[29]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 203-208. doi: 10.1002/cpa.3160410704. Google Scholar

[30]

B. Khesin, J. Lenells and G. Misiolek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342 (2008), 617-656. doi: 10.1007/s00208-008-0250-3. Google Scholar

[31]

B. Kolev, Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst., 19 (2007), 555-574. doi: 10.3934/dcds.2007.19.555. Google Scholar

[32]

J. Lenells, G. Misiolek and F. Ti${\rm{\ddot g}}$ay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Commun. Math. Phys., 299 (2010), 129-161. doi: 10.1007/s00220-010-1069-9. Google Scholar

[33]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Diff. Equ., 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. Google Scholar

[34]

J. Liu and Z. Yin, On the Cauchy problem of a weakly dissipative μ-Hunter-Saxton equation, Ann. I. H. Poincaré-AN., 31 (2014), 267-279. doi: 10.1016/j.anihpc.2013.02.008. Google Scholar

[35]

R. McLachlan and X. Zhang, Well-posedness of a modified Camassa-Holm equations, J. Diff. Equ., 246 (2009), 3241-3259. doi: 10.1016/j.jde.2009.01.039. Google Scholar

[36]

Z. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Commun. Math. Phys., 239 (2003), 309-341. doi: 10.1007/s00220-003-0880-y. Google Scholar

[37]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[38] M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991. doi: 10.1007/978-1-4612-0431-2. Google Scholar
[39]

M. Taylor, Commutator estimates, Proc. Amer. Math. Soc., 131 (2003), 1501-1507. doi: 10.1090/S0002-9939-02-06723-0. Google Scholar

[40]

L. Tian, P. Zhang and L. Xia, Global existence for the higher-order Camassa-Holm shallow water equation, Nonlinear Anal., 74 (2011), 2468-2474. doi: 10.1016/j.na.2010.12.002. Google Scholar

[41]

F. Wang, F. Li and Z. Qiao, Well-posedness and peakons for a higher-order µ-Camassa-Holm equation, arXiv: 1712.07996.Google Scholar

[42]

S. Wu and Z. Yin, Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation, J. Diff. Equ., 246 (2009), 4309-4321. doi: 10.1016/j.jde.2008.12.008. Google Scholar

[43]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar

show all references

References:
[1]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 5 (2007), 1-27. doi: 10.1142/S0219530507000857. Google Scholar

[2]

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. Google Scholar

[3]

R. Chen, J. Lenells and Y. Liu, Stability of the μ-Camassa-Holm peakons, J. Nonlinear Sci., 23 (2013), 97-112. doi: 10.1007/s00332-012-9141-6. Google Scholar

[4]

G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation, SIAM J. Math. Anal., 37 (2005), 1044-1069. doi: 10.1137/040616711. Google Scholar

[5]

G. M. Coclite, H. Holden and K. H. Karlsen, Well-posedness of higher-order Camassa-Holm equations, J. Diff. Equ., 246 (2009), 929-963. doi: 10.1016/j.jde.2008.04.014. Google Scholar

[6]

G. M. Coclite and K. H. Karlsen, A note on the Camassa-Holm equation, J. Diff. Equ., 259 (2015), 2158-2166. doi: 10.1016/j.jde.2015.03.020. Google Scholar

[7]

G. M. Coclite and L. Ruvo, A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law, Discrete Contin. Dyn. Syst., 37 (2017), 1247-1282. doi: 10.3934/dcds.2017052. Google Scholar

[8]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Diff. Equ., 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. Google Scholar

[9]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231. Google Scholar

[10]

A. Constantin, On the blow-up of solutions of a periodic shallow water equation, J. Nonlinear Sci., 10 (2000), 391-399. doi: 10.1007/s003329910017. Google Scholar

[11]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. Google Scholar

[12]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328. Google Scholar

[13]

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. Google Scholar

[14]

A. Constantin and B. Kolev, On the geometric approach to the motion of inertial mechanical systems, J. Phys. A., 35 (2002), R51-R79. doi: 10.1088/0305-4470/35/32/201. Google Scholar

[15]

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. Google Scholar

[16]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math., 52 (1999), 949-982. doi: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D. Google Scholar

[17]

A. Constantin and L. Molinet, Global weak solution solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. Google Scholar

[18]

A. Constantin and W. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. Google Scholar

[19]

R. Danchin, A few remarks on the Camassa-Holm equation, Diff. Integ. Equ., 14 (2001), 953-988. Google Scholar

[20]

D. Ding, Traveling solutions and evolution properties of the higher order Camassa-Holm equation, Nonlinear Anal., 152 (2017), 1-11. doi: 10.1016/j.na.2016.12.010. Google Scholar

[21]

D. Ding and P. Lv, Conservative solutions for higher-order Camassa-Holm equations, J. Math.Phys., 51 (2010), 072701, 15pp. doi: 10.1063/1.3462917. Google Scholar

[22]

D. Ding and S. Zhang, Lipschitz metric for the periodic second-order Camassa-Holm equation, J. Math. Anal. Appl., 451 (2017), 990-1025. doi: 10.1016/j.jmaa.2017.02.018. Google Scholar

[23]

J. Escher and B. Kolev, Geodesic completeness for Sobolev H^{s}$-metrics on the diffeomorphism group of the circle, J. Evol. Equ., 14 (2014), 949-968. doi: 10.1007/s00028-014-0245-3. Google Scholar

[24]

J. Escher and B. Kolev, Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle, J. Geom. Mech., 6 (2014), 335-372. doi: 10.3934/jgm.2014.6.335. Google Scholar

[25]

A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. doi: 10.1016/0167-2789(81)90004-X. Google Scholar

[26]

A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Diff. Int. Eqs., 22 (2009), 201-224. Google Scholar

[27]

A. Himonas, C. Kenig and G. Misiolek, Non-uniform dependence for the periodic CH equation, Comm. Partial Differential Equations, 35 (2010), 1145-1162. doi: 10.1080/03605300903436746. Google Scholar

[28]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, In: Spectral Theory and Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 448 (1975), 25-70. Google Scholar

[29]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 203-208. doi: 10.1002/cpa.3160410704. Google Scholar

[30]

B. Khesin, J. Lenells and G. Misiolek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann., 342 (2008), 617-656. doi: 10.1007/s00208-008-0250-3. Google Scholar

[31]

B. Kolev, Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst., 19 (2007), 555-574. doi: 10.3934/dcds.2007.19.555. Google Scholar

[32]

J. Lenells, G. Misiolek and F. Ti${\rm{\ddot g}}$ay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Commun. Math. Phys., 299 (2010), 129-161. doi: 10.1007/s00220-010-1069-9. Google Scholar

[33]

Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Diff. Equ., 162 (2000), 27-63. doi: 10.1006/jdeq.1999.3683. Google Scholar

[34]

J. Liu and Z. Yin, On the Cauchy problem of a weakly dissipative μ-Hunter-Saxton equation, Ann. I. H. Poincaré-AN., 31 (2014), 267-279. doi: 10.1016/j.anihpc.2013.02.008. Google Scholar

[35]

R. McLachlan and X. Zhang, Well-posedness of a modified Camassa-Holm equations, J. Diff. Equ., 246 (2009), 3241-3259. doi: 10.1016/j.jde.2009.01.039. Google Scholar

[36]

Z. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Commun. Math. Phys., 239 (2003), 309-341. doi: 10.1007/s00220-003-0880-y. Google Scholar

[37]

J. Simon, Compact sets in the space $L^{p}(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360. Google Scholar

[38] M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991. doi: 10.1007/978-1-4612-0431-2. Google Scholar
[39]

M. Taylor, Commutator estimates, Proc. Amer. Math. Soc., 131 (2003), 1501-1507. doi: 10.1090/S0002-9939-02-06723-0. Google Scholar

[40]

L. Tian, P. Zhang and L. Xia, Global existence for the higher-order Camassa-Holm shallow water equation, Nonlinear Anal., 74 (2011), 2468-2474. doi: 10.1016/j.na.2010.12.002. Google Scholar

[41]

F. Wang, F. Li and Z. Qiao, Well-posedness and peakons for a higher-order µ-Camassa-Holm equation, arXiv: 1712.07996.Google Scholar

[42]

S. Wu and Z. Yin, Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation, J. Diff. Equ., 246 (2009), 4309-4321. doi: 10.1016/j.jde.2008.12.008. Google Scholar

[43]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar

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