# American Institute of Mathematical Sciences

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

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

 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.

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
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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. 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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. 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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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