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
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. ChenJ. 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. CocliteH. 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. CocliteH. 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

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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. HimonasC. 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. KhesinJ. 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. LenellsG. 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. TianP. 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. ChenJ. 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. CocliteH. 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. CocliteH. 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. HimonasC. 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. KhesinJ. 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. LenellsG. 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. TianP. 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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