2016, 36(5): 2781-2801. doi: 10.3934/dcds.2016.36.2781

Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions

1. 

Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

2. 

Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou

Received  April 2015 Revised  September 2015 Published  October 2015

In this paper we mainly study the Cauchy problem for a generalized Camassa-Holm equation. First, by using the Littlewood-Paley decomposition and transport equations theory, we establish the local well-posedness for the Cauchy problem of the equation in Besov spaces. Then we give a blow-up criterion for the Cauchy problem of the equation. we present a blow-up result and the exact blow-up rate of strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion. Finally, we verify that the system possesses peakon solutions.
Citation: Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften, (2011). doi: 10.1007/978-3-642-16830-7.

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z.

[3]

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

[4]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661.

[5]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. doi: 10.1016/S0065-2156(08)70254-0.

[6]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation,, J. Funct. Anal., 233 (2006), 60. doi: 10.1016/j.jfa.2005.07.008.

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53.

[8]

A. Constantin, On the scattering problem for the Camassa-Holm equation},, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953. doi: 10.1098/rspa.2000.0701.

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757.

[10]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5.

[11]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303.

[12]

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. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.

[13]

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

[14]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7.

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12.

[16]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2.

[17]

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

[18]

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

[19]

A. Constantin, R. I. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation,, Nonlinearity, 23 (2010), 2559. doi: 10.1088/0951-7715/23/10/012.

[20]

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

[21]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.

[22]

A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,, Theoret. and Math. Phys., 133 (2002), 1463. doi: 10.1023/A:1021186408422.

[23]

A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and Perturbation Theory (Rome, (1998), 23.

[24]

H. R. Dullin, G. A. Gottwald and D. D. Holm, On asymptotically equivalent shallow water wave equations,, Phys. D, 190 (2004), 1. doi: 10.1016/j.physd.2003.11.004.

[25]

J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241 (2006), 457. doi: 10.1016/j.jfa.2006.03.022.

[26]

J. Escher, Y. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation,, Indiana Univ. Math. J., 56 (2007), 87. doi: 10.1512/iumj.2007.56.3040.

[27]

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

[28]

G. Gui and Y. Liu, On the Cauchy problem for the Degasperis-Procesi equation,, Quart. Appl. Math., 69 (2011), 445. doi: 10.1090/S0033-569X-2011-01216-5.

[29]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449. doi: 10.1088/0951-7715/25/2/449.

[30]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys. A, 41 (2008). doi: 10.1088/1751-8113/41/37/372002.

[31]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation,, Comm. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704.

[32]

S. Lai, Global weak solutions to the Novikov equation,, J. Funct. Anal., 265 (2013), 520. doi: 10.1016/j.jfa.2013.05.022.

[33]

J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation,, J. Math. Anal. Appl., 306 (2005), 72. doi: 10.1016/j.jmaa.2004.11.038.

[34]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801. doi: 10.1007/s00220-006-0082-5.

[35]

Y. Liu and Z. Yin, On the blow-up phenomena for the Degasperis-Procesi equation,, Int. Math. Res. Not. IMRN, 23 (2007). doi: 10.1093/imrn/rnm117.

[36]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17 (2007), 169. doi: 10.1007/s00332-006-0803-3.

[37]

H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation,, Inverse Problems, 19 (2003), 1241. doi: 10.1088/0266-5611/19/6/001.

[38]

W. Luo and Z. Yin, Well-posedness and persistence property for a four-component Novikov system with peakon solutions,, Monatsh. Math., (2015), 1. doi: 10.1007/s00605-015-0809-5.

[39]

V. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/34/342002.

[40]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X.

[41]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.

[42]

V. O. Vakhnenko and E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation,, Chaos Solitons Fractals, 20 (2004), 1059. doi: 10.1016/j.chaos.2003.09.043.

[43]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation,, J. Phys. A, 44 (2011). doi: 10.1088/1751-8113/44/5/055202.

[44]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 707.

[45]

X. Wu and Z. Yin, A note on the Cauchy problem of the Novikov equation,, Appl. Anal., 92 (2013), 1116. doi: 10.1080/00036811.2011.649735.

[46]

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

[47]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the integrable Novikov equation,, J. Differential Equations, 253 (2012), 298. doi: 10.1016/j.jde.2012.03.015.

[48]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the Novikov equation,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1157. doi: 10.1007/s00030-012-0202-1.

[49]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions,, Illinois J. Math., 47 (2003), 649.

[50]

Z. Yin, Global existence for a new periodic integrable equation,, J. Math. Anal. Appl., 283 (2003), 129. doi: 10.1016/S0022-247X(03)00250-6.

[51]

Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions,, J. Funct. Anal., 212 (2004), 182. doi: 10.1016/j.jfa.2003.07.010.

[52]

Z. Yin, Global solutions to a new integrable equation with peakons,, Indiana Univ. Math. J., 53 (2004), 1189. doi: 10.1512/iumj.2004.53.2479.

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften, (2011). doi: 10.1007/978-3-642-16830-7.

[2]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z.

[3]

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

[4]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661.

[5]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. doi: 10.1016/S0065-2156(08)70254-0.

[6]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation,, J. Funct. Anal., 233 (2006), 60. doi: 10.1016/j.jfa.2005.07.008.

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53.

[8]

A. Constantin, On the scattering problem for the Camassa-Holm equation},, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953. doi: 10.1098/rspa.2000.0701.

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757.

[10]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5.

[11]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303.

[12]

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. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.

[13]

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

[14]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7.

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12.

[16]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2.

[17]

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

[18]

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

[19]

A. Constantin, R. I. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation,, Nonlinearity, 23 (2010), 2559. doi: 10.1088/0951-7715/23/10/012.

[20]

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

[21]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.

[22]

A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,, Theoret. and Math. Phys., 133 (2002), 1463. doi: 10.1023/A:1021186408422.

[23]

A. Degasperis and M. Procesi, Asymptotic integrability,, in Symmetry and Perturbation Theory (Rome, (1998), 23.

[24]

H. R. Dullin, G. A. Gottwald and D. D. Holm, On asymptotically equivalent shallow water wave equations,, Phys. D, 190 (2004), 1. doi: 10.1016/j.physd.2003.11.004.

[25]

J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation,, J. Funct. Anal., 241 (2006), 457. doi: 10.1016/j.jfa.2006.03.022.

[26]

J. Escher, Y. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation,, Indiana Univ. Math. J., 56 (2007), 87. doi: 10.1512/iumj.2007.56.3040.

[27]

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

[28]

G. Gui and Y. Liu, On the Cauchy problem for the Degasperis-Procesi equation,, Quart. Appl. Math., 69 (2011), 445. doi: 10.1090/S0033-569X-2011-01216-5.

[29]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation,, Nonlinearity, 25 (2012), 449. doi: 10.1088/0951-7715/25/2/449.

[30]

A. N. W. Hone and J. P. Wang, Integrable peakon equations with cubic nonlinearity,, J. Phys. A, 41 (2008). doi: 10.1088/1751-8113/41/37/372002.

[31]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equation,, Comm. Pure Appl. Math., 41 (1988), 891. doi: 10.1002/cpa.3160410704.

[32]

S. Lai, Global weak solutions to the Novikov equation,, J. Funct. Anal., 265 (2013), 520. doi: 10.1016/j.jfa.2013.05.022.

[33]

J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation,, J. Math. Anal. Appl., 306 (2005), 72. doi: 10.1016/j.jmaa.2004.11.038.

[34]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation,, Comm. Math. Phys., 267 (2006), 801. doi: 10.1007/s00220-006-0082-5.

[35]

Y. Liu and Z. Yin, On the blow-up phenomena for the Degasperis-Procesi equation,, Int. Math. Res. Not. IMRN, 23 (2007). doi: 10.1093/imrn/rnm117.

[36]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation,, J. Nonlinear Sci., 17 (2007), 169. doi: 10.1007/s00332-006-0803-3.

[37]

H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation,, Inverse Problems, 19 (2003), 1241. doi: 10.1088/0266-5611/19/6/001.

[38]

W. Luo and Z. Yin, Well-posedness and persistence property for a four-component Novikov system with peakon solutions,, Monatsh. Math., (2015), 1. doi: 10.1007/s00605-015-0809-5.

[39]

V. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A, 42 (2009). doi: 10.1088/1751-8113/42/34/342002.

[40]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X.

[41]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.

[42]

V. O. Vakhnenko and E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation,, Chaos Solitons Fractals, 20 (2004), 1059. doi: 10.1016/j.chaos.2003.09.043.

[43]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation,, J. Phys. A, 44 (2011). doi: 10.1088/1751-8113/44/5/055202.

[44]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation,, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 707.

[45]

X. Wu and Z. Yin, A note on the Cauchy problem of the Novikov equation,, Appl. Anal., 92 (2013), 1116. doi: 10.1080/00036811.2011.649735.

[46]

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

[47]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the integrable Novikov equation,, J. Differential Equations, 253 (2012), 298. doi: 10.1016/j.jde.2012.03.015.

[48]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the Novikov equation,, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1157. doi: 10.1007/s00030-012-0202-1.

[49]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions,, Illinois J. Math., 47 (2003), 649.

[50]

Z. Yin, Global existence for a new periodic integrable equation,, J. Math. Anal. Appl., 283 (2003), 129. doi: 10.1016/S0022-247X(03)00250-6.

[51]

Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions,, J. Funct. Anal., 212 (2004), 182. doi: 10.1016/j.jfa.2003.07.010.

[52]

Z. Yin, Global solutions to a new integrable equation with peakons,, Indiana Univ. Math. J., 53 (2004), 1189. doi: 10.1512/iumj.2004.53.2479.

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