October  2016, 36(10): 5493-5508. doi: 10.3934/dcds.2016042

Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation

1. 

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

2. 

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

Received  November 2015 Revised  December 2015 Published  July 2016

We first establish the local existence and uniqueness of strong solutions for the Cauchy problem of a generalized Camassa-Holm equation in nonhomogeneous Besov spaces by using the Littlewood-Paley theory. Then, we prove that the solution depends continuously on the initial data in the corresponding Besov space. Finally, we derive a blow-up criterion and present a blow-up result and a blow-up rate of the blow-up solutions to the equation.
Citation: Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 343, Berlin-Heidelberg-NewYork: Springer, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

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

[5]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[6]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91. doi: 10.1016/j.jfa.2005.07.008.  Google Scholar

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85.  Google Scholar

[8]

A. Constantin, On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.  Google Scholar

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.  Google Scholar

[10]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[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), 303-328.  Google Scholar

[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), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.  Google Scholar

[13]

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

[14]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[16]

A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[17]

A. Constantin, R. I. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity 23 (2010), 2559-2575. doi: 10.1088/0951-7715/23/10/012.  Google Scholar

[18]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar

[19]

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

[20]

A. Constantin and W. A. 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

[21]

R. Danchin, A few remarks on the Camassa-Holm equation. Differential Integral Equations, 14 (2001), 953-988.  Google Scholar

[22]

R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2.  Google Scholar

[23]

A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theoret. and Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422.  Google Scholar

[24]

A. Degasperis and M. Procesi, Asymptotic integrability. Symmetry and perturbation theory, World Sci. Publ., River Edge, NJ, 1999, 23-37.  Google Scholar

[25]

H. R. Dullin, G. A. Gottwald and D. D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D, 190 (2004), 1-14. doi: 10.1016/j.physd.2003.11.004.  Google Scholar

[26]

J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. doi: 10.1016/j.jfa.2006.03.022.  Google Scholar

[27]

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), 87-117. doi: 10.1512/iumj.2007.56.3040.  Google Scholar

[28]

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

[29]

G. Gui and Y. Liu, On the Cauchy problem for the Degasperis-Procesi equation, Quart. Appl. Math., 69 (2011), 445-464. doi: 10.1090/S0033-569X-2011-01216-5.  Google Scholar

[30]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479. doi: 10.1088/0951-7715/25/2/449.  Google Scholar

[31]

A. N. W. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 372002, 10 pp. doi: 10.1088/1751-8113/41/37/372002.  Google Scholar

[32]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[33]

J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82. doi: 10.1016/j.jmaa.2004.11.038.  Google Scholar

[34]

J. Li and Z. Yin, Well-posedness and global existence for a generalized Degasperis-Procesi equation, Nonlinear Anal. Real World Appl., 28 (2016), 72-90. doi: 10.1016/j.nonrwa.2015.09.003.  Google Scholar

[35]

Y. Liu and Z. Yin, Global Existence and Blow-up Phenomena for the Degasperis-Procesi Equation, Comm. Math. Phys., 267 (2006), 801-820. doi: 10.1007/s00220-006-0082-5.  Google Scholar

[36]

Y. Liu and Z. Yin, On the blow-up phenomena for the Degasperis-Procesi equation, Int. Math. Res. Not. IMRN, 23 (2007), Art. ID rnm117, 22 pp. doi: 10.1093/imrn/rnm117.  Google Scholar

[37]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198. doi: 10.1007/s00332-006-0803-3.  Google Scholar

[38]

H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Problems, 19 (2003), 1241-1245. doi: 10.1088/0266-5611/19/6/001.  Google Scholar

[39]

V. Novikov, Generalization of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14 pp. doi: 10.1088/1751-8113/42/34/342002.  Google Scholar

[40]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar

[41]

V. O. Vakhnenko and E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos Solitons Fractals, 20 (2004), 1059-1073. doi: 10.1016/j.chaos.2003.09.043.  Google Scholar

[42]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation, J. Phys. A, 44 (2011), 055202, 17pp. doi: 10.1088/1751-8113/44/5/055202.  Google Scholar

[43]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 707-727.  Google Scholar

[44]

X. Wu and Z. Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137. doi: 10.1080/00036811.2011.649735.  Google Scholar

[45]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the integrable Novikov equation, J. Differential Equations, 253 (2012), 298-318. doi: 10.1016/j.jde.2012.03.015.  Google Scholar

[46]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the Novikov equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1157-1169. doi: 10.1007/s00030-012-0202-1.  Google Scholar

[47]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.  Google Scholar

[48]

Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139. doi: 10.1016/S0022-247X(03)00250-6.  Google Scholar

[49]

Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194. doi: 10.1016/j.jfa.2003.07.010.  Google Scholar

[50]

Z. Yin, Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J., 53 (2004), 1189-1209. doi: 10.1512/iumj.2004.53.2479.  Google Scholar

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Vol. 343, Berlin-Heidelberg-NewYork: Springer, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

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.  Google Scholar

[3]

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

[4]

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

[5]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[6]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 233 (2006), 60-91. doi: 10.1016/j.jfa.2005.07.008.  Google Scholar

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), 53-85.  Google Scholar

[8]

A. Constantin, On the scattering problem for the Camassa-Holm equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.  Google Scholar

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach, Ann. Inst. Fourier (Grenoble), 50 (2000), 321-362. doi: 10.5802/aif.1757.  Google Scholar

[10]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar

[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), 303-328.  Google Scholar

[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), 475-504. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.  Google Scholar

[13]

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

[14]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431. doi: 10.1090/S0273-0979-07-01159-7.  Google Scholar

[15]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[16]

A. Constantin, V. S. Gerdjikov and R. I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, 22 (2006), 2197-2207. doi: 10.1088/0266-5611/22/6/017.  Google Scholar

[17]

A. Constantin, R. I. Ivanov and J. Lenells, Inverse scattering transform for the Degasperis-Procesi equation, Nonlinearity 23 (2010), 2559-2575. doi: 10.1088/0951-7715/23/10/012.  Google Scholar

[18]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.  Google Scholar

[19]

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

[20]

A. Constantin and W. A. 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

[21]

R. Danchin, A few remarks on the Camassa-Holm equation. Differential Integral Equations, 14 (2001), 953-988.  Google Scholar

[22]

R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2.  Google Scholar

[23]

A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theoret. and Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422.  Google Scholar

[24]

A. Degasperis and M. Procesi, Asymptotic integrability. Symmetry and perturbation theory, World Sci. Publ., River Edge, NJ, 1999, 23-37.  Google Scholar

[25]

H. R. Dullin, G. A. Gottwald and D. D. Holm, On asymptotically equivalent shallow water wave equations, Phys. D, 190 (2004), 1-14. doi: 10.1016/j.physd.2003.11.004.  Google Scholar

[26]

J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. doi: 10.1016/j.jfa.2006.03.022.  Google Scholar

[27]

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), 87-117. doi: 10.1512/iumj.2007.56.3040.  Google Scholar

[28]

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

[29]

G. Gui and Y. Liu, On the Cauchy problem for the Degasperis-Procesi equation, Quart. Appl. Math., 69 (2011), 445-464. doi: 10.1090/S0033-569X-2011-01216-5.  Google Scholar

[30]

A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479. doi: 10.1088/0951-7715/25/2/449.  Google Scholar

[31]

A. N. W. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 372002, 10 pp. doi: 10.1088/1751-8113/41/37/372002.  Google Scholar

[32]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[33]

J. Lenells, Traveling wave solutions of the Degasperis-Procesi equation, J. Math. Anal. Appl., 306 (2005), 72-82. doi: 10.1016/j.jmaa.2004.11.038.  Google Scholar

[34]

J. Li and Z. Yin, Well-posedness and global existence for a generalized Degasperis-Procesi equation, Nonlinear Anal. Real World Appl., 28 (2016), 72-90. doi: 10.1016/j.nonrwa.2015.09.003.  Google Scholar

[35]

Y. Liu and Z. Yin, Global Existence and Blow-up Phenomena for the Degasperis-Procesi Equation, Comm. Math. Phys., 267 (2006), 801-820. doi: 10.1007/s00220-006-0082-5.  Google Scholar

[36]

Y. Liu and Z. Yin, On the blow-up phenomena for the Degasperis-Procesi equation, Int. Math. Res. Not. IMRN, 23 (2007), Art. ID rnm117, 22 pp. doi: 10.1093/imrn/rnm117.  Google Scholar

[37]

H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198. doi: 10.1007/s00332-006-0803-3.  Google Scholar

[38]

H. Lundmark and J. Szmigielski, Multi-peakon solutions of the Degasperis-Procesi equation, Inverse Problems, 19 (2003), 1241-1245. doi: 10.1088/0266-5611/19/6/001.  Google Scholar

[39]

V. Novikov, Generalization of the Camassa-Holm equation, J. Phys. A, 42 (2009), 342002, 14 pp. doi: 10.1088/1751-8113/42/34/342002.  Google Scholar

[40]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  Google Scholar

[41]

V. O. Vakhnenko and E. J. Parkes, Periodic and solitary-wave solutions of the Degasperis-Procesi equation, Chaos Solitons Fractals, 20 (2004), 1059-1073. doi: 10.1016/j.chaos.2003.09.043.  Google Scholar

[42]

X. Wu and Z. Yin, Global weak solutions for the Novikov equation, J. Phys. A, 44 (2011), 055202, 17pp. doi: 10.1088/1751-8113/44/5/055202.  Google Scholar

[43]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 707-727.  Google Scholar

[44]

X. Wu and Z. Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137. doi: 10.1080/00036811.2011.649735.  Google Scholar

[45]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the integrable Novikov equation, J. Differential Equations, 253 (2012), 298-318. doi: 10.1016/j.jde.2012.03.015.  Google Scholar

[46]

W. Yan, Y. Li and Y. Zhang, The Cauchy problem for the Novikov equation, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1157-1169. doi: 10.1007/s00030-012-0202-1.  Google Scholar

[47]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.  Google Scholar

[48]

Z. Yin, Global existence for a new periodic integrable equation, J. Math. Anal. Appl., 283 (2003), 129-139. doi: 10.1016/S0022-247X(03)00250-6.  Google Scholar

[49]

Z. Yin, Global weak solutions for a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194. doi: 10.1016/j.jfa.2003.07.010.  Google Scholar

[50]

Z. Yin, Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J., 53 (2004), 1189-1209. doi: 10.1512/iumj.2004.53.2479.  Google Scholar

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