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December  2016, 9(6): 2149-2165. doi: 10.3934/dcdss.2016089

Wave breaking and persistent decay of solution to a shallow water wave equation

1. 

Business School of Central South University, Changsha 410012, China

2. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

Received  June 2015 Revised  September 2016 Published  November 2016

As we all know, wave breaking of the water wave is important and interesting to physicist and mathematician. In the article, we devote to the study of blow-up phenomena, the decay of solution and traveling wave solution to a shallow water wave equation. First, based on the blow-up scenario, some new blow-up phenomena is derived. By virtue of a weighted function, the persistent decay of solution is established. Finally, we explore the analytic solutions and traveling wave solutions.
Citation: Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089
References:
[1]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Notices, 22 (2012), 5161-5181.

[2]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.

[3]

R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.

[4]

G. M. Coclite, K. H. Karlsen and N. H. Risebro, Numberical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, IMA J. Numer. Anal., 28 (2008), 80-105. doi: 10.1093/imanum/drm003.

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.

[6]

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

[7]

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

[8]

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

[9]

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.

[10]

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

[11]

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.

[12]

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

[13]

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

[14]

A. Degasperis and M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Scientific, (1999), 23-37.

[15]

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.

[16]

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.

[17]

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

[18]

A. Geyer, Solitary traveling waves of moderate amplitude, J. Nonlinear Math. Phys., 19 (2012), 1240010, 12pp. doi: 10.1142/S1402925112400104.

[19]

A. A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584. doi: 10.1007/s00208-003-0466-1.

[20]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380. doi: 10.1137/S1111111102410943.

[21]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity, I. Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 455-467.

[22]

T. Kato and G. Ponce, Commutator estimation and the Euler and Navier-Stokes Equation, Comm. Pure Appl. Math., 41 (1998), 891-907. doi: 10.1002/cpa.3160410704.

[23]

M. C. Lombardo, M. Sammartino and V. Sciacca, A Note on the analytic solutions of the Camassa-Holm equation, C. R. Math. Acad. Sci. Paris, 341 (2005), 659-664. doi: 10.1016/j.crma.2005.10.006.

[24]

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.

[25]

N. D. Mutlubas and A. Geyer, Orbital stability of solitary waves of moderate amplitude, J. Differential Equations, 255 (2013), 254-263. doi: 10.1016/j.jde.2013.04.010.

[26]

G. Rodrigues Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X.

[27]

M. V. Safonov, An abstract Cauchy-Kovalevskaya thm in a weighted Banach space, Comm. Pure Appl. Math., 48 (1995), 629-637. doi: 10.1002/cpa.3160480604.

[28]

E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math., 30 (1977), 321-337. doi: 10.1002/cpa.3160300305.

[29]

X. Wu, On some wave breaking for the nonlinear integrable shallow water wave equations, Nonlinear Analysis, TMA., 127 (2015), 352-361. doi: 10.1016/j.na.2015.07.015.

[30]

X. Wu, Global Analytic Solutions and Traveling wave Solutions of the Cauchy problem for the Novikov Equation, Pro. AMS.

[31]

X. Wu and B. Guo, The exponential decay of solutions and traveling wave solutions for a modified Camassa-Holm equation with cubic nonlinearity, J. Math. Phys., 55 (2014), 081504, 17pp. doi: 10.1063/1.4891989.

[32]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Annali Sc. Norm. Sup. Pisa, 11 (2012), 707-727.

[33]

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.

[34]

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

[35]

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

show all references

References:
[1]

L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Notices, 22 (2012), 5161-5181.

[2]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.

[3]

R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.

[4]

G. M. Coclite, K. H. Karlsen and N. H. Risebro, Numberical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, IMA J. Numer. Anal., 28 (2008), 80-105. doi: 10.1093/imanum/drm003.

[5]

A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701.

[6]

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

[7]

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

[8]

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

[9]

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.

[10]

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

[11]

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.

[12]

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

[13]

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

[14]

A. Degasperis and M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Scientific, (1999), 23-37.

[15]

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.

[16]

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.

[17]

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

[18]

A. Geyer, Solitary traveling waves of moderate amplitude, J. Nonlinear Math. Phys., 19 (2012), 1240010, 12pp. doi: 10.1142/S1402925112400104.

[19]

A. A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584. doi: 10.1007/s00208-003-0466-1.

[20]

D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380. doi: 10.1137/S1111111102410943.

[21]

T. Kato and K. Masuda, Nonlinear evolution equations and analyticity, I. Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 455-467.

[22]

T. Kato and G. Ponce, Commutator estimation and the Euler and Navier-Stokes Equation, Comm. Pure Appl. Math., 41 (1998), 891-907. doi: 10.1002/cpa.3160410704.

[23]

M. C. Lombardo, M. Sammartino and V. Sciacca, A Note on the analytic solutions of the Camassa-Holm equation, C. R. Math. Acad. Sci. Paris, 341 (2005), 659-664. doi: 10.1016/j.crma.2005.10.006.

[24]

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.

[25]

N. D. Mutlubas and A. Geyer, Orbital stability of solitary waves of moderate amplitude, J. Differential Equations, 255 (2013), 254-263. doi: 10.1016/j.jde.2013.04.010.

[26]

G. Rodrigues Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X.

[27]

M. V. Safonov, An abstract Cauchy-Kovalevskaya thm in a weighted Banach space, Comm. Pure Appl. Math., 48 (1995), 629-637. doi: 10.1002/cpa.3160480604.

[28]

E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math., 30 (1977), 321-337. doi: 10.1002/cpa.3160300305.

[29]

X. Wu, On some wave breaking for the nonlinear integrable shallow water wave equations, Nonlinear Analysis, TMA., 127 (2015), 352-361. doi: 10.1016/j.na.2015.07.015.

[30]

X. Wu, Global Analytic Solutions and Traveling wave Solutions of the Cauchy problem for the Novikov Equation, Pro. AMS.

[31]

X. Wu and B. Guo, The exponential decay of solutions and traveling wave solutions for a modified Camassa-Holm equation with cubic nonlinearity, J. Math. Phys., 55 (2014), 081504, 17pp. doi: 10.1063/1.4891989.

[32]

X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Annali Sc. Norm. Sup. Pisa, 11 (2012), 707-727.

[33]

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.

[34]

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

[35]

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

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