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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 & 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.   Google Scholar

[2]

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

[3]

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

[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.  doi: 10.1093/imanum/drm003.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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.  Google Scholar

[10]

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

[11]

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.  Google Scholar

[12]

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

[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.  doi: 10.1023/A:1021186408422.  Google Scholar

[14]

A. Degasperis and M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.),, Symmetry and Perturbation Theory, (1999), 23.   Google Scholar

[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.  doi: 10.1016/j.jfa.2006.03.022.  Google Scholar

[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.  doi: 10.1512/iumj.2007.56.3040.  Google Scholar

[17]

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

[18]

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

[19]

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

[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.  doi: 10.1137/S1111111102410943.  Google Scholar

[21]

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

[22]

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

[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.  doi: 10.1016/j.crma.2005.10.006.  Google Scholar

[24]

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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[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).  doi: 10.1063/1.4891989.  Google Scholar

[32]

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

[33]

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.  Google Scholar

[34]

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

[35]

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

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.   Google Scholar

[2]

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

[3]

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

[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.  doi: 10.1093/imanum/drm003.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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.  Google Scholar

[10]

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

[11]

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.  Google Scholar

[12]

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

[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.  doi: 10.1023/A:1021186408422.  Google Scholar

[14]

A. Degasperis and M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.),, Symmetry and Perturbation Theory, (1999), 23.   Google Scholar

[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.  doi: 10.1016/j.jfa.2006.03.022.  Google Scholar

[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.  doi: 10.1512/iumj.2007.56.3040.  Google Scholar

[17]

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

[18]

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

[19]

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

[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.  doi: 10.1137/S1111111102410943.  Google Scholar

[21]

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

[22]

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

[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.  doi: 10.1016/j.crma.2005.10.006.  Google Scholar

[24]

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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[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).  doi: 10.1063/1.4891989.  Google Scholar

[32]

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

[33]

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.  Google Scholar

[34]

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

[35]

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

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