2015, 35(11): 5171-5184. doi: 10.3934/dcds.2015.35.5171

Global existence for the stochastic Degasperis-Procesi equation

1. 

School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China

2. 

Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, School of Mathematical Science, Nanjing Normal University, Nanjing 210023

Received  March 2014 Revised  May 2014 Published  May 2015

This paper is concerned with the Cauchy problem of stochastic Degasperis-Procesi equation. Firstly, the local well-posedness for this system is established. Then the precise blow-up scenario for solutions to the system is derived. Finally, the gloabl well-posedness to the system is presented.
Citation: Yong Chen, Hongjun Gao. Global existence for the stochastic Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5171-5184. doi: 10.3934/dcds.2015.35.5171
References:
[1]

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

[2]

Y. Chen, H. J. Gao and B. L. Guo, Well posedness for stochastic Camassa-Holm equation,, J. Differential Equations, 253 (2012), 2353. doi: 10.1016/j.jde.2012.06.023.

[3]

P. L. Chow, Stochastic Partial Differential Equation,, Chapman & Hall/CRC, (2007).

[4]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation,, J. Funct. Anal., 223 (2006), 60. doi: 10.1016/j.jfa.2005.07.008.

[5]

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.

[6]

R. Danchin, Fourier Analysis Methods for PDEs,, Lecture Notes, (2003).

[7]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223.

[8]

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

[9]

H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid Dynam. Res., 33 (2003), 73. doi: 10.1016/S0169-5983(03)00046-7.

[10]

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.

[11]

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.

[12]

J. Escher and J. Seiler, The periodic b-equation and Euler equations on the circle,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3405494.

[13]

J. Feng and D. Nualart, Stochastic scalar conservation laws,, J. Funct. Anal., 255 (2008), 313. doi: 10.1016/j.jfa.2008.02.004.

[14]

F. Flandoli, M. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation,, Invent. Math., 180 (2010), 1. doi: 10.1007/s00222-009-0224-4.

[15]

J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation,, SIAM J. Math. Anal., 20 (1989), 1388. doi: 10.1137/0520091.

[16]

D. Henry, Infinite propagation speed for the Degasperis-Procesi equation,, J. Math. Anal. Appl., 311 (2005), 755. doi: 10.1016/j.jmaa.2005.03.001.

[17]

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.

[18]

D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1-1 nonlinear evolutionary PDE,, Phys. Lett. A, 308 (2003), 437. doi: 10.1016/S0375-9601(03)00114-2.

[19]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224.

[20]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices,, Duke Math. J., 71 (1993), 1. doi: 10.1215/S0012-7094-93-07101-3.

[21]

J. U. Kim, On the Cauchy problem for the transport equation with random noise,, J. Funct. Anal., 259 (2010), 3328. doi: 10.1016/j.jfa.2010.08.017.

[22]

J. U. Kim, On the stochastic quasi-linear symmertic hyperbolic system,, J. Differential Equations, 250 (2011), 1650. doi: 10.1016/j.jde.2010.09.025.

[23]

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.

[24]

Z. W. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation,, Comm. Pure Appl. Math., 62 (2009), 125. doi: 10.1002/cpa.20239.

[25]

Y. Liu and Z. Y. 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.

[26]

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.

[27]

Y. Matsuno, Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit,, Inverse Problems, 21 (2005), 1553. doi: 10.1088/0266-5611/21/5/004.

[28]

O. G. Mustafa, A note on the Degasperis-Procesi equation,, J. Nonlinear Math. Phys., 12 (2005), 10. doi: 10.2991/jnmp.2005.12.1.2.

[29]

T. Tao, Multilinear weighted convolution of $L^{2}$ functions, and applications to nonlinear dispersive equation,, Amer. J. Math., 123 (2001), 839. doi: 10.1353/ajm.2001.0035.

[30]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, Second edition, (1995).

[31]

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.

[32]

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

[33]

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

[34]

Z. Y. 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.

[35]

Z. Y. 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]

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

[2]

Y. Chen, H. J. Gao and B. L. Guo, Well posedness for stochastic Camassa-Holm equation,, J. Differential Equations, 253 (2012), 2353. doi: 10.1016/j.jde.2012.06.023.

[3]

P. L. Chow, Stochastic Partial Differential Equation,, Chapman & Hall/CRC, (2007).

[4]

G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation,, J. Funct. Anal., 223 (2006), 60. doi: 10.1016/j.jfa.2005.07.008.

[5]

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.

[6]

R. Danchin, Fourier Analysis Methods for PDEs,, Lecture Notes, (2003).

[7]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions,, Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223.

[8]

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

[9]

H. R. Dullin, G. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves,, Fluid Dynam. Res., 33 (2003), 73. doi: 10.1016/S0169-5983(03)00046-7.

[10]

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.

[11]

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.

[12]

J. Escher and J. Seiler, The periodic b-equation and Euler equations on the circle,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3405494.

[13]

J. Feng and D. Nualart, Stochastic scalar conservation laws,, J. Funct. Anal., 255 (2008), 313. doi: 10.1016/j.jfa.2008.02.004.

[14]

F. Flandoli, M. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation,, Invent. Math., 180 (2010), 1. doi: 10.1007/s00222-009-0224-4.

[15]

J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation,, SIAM J. Math. Anal., 20 (1989), 1388. doi: 10.1137/0520091.

[16]

D. Henry, Infinite propagation speed for the Degasperis-Procesi equation,, J. Math. Anal. Appl., 311 (2005), 755. doi: 10.1016/j.jmaa.2005.03.001.

[17]

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.

[18]

D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1-1 nonlinear evolutionary PDE,, Phys. Lett. A, 308 (2003), 437. doi: 10.1016/S0375-9601(03)00114-2.

[19]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224.

[20]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices,, Duke Math. J., 71 (1993), 1. doi: 10.1215/S0012-7094-93-07101-3.

[21]

J. U. Kim, On the Cauchy problem for the transport equation with random noise,, J. Funct. Anal., 259 (2010), 3328. doi: 10.1016/j.jfa.2010.08.017.

[22]

J. U. Kim, On the stochastic quasi-linear symmertic hyperbolic system,, J. Differential Equations, 250 (2011), 1650. doi: 10.1016/j.jde.2010.09.025.

[23]

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.

[24]

Z. W. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation,, Comm. Pure Appl. Math., 62 (2009), 125. doi: 10.1002/cpa.20239.

[25]

Y. Liu and Z. Y. 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.

[26]

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.

[27]

Y. Matsuno, Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit,, Inverse Problems, 21 (2005), 1553. doi: 10.1088/0266-5611/21/5/004.

[28]

O. G. Mustafa, A note on the Degasperis-Procesi equation,, J. Nonlinear Math. Phys., 12 (2005), 10. doi: 10.2991/jnmp.2005.12.1.2.

[29]

T. Tao, Multilinear weighted convolution of $L^{2}$ functions, and applications to nonlinear dispersive equation,, Amer. J. Math., 123 (2001), 839. doi: 10.1353/ajm.2001.0035.

[30]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, Second edition, (1995).

[31]

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.

[32]

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

[33]

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

[34]

Z. Y. 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.

[35]

Z. Y. 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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