Advanced Search
Article Contents
Article Contents

Global existence for the stochastic Degasperis-Procesi equation

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: 60H15, 35L05, 35L70.


    \begin{equation} \\ \end{equation}
  • [1]

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


    Y. Chen, H. J. Gao and B. L. Guo, Well posedness for stochastic Camassa-Holm equation, J. Differential Equations, 253 (2012), 2353-2379.doi: 10.1016/j.jde.2012.06.023.


    P. L. Chow, Stochastic Partial Differential Equation, Chapman & Hall/CRC, Boca Raton, London, New York, 2007.


    G. M. Coclite and K. H. Karlsen, On the well-posedness of the Degasperis-Procesi equation, J. Funct. Anal., 223 (2006), 60-91.doi: 10.1016/j.jfa.2005.07.008.


    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.


    R. Danchin, Fourier Analysis Methods for PDEs, Lecture Notes, 14 November, 2003.


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


    A. Degasperis and M. Procesi, Asymptotic integrability, in Symmetry and Perturbation Theory (Rome, 1998), World Sci. Publishing, River Edge, NJ, 1999, 23-37.


    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), 73-95.doi: 10.1016/S0169-5983(03)00046-7.


    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.


    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.


    J. Escher and J. Seiler, The periodic b-equation and Euler equations on the circle, J. Math. Phys., 51 (2010), 053101, 6pp.doi: 10.1063/1.3405494.


    J. Feng and D. Nualart, Stochastic scalar conservation laws, J. Funct. Anal., 255 (2008), 313-373.doi: 10.1016/j.jfa.2008.02.004.


    F. Flandoli, M. Gubinelli and E. Priola, Well-posedness of the transport equation by stochastic perturbation, Invent. Math., 180 (2010), 1-53.doi: 10.1007/s00222-009-0224-4.


    J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal., 20 (1989), 1388-1425.doi: 10.1137/0520091.


    D. Henry, Infinite propagation speed for the Degasperis-Procesi equation, J. Math. Anal. Appl., 311 (2005), 755-759.doi: 10.1016/j.jmaa.2005.03.001.


    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.


    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), 437-444.doi: 10.1016/S0375-9601(03)00114-2.


    R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.doi: 10.1017/S0022112001007224.


    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), 1-21.doi: 10.1215/S0012-7094-93-07101-3.


    J. U. Kim, On the Cauchy problem for the transport equation with random noise, J. Funct. Anal., 259 (2010), 3328-3359.doi: 10.1016/j.jfa.2010.08.017.


    J. U. Kim, On the stochastic quasi-linear symmertic hyperbolic system, J. Differential Equations, 250 (2011), 1650-1684.doi: 10.1016/j.jde.2010.09.025.


    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.


    Z. W. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.doi: 10.1002/cpa.20239.


    Y. Liu and Z. Y. 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.


    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.


    Y. Matsuno, Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit, Inverse Problems, 21 (2005), 1553-1570.doi: 10.1088/0266-5611/21/5/004.


    O. G. Mustafa, A note on the Degasperis-Procesi equation, J. Nonlinear Math. Phys., 12 (2005), 10-14.doi: 10.2991/jnmp.2005.12.1.2.


    T. Tao, Multilinear weighted convolution of $L^{2}$ functions, and applications to nonlinear dispersive equation, Amer. J. Math., 123 (2001), 839-908.doi: 10.1353/ajm.2001.0035.


    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Second edition, Johann Ambrosius Barth, Heidelberg, 1995.


    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.


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


    Z. Y. 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.


    Z. Y. 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.


    Z. Y. 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.

  • 加载中

Article Metrics

HTML views() PDF downloads(137) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint