\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On well-posedness of the Degasperis-Procesi equation

Abstract Related Papers Cited by
  • It is shown in both the periodic and the non-periodic cases that the data-to-solution map for the Degasperis-Procesi (DP) equation is not a uniformly continuous map on bounded subsets of Sobolev spaces with exponent greater than 3/2. This shows that continuous dependence on initial data of solutions to the DP equation is sharp. The proof is based on well-posedness results and approximate solutions. It also exploits the fact that DP solutions conserve a quantity which is equivalent to the $L^2$ norm. Finally, it provides an outline of the local well-posedness proof including the key estimates for the size of the solution and for the solution's lifespan that are needed in the proof of the main result.
    Mathematics Subject Classification: Primary: 35Q53, 58F17.

    Citation:

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

    J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252.

    [2]

    A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239.doi: 10.1007/s00205-006-0010-z.

    [3]

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

    [4]

    M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.doi: 10.1353/ajm.2003.0040.

    [5]

    O. Christov and S. Hakkaev, On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of the Degasperis-Procesi equation, J. Math. Anal. Appl., 360 (2009), 47-56.doi: 10.1016/j.jmaa.2009.06.035.

    [6]

    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.

    [7]

    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.

    [8]

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

    [9]

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

    [10]

    A. Degasperis and M. Procesi, Asymptotic integrability symmetry and perturbation theory, World Sci. Publ., (1999), 23-37.

    [11]

    C. deLellis, T. Kappeler and P. Topalov, Low-regularity solutions of the periodic Camassa-Holm equation, Comm. Partial Differential Equations, 32 (2007), 87-126.doi: 10.1080/03605300601091470.

    [12]

    Dieudonne, "Foundations of Modern Analysis," Academic Press, 1960.

    [13]

    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.

    [14]

    J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 257-485.doi: 10.1016/j.jfa.2006.03.022.

    [15]

    A. Fokas and B. FuchssteinerSymplectic structures, their Bäklund transformations and hereditary symmetries, Phys. D, 4 (1981/82), 47-66. doi: 10.1016/0167-2789(81)90004-X.

    [16]

    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.

    [17]

    A. Himonas and C. Kenig, Non-uniform dependence on initial data for the CH equation on the line, Differential and Integral Equations, 22 (2009), 201-224.

    [18]

    A. Himonas, C. Kenig and G. Misiołek, Non-uniform dependence for the periodic CH equation, Comm. Partial Differential Equations, 35 (2010), 1145-1162.doi: 10.1080/03605300903436746.

    [19]

    A. Himonas and G. Misiołek, High-frequency smooth solutions and well-posedness of the Camassa-Holm equation, Int. Math. Res. Not., 51 (2005), 3135-3151.doi: 10.1155/IMRN.2005.3135.

    [20]

    A. Himonas and G. Misiołek, The Cauchy problem for an integrable shallow water equation, Differential Integral Equations, 14 (2001), 821-831.

    [21]

    A. Himonas and G. Misiołek, Non-uniform dependence on initial data of solutions to the Euler equations of hydrodynamics, Comm. Math. Phys., 296 (2010), 285-301.doi: 10.1007/s00220-010-0991-1.

    [22]

    A. Himonas, G. Misiołek and G. Ponce, Non-uniform continuity in $H^1$ of the solution map of the CH equation, Asian J. Math., 11 (2007), 141-150.

    [23]

    H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation, Ann. Inst. Fourier (Grenoble), 58 (2008), 945-988.

    [24]

    C. Holliman, Non-uniform dependence and well-posedness for the periodic Hunter-Saxton equation, J. Diff. Int. Eq., 23 (2010), 1150-1194.

    [25]

    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.

    [26]

    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.

    [27]

    C. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. J., 106 (2001), 617-633.doi: 10.1215/S0012-7094-01-10638-8.

    [28]

    H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not., 30 (2005), 1833-1847.doi: 10.1155/IMRN.2005.1833.

    [29]

    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.

    [30]

    Y. Li and P. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations, 162 (2000), 27-63.doi: 10.1006/jdeq.1999.3683.

    [31]

    Y. Liu and Z. Yin, Global existence and blow-up phenomona for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.doi: 10.1007/s00220-006-0082-5.

    [32]

    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.

    [33]

    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.

    [34]

    G. Misiołek, A shallow water equation as a geodesic flow on the Bott-Virasoro group, J. Geom. Phys., 24 (1998), 203-208.doi: 10.1016/S0393-0440(97)00010-7.

    [35]

    L. Molinet, On well-posedness results for the Camassa-Holm equation on the line: A survey, J. Nonlin. Math. Phys., 11 (2004), 521-533.doi: 10.2991/jnmp.2004.11.4.8.

    [36]

    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.

    [37]

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

    [38]

    M. Taylor, Commutator estimates, Proc. Amer. Math. Soc., 131 (2003), 1501-1507.doi: 10.1090/S0002-9939-02-06723-0.

    [39]

    M. E. Taylor, "Pseudodifferential Operators and Nonlinear PDE," Birkhauser, Boston 1991.

    [40]

    M. E. Taylor, "Partial Differential Equations III, Nonlinear Equations," Springer, 1996.

    [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.

    [42]

    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.

    [43]

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

    [44]

    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.

    [45]

    Z. Yin, Global solutions to a new integrable equation with peakons, Ind. Univ. Math. J., 53 (2004), 1189-1210.doi: 10.1512/iumj.2004.53.2479.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(159) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return