# American Institute of Mathematical Sciences

2011, 31(2): 469-488. doi: 10.3934/dcds.2011.31.469

## On well-posedness of the Degasperis-Procesi equation

 1 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States, United States

Received  March 2010 Revised  February 2011 Published  June 2011

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.
Citation: A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469
##### References:
 [1] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete Contin. Dyn. Syst., 23 (2009), 1241. [2] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Rat. Mech. Anal., 183 (2007), 215. 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. 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. 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. 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. 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. doi: 10.1007/s00205-008-0128-2. [8] R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953. [9] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integral equation with peakon solutions,, Theoret. Math. Phys., 133 (2002), 1463. doi: 10.1023/A:1021186408422. [10] A. Degasperis and M. Procesi, Asymptotic integrability symmetry and perturbation theory,, World Sci. Publ., (1999), 23. [11] C. deLellis, T. Kappeler and P. Topalov, Low-regularity solutions of the periodic Camassa-Holm equation,, Comm. Partial Differential Equations, 32 (2007), 87. 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. 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. doi: 10.1016/j.jfa.2006.03.022. [15] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditary symmetries,, Phys. D, 4 (): 47. 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. 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. [18] A. Himonas, C. Kenig and G. Misiołek, Non-uniform dependence for the periodic CH equation,, Comm. Partial Differential Equations, 35 (2010), 1145. 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. 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. [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. 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. [23] H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation,, Ann. Inst. Fourier (Grenoble), 58 (2008), 945. [24] C. Holliman, Non-uniform dependence and well-posedness for the periodic Hunter-Saxton equation,, J. Diff. Int. Eq., 23 (2010), 1150. [25] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. 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. 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. 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. doi: 10.1155/IMRN.2005.1833. [29] 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. [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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0362-546X(01)00791-X. [38] M. Taylor, Commutator estimates,, Proc. Amer. Math. Soc., 131 (2003), 1501. doi: 10.1090/S0002-9939-02-06723-0. [39] M. E. Taylor, "Pseudodifferential Operators and Nonlinear PDE,", Birkhauser, (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. 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. 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. [44] 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. [45] Z. Yin, Global solutions to a new integrable equation with peakons,, Ind. Univ. Math. J., 53 (2004), 1189. doi: 10.1512/iumj.2004.53.2479.

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##### References:
 [1] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete Contin. Dyn. Syst., 23 (2009), 1241. [2] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Rat. Mech. Anal., 183 (2007), 215. 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. 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. 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. 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. 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. doi: 10.1007/s00205-008-0128-2. [8] R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953. [9] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integral equation with peakon solutions,, Theoret. Math. Phys., 133 (2002), 1463. doi: 10.1023/A:1021186408422. [10] A. Degasperis and M. Procesi, Asymptotic integrability symmetry and perturbation theory,, World Sci. Publ., (1999), 23. [11] C. deLellis, T. Kappeler and P. Topalov, Low-regularity solutions of the periodic Camassa-Holm equation,, Comm. Partial Differential Equations, 32 (2007), 87. 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. 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. doi: 10.1016/j.jfa.2006.03.022. [15] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäklund transformations and hereditary symmetries,, Phys. D, 4 (): 47. 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. 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. [18] A. Himonas, C. Kenig and G. Misiołek, Non-uniform dependence for the periodic CH equation,, Comm. Partial Differential Equations, 35 (2010), 1145. 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. 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. [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. 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. [23] H. Holden and X. Raynaud, Periodic conservative solutions of the Camassa-Holm equation,, Ann. Inst. Fourier (Grenoble), 58 (2008), 945. [24] C. Holliman, Non-uniform dependence and well-posedness for the periodic Hunter-Saxton equation,, J. Diff. Int. Eq., 23 (2010), 1150. [25] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. 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. 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. 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. doi: 10.1155/IMRN.2005.1833. [29] 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. [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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0362-546X(01)00791-X. [38] M. Taylor, Commutator estimates,, Proc. Amer. Math. Soc., 131 (2003), 1501. doi: 10.1090/S0002-9939-02-06723-0. [39] M. E. Taylor, "Pseudodifferential Operators and Nonlinear PDE,", Birkhauser, (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. 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. 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. [44] 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. [45] Z. Yin, Global solutions to a new integrable equation with peakons,, Ind. Univ. Math. J., 53 (2004), 1189. doi: 10.1512/iumj.2004.53.2479.
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