American Institute of Mathematical Sciences

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

On well-posedness of the Degasperis-Procesi equation

A. Alexandrou Himonas 1, and  Curtis Holliman 1,

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.
Keywords: non-uniform dependence on initial data, conserved quantities., DP equation, well-posedness, approximate solutions, Sobolev spaces, commutator estimate, Cauchy problem.
Mathematics Subject Classification: Primary: 35Q53, 58F1.
Citation: A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems, 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), 1241-1252.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[39]

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

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[39]

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

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[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.  Google Scholar

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