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On well-posedness of the Degasperis-Procesi equation
1. | Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States, United States |
References:
[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. Fuchssteiner, Symplectic 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. |
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. |
[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. Fuchssteiner, Symplectic 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. |
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