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Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations
Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces
1. | Universidad Nacional de Colombia, Bogotá, Colombia, Colombia, Colombia |
References:
[1] |
J. Angulo, M. Scialom and C. Banquet, The regularized Benjamin-Ono and BBM equations: Well-posedness and nonlinear stability, J. Diff. Eqs., 250 (2011), 4011-4036.
doi: 10.1016/j.jde.2010.12.016. |
[2] |
J. P. Albert and J. L. Bona, Comparisons between model equations for long waves, J. Nonlinear Sci., 1 (1991), 345-374.
doi: 10.1007/BF01238818. |
[3] |
T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. |
[4] |
J. Bona and H. Kalisch, Models for internal waves in deep water, Discrete Contin. Dyn. Syst., 6 (2000), 1-20.
doi: 10.3934/dcds.2000.6.1. |
[5] |
H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal. TMA., 4 (1980), 677-681.
doi: 10.1016/0362-546X(80)90068-1. |
[6] |
J. Duoandikoetxea, Fourier Analysis, Grad. Studies in Math., 29 Amer. Math. Soc., 2001. |
[7] |
L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of the k-generalized KdV equations, J. Funct. Anal., 244 (2007), 504-535.
doi: 10.1016/j.jfa.2006.11.004. |
[8] |
L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, The sharp hardy uncertainty principle for Schrödinger evolutions, Duke Math. J., 155 (2010), 163-187
doi: 10.1215/00127094-2010-053. |
[9] |
G. Fonseca and G. Ponce, The I.V.P for the Benjamin-Ono equation in weighted Sobolev spaces, J. Funct. Anal., 260 (2011), 436-459.
doi: 10.1016/j.jfa.2010.09.010. |
[10] |
G. Fonseca, F. Linares and G. Ponce, The I.V.P for the Benjamin-Ono equation in weighted Sobolev spaces II, J. Funct. Anal., 262 (2012), 2031-2049.
doi: 10.1016/j.jfa.2011.12.017. |
[11] |
G. Fonseca, F. Linares and G. Ponce, The IVP for the dispersion generalized Benjamin-Ono equation in weighted Sobolev spaces, Ann. I. H. Poincaré-AN, 30 (2013), 763-790.
doi: 10.1016/j.anihpc.2012.06.006. |
[12] |
R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. AMS., 176 (1973), 227-251. |
[13] |
R. J. Iorio, On the Cauchy problem for the Benjamin-Ono equation, Comm. P. D. E., 11 (1986), 1031-1081.
doi: 10.1080/03605308608820456. |
[14] |
R. J. Iorio, Unique continuation principle for the Benjamin-Ono equation, Diff. and Int. Eqs., 16 (2003), 1281-1291. |
[15] |
R. J. Iorio and V. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge University Press, 2001.
doi: 10.1017/CBO9780511623745. |
[16] |
H. Kalisch, Error analysis of a spectral projection of the regularized Benjamin-Ono equation, BIT, 45 (2005), 69-89.
doi: 10.1007/s10543-005-2636-x. |
[17] |
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math., 8 (1983), 93-128. |
[18] |
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. |
[19] |
C. E. Kenig, G. Ponce and L. Vega, On the unique continuation of solutions to the generalized KdV equation, Math. Res. Letters, 10 (2003), 833-846.
doi: 10.4310/MRL.2003.v10.n6.a10. |
[20] |
D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 5, 39 (1895), 22-443. |
[21] |
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. |
[22] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext. Springer, 2009. |
[23] |
L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988.
doi: 10.1137/S0036141001385307. |
[24] |
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. AMS., 165 (1972), 207-226. |
[25] |
J. Nahas and G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation, Comm. P.D.E., 34 (2009), 1208-1227.
doi: 10.1080/03605300903129044. |
[26] |
J. Nahas and G. Ponce, On the persistent properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces, RIMS Kokyuroku Bessatsu (RIMS Proceedings), (2011), 23-36. |
[27] |
H. Ono, Algebraic solitary waves on stratified fluids, J. Phy. Soc. Japan, 39 (1975), 1082-1091. |
[28] |
G. Ponce, On the global well-posedness of the Benjamin-Ono equation, Diff. Int. Eqs., 4 (1991), 527-542. |
[29] |
E. M. Stein, The characterization of functions arising as potentials, Bull. Amer. Math. Soc., 67 (1961), 102-104. |
[30] |
E. M. Stein, Harmonic Analysis, Princeton University Press, 1993. |
show all references
References:
[1] |
J. Angulo, M. Scialom and C. Banquet, The regularized Benjamin-Ono and BBM equations: Well-posedness and nonlinear stability, J. Diff. Eqs., 250 (2011), 4011-4036.
doi: 10.1016/j.jde.2010.12.016. |
[2] |
J. P. Albert and J. L. Bona, Comparisons between model equations for long waves, J. Nonlinear Sci., 1 (1991), 345-374.
doi: 10.1007/BF01238818. |
[3] |
T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. |
[4] |
J. Bona and H. Kalisch, Models for internal waves in deep water, Discrete Contin. Dyn. Syst., 6 (2000), 1-20.
doi: 10.3934/dcds.2000.6.1. |
[5] |
H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal. TMA., 4 (1980), 677-681.
doi: 10.1016/0362-546X(80)90068-1. |
[6] |
J. Duoandikoetxea, Fourier Analysis, Grad. Studies in Math., 29 Amer. Math. Soc., 2001. |
[7] |
L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of the k-generalized KdV equations, J. Funct. Anal., 244 (2007), 504-535.
doi: 10.1016/j.jfa.2006.11.004. |
[8] |
L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, The sharp hardy uncertainty principle for Schrödinger evolutions, Duke Math. J., 155 (2010), 163-187
doi: 10.1215/00127094-2010-053. |
[9] |
G. Fonseca and G. Ponce, The I.V.P for the Benjamin-Ono equation in weighted Sobolev spaces, J. Funct. Anal., 260 (2011), 436-459.
doi: 10.1016/j.jfa.2010.09.010. |
[10] |
G. Fonseca, F. Linares and G. Ponce, The I.V.P for the Benjamin-Ono equation in weighted Sobolev spaces II, J. Funct. Anal., 262 (2012), 2031-2049.
doi: 10.1016/j.jfa.2011.12.017. |
[11] |
G. Fonseca, F. Linares and G. Ponce, The IVP for the dispersion generalized Benjamin-Ono equation in weighted Sobolev spaces, Ann. I. H. Poincaré-AN, 30 (2013), 763-790.
doi: 10.1016/j.anihpc.2012.06.006. |
[12] |
R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. AMS., 176 (1973), 227-251. |
[13] |
R. J. Iorio, On the Cauchy problem for the Benjamin-Ono equation, Comm. P. D. E., 11 (1986), 1031-1081.
doi: 10.1080/03605308608820456. |
[14] |
R. J. Iorio, Unique continuation principle for the Benjamin-Ono equation, Diff. and Int. Eqs., 16 (2003), 1281-1291. |
[15] |
R. J. Iorio and V. Iorio, Fourier Analysis and Partial Differential Equations, Cambridge University Press, 2001.
doi: 10.1017/CBO9780511623745. |
[16] |
H. Kalisch, Error analysis of a spectral projection of the regularized Benjamin-Ono equation, BIT, 45 (2005), 69-89.
doi: 10.1007/s10543-005-2636-x. |
[17] |
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math., 8 (1983), 93-128. |
[18] |
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. |
[19] |
C. E. Kenig, G. Ponce and L. Vega, On the unique continuation of solutions to the generalized KdV equation, Math. Res. Letters, 10 (2003), 833-846.
doi: 10.4310/MRL.2003.v10.n6.a10. |
[20] |
D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. 5, 39 (1895), 22-443. |
[21] |
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. |
[22] |
F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Universitext. Springer, 2009. |
[23] |
L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal., 33 (2001), 982-988.
doi: 10.1137/S0036141001385307. |
[24] |
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. AMS., 165 (1972), 207-226. |
[25] |
J. Nahas and G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation, Comm. P.D.E., 34 (2009), 1208-1227.
doi: 10.1080/03605300903129044. |
[26] |
J. Nahas and G. Ponce, On the persistent properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces, RIMS Kokyuroku Bessatsu (RIMS Proceedings), (2011), 23-36. |
[27] |
H. Ono, Algebraic solitary waves on stratified fluids, J. Phy. Soc. Japan, 39 (1975), 1082-1091. |
[28] |
G. Ponce, On the global well-posedness of the Benjamin-Ono equation, Diff. Int. Eqs., 4 (1991), 527-542. |
[29] |
E. M. Stein, The characterization of functions arising as potentials, Bull. Amer. Math. Soc., 67 (1961), 102-104. |
[30] |
E. M. Stein, Harmonic Analysis, Princeton University Press, 1993. |
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