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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,, \emph{J. Diff. Eqs.}, 250 (2011), 4011.
doi: 10.1016/j.jde.2010.12.016. |
[2] |
J. P. Albert and J. L. Bona, Comparisons between model equations for long waves,, \emph{J. Nonlinear Sci.}, 1 (1991), 345.
doi: 10.1007/BF01238818. |
[3] |
T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, \emph{J. Fluid Mech.}, 29 (1967), 559. Google Scholar |
[4] |
J. Bona and H. Kalisch, Models for internal waves in deep water,, \emph{Discrete Contin. Dyn. Syst.}, 6 (2000), 1.
doi: 10.3934/dcds.2000.6.1. |
[5] |
H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations,, \emph{Nonlinear Anal. TMA.}, 4 (1980), 677.
doi: 10.1016/0362-546X(80)90068-1. |
[6] |
J. Duoandikoetxea, Fourier Analysis,, Grad. Studies in Math., 29 (2001).
|
[7] |
L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of the k-generalized KdV equations,, \emph{J. Funct. Anal.}, 244 (2007), 504.
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,, \emph{Duke Math. J.}, 155 (2010), 163.
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,, \emph{J. Funct. Anal.}, 260 (2011), 436.
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,, \emph{J. Funct. Anal.}, 262 (2012), 2031.
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,, \emph{Ann. I. H. Poincar\'e-AN}, 30 (2013), 763.
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,, \emph{Trans. AMS.}, 176 (1973), 227.
|
[13] |
R. J. Iorio, On the Cauchy problem for the Benjamin-Ono equation,, \emph{Comm. P. D. E.}, 11 (1986), 1031.
doi: 10.1080/03605308608820456. |
[14] |
R. J. Iorio, Unique continuation principle for the Benjamin-Ono equation,, \emph{Diff. and Int. Eqs.}, 16 (2003), 1281.
|
[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,, \emph{BIT}, 45 (2005), 69.
doi: 10.1007/s10543-005-2636-x. |
[17] |
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, \emph{Advances in Mathematics Supplementary Studies, 8 (1983), 93.
|
[18] |
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-stokes equations,, \emph{Comm. Pure Appl. Math.}, 41 (1988), 891.
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,, \emph{Math. Res. Letters}, 10 (2003), 833.
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,, \emph{Philos. Mag. 5}, 39 (1895), 22. Google Scholar |
[21] |
H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation,, \emph{Int. Math. Res. Not.}, 30 (2005), 1833.
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,, \emph{SIAM J. Math. Anal.}, 33 (2001), 982.
doi: 10.1137/S0036141001385307. |
[24] |
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function,, \emph{Trans. AMS.}, 165 (1972), 207.
|
[25] |
J. Nahas and G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation,, \emph{Comm. P.D.E.}, 34 (2009), 1208.
doi: 10.1080/03605300903129044. |
[26] |
J. Nahas and G. Ponce, On the persistent properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces,, \emph{RIMS Kokyuroku Bessatsu (RIMS Proceedings)}, (2011), 23.
|
[27] |
H. Ono, Algebraic solitary waves on stratified fluids,, \emph{J. Phy. Soc. Japan}, 39 (1975), 1082.
|
[28] |
G. Ponce, On the global well-posedness of the Benjamin-Ono equation,, \emph{Diff. Int. Eqs.}, 4 (1991), 527.
|
[29] |
E. M. Stein, The characterization of functions arising as potentials,, \emph{Bull. Amer. Math. Soc.}, 67 (1961), 102.
|
[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,, \emph{J. Diff. Eqs.}, 250 (2011), 4011.
doi: 10.1016/j.jde.2010.12.016. |
[2] |
J. P. Albert and J. L. Bona, Comparisons between model equations for long waves,, \emph{J. Nonlinear Sci.}, 1 (1991), 345.
doi: 10.1007/BF01238818. |
[3] |
T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, \emph{J. Fluid Mech.}, 29 (1967), 559. Google Scholar |
[4] |
J. Bona and H. Kalisch, Models for internal waves in deep water,, \emph{Discrete Contin. Dyn. Syst.}, 6 (2000), 1.
doi: 10.3934/dcds.2000.6.1. |
[5] |
H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations,, \emph{Nonlinear Anal. TMA.}, 4 (1980), 677.
doi: 10.1016/0362-546X(80)90068-1. |
[6] |
J. Duoandikoetxea, Fourier Analysis,, Grad. Studies in Math., 29 (2001).
|
[7] |
L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of the k-generalized KdV equations,, \emph{J. Funct. Anal.}, 244 (2007), 504.
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,, \emph{Duke Math. J.}, 155 (2010), 163.
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,, \emph{J. Funct. Anal.}, 260 (2011), 436.
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,, \emph{J. Funct. Anal.}, 262 (2012), 2031.
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,, \emph{Ann. I. H. Poincar\'e-AN}, 30 (2013), 763.
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,, \emph{Trans. AMS.}, 176 (1973), 227.
|
[13] |
R. J. Iorio, On the Cauchy problem for the Benjamin-Ono equation,, \emph{Comm. P. D. E.}, 11 (1986), 1031.
doi: 10.1080/03605308608820456. |
[14] |
R. J. Iorio, Unique continuation principle for the Benjamin-Ono equation,, \emph{Diff. and Int. Eqs.}, 16 (2003), 1281.
|
[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,, \emph{BIT}, 45 (2005), 69.
doi: 10.1007/s10543-005-2636-x. |
[17] |
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, \emph{Advances in Mathematics Supplementary Studies, 8 (1983), 93.
|
[18] |
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-stokes equations,, \emph{Comm. Pure Appl. Math.}, 41 (1988), 891.
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,, \emph{Math. Res. Letters}, 10 (2003), 833.
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,, \emph{Philos. Mag. 5}, 39 (1895), 22. Google Scholar |
[21] |
H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation,, \emph{Int. Math. Res. Not.}, 30 (2005), 1833.
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,, \emph{SIAM J. Math. Anal.}, 33 (2001), 982.
doi: 10.1137/S0036141001385307. |
[24] |
B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function,, \emph{Trans. AMS.}, 165 (1972), 207.
|
[25] |
J. Nahas and G. Ponce, On the persistent properties of solutions to semi-linear Schrödinger equation,, \emph{Comm. P.D.E.}, 34 (2009), 1208.
doi: 10.1080/03605300903129044. |
[26] |
J. Nahas and G. Ponce, On the persistent properties of solutions of nonlinear dispersive equations in weighted Sobolev spaces,, \emph{RIMS Kokyuroku Bessatsu (RIMS Proceedings)}, (2011), 23.
|
[27] |
H. Ono, Algebraic solitary waves on stratified fluids,, \emph{J. Phy. Soc. Japan}, 39 (1975), 1082.
|
[28] |
G. Ponce, On the global well-posedness of the Benjamin-Ono equation,, \emph{Diff. Int. Eqs.}, 4 (1991), 527.
|
[29] |
E. M. Stein, The characterization of functions arising as potentials,, \emph{Bull. Amer. Math. Soc.}, 67 (1961), 102.
|
[30] |
E. M. Stein, Harmonic Analysis,, Princeton University Press, (1993).
|
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