Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces

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July 2015, 14(4): 1327-1341. doi: 10.3934/cpaa.2015.14.1327

Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces

G. Fonseca ^1, , G. Rodríguez-Blanco ^1, and W. Sandoval ^1,

1.	Universidad Nacional de Colombia, Bogotá, Colombia, Colombia, Colombia

Received May 2013 Revised September 2013 Published April 2015

Abstract
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We consider the initial value problem associated to the regularized Benjamin-Ono equation, rBO. Our aim is to establish local and global well-posedness results in weighted Sobolev spaces via contraction principle. We also prove a unique continuation property that implies that arbitrary polynomial type decay is not preserved yielding sharp results regarding well-posedness of the initial value problem in most weighted Sobolev spaces.

Keywords: Benjamin-Ono equation, well-posedness, weighted Sobolev spaces..

Mathematics Subject Classification: Primary: 35B05; Secondary: 35B6.

Citation: G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327

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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform,, \emph{Trans. AMS.}, 176 (1973), 227. Google Scholar
[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. Google Scholar
[14]	R. J. Iorio, Unique continuation principle for the Benjamin-Ono equation,, \emph{Diff. and Int. Eqs.}, 16 (2003), 1281. Google Scholar
[15]	R. J. Iorio and V. Iorio, Fourier Analysis and Partial Differential Equations,, Cambridge University Press, (2001). doi: 10.1017/CBO9780511623745. Google Scholar
[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. Google Scholar
[17]	T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, \emph{Advances in Mathematics Supplementary Studies, 8 (1983), 93. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Universitext. Springer, (2009). Google Scholar
[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. Google Scholar
[24]	B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function,, \emph{Trans. AMS.}, 165 (1972), 207. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	H. Ono, Algebraic solitary waves on stratified fluids,, \emph{J. Phy. Soc. Japan}, 39 (1975), 1082. Google Scholar
[28]	G. Ponce, On the global well-posedness of the Benjamin-Ono equation,, \emph{Diff. Int. Eqs.}, 4 (1991), 527. Google Scholar
[29]	E. M. Stein, The characterization of functions arising as potentials,, \emph{Bull. Amer. Math. Soc.}, 67 (1961), 102. Google Scholar
[30]	E. M. Stein, Harmonic Analysis,, Princeton University Press, (1993). Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	J. Duoandikoetxea, Fourier Analysis,, Grad. Studies in Math., 29 (2001). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	R. Hunt, B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform,, \emph{Trans. AMS.}, 176 (1973), 227. Google Scholar
[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. Google Scholar
[14]	R. J. Iorio, Unique continuation principle for the Benjamin-Ono equation,, \emph{Diff. and Int. Eqs.}, 16 (2003), 1281. Google Scholar
[15]	R. J. Iorio and V. Iorio, Fourier Analysis and Partial Differential Equations,, Cambridge University Press, (2001). doi: 10.1017/CBO9780511623745. Google Scholar
[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. Google Scholar
[17]	T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation,, \emph{Advances in Mathematics Supplementary Studies, 8 (1983), 93. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations,, Universitext. Springer, (2009). Google Scholar
[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. Google Scholar
[24]	B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function,, \emph{Trans. AMS.}, 165 (1972), 207. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	H. Ono, Algebraic solitary waves on stratified fluids,, \emph{J. Phy. Soc. Japan}, 39 (1975), 1082. Google Scholar
[28]	G. Ponce, On the global well-posedness of the Benjamin-Ono equation,, \emph{Diff. Int. Eqs.}, 4 (1991), 527. Google Scholar
[29]	E. M. Stein, The characterization of functions arising as potentials,, \emph{Bull. Amer. Math. Soc.}, 67 (1961), 102. Google Scholar
[30]	E. M. Stein, Harmonic Analysis,, Princeton University Press, (1993). Google Scholar

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