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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

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

[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

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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