American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On a system of semirelativistic equations in the energy space
  • CPAA Home
  • This Issue
  • Next Article
    Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations
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

[1]

Luc Molinet, Francis Ribaud. Well-posedness in $ H^1 $ for generalized Benjamin-Ono equations on the circle. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1295-1311. doi: 10.3934/dcds.2009.23.1295
[2]

Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 449-483. doi: 10.3934/dcds.2017019
[3]

Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941
[4]

Jerry Bona, H. Kalisch. Singularity formation in the generalized Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 27-45. doi: 10.3934/dcds.2004.11.27
[5]

Amin Esfahani, Steve Levandosky. Solitary waves of the rotation-generalized Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 663-700. doi: 10.3934/dcds.2013.33.663
[6]

Sondre Tesdal Galtung. A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1243-1268. doi: 10.3934/dcds.2018051
[7]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521
[8]

Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053
[9]

Jaime Angulo, Carlos Matheus, Didier Pilod. Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system. Communications on Pure & Applied Analysis, 2009, 8 (3) : 815-844. doi: 10.3934/cpaa.2009.8.815
[10]

Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237
[11]

Kenta Ohi, Tatsuo Iguchi. A two-phase problem for capillary-gravity waves and the Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1205-1240. doi: 10.3934/dcds.2009.23.1205
[12]

Lufang Mi, Kangkang Zhang. Invariant Tori for Benjamin-Ono Equation with Unbounded quasi-periodically forced Perturbation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 689-707. doi: 10.3934/dcds.2014.34.689
[13]

C. H. Arthur Cheng, John M. Hong, Ying-Chieh Lin, Jiahong Wu, Juan-Ming Yuan. Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 763-779. doi: 10.3934/dcdsb.2016.21.763
[14]

Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171
[15]

Hongjie Dong. Dissipative quasi-geostrophic equations in critical Sobolev spaces: Smoothing effect and global well-posedness. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1197-1211. doi: 10.3934/dcds.2010.26.1197
[16]

P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691
[17]

Alan Compelli, Rossen Ivanov. Benjamin-Ono model of an internal wave under a flat surface. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4519-4532. doi: 10.3934/dcds.2019185
[18]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078
[19]

Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072
[20]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences