• 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

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.
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 and 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, 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.

[1]

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

[2]

Baoyan Sun, Kung-Chien Wu. Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2537-2562. doi: 10.3934/dcdsb.2021147

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

Wei Yan, Yimin Zhang, Yongsheng Li, Jinqiao Duan. Sharp well-posedness of the Cauchy problem for the rotation-modified Kadomtsev-Petviashvili equation in anisotropic Sobolev spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5825-5849. doi: 10.3934/dcds.2021097

[11]

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 and Applied Analysis, 2009, 8 (3) : 815-844. doi: 10.3934/cpaa.2009.8.815

[12]

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

[13]

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

[14]

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

[15]

Thomas Kappeler, Riccardo Montalto. Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022048

[16]

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 and Continuous Dynamical Systems - B, 2016, 21 (3) : 763-779. doi: 10.3934/dcdsb.2016.21.763

[17]

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

[18]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393

[19]

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

[20]

Fabio S. Bemfica, Marcelo M. Disconzi, P. Jameson Graber. Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2885-2914. doi: 10.3934/cpaa.2021068

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]