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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 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.

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