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

G. Fonseca; G. Rodríguez-Blanco; W. Sandoval

doi:10.3934/cpaa.2015.14.1327

Article Contents

2015, Volume 14, Issue 4: 1327-1341. Doi: 10.3934/cpaa.2015.14.1327

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

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

1.
Universidad Nacional de Colombia, Bogotá

Received: May 2013

Revised: September 2013

Published: July 2015

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35B05; Secondary: 35B60.

Citation:

Related Papers

Cited by

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), 163-187doi: 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.