Ill-Posedness Issues for the Benjamin-Ono and Related Equations

We establish that the Cauchy problem for the Benjamin--Ono equation and for a rather general class of nonlinear dispersive equations with dispersion slightly weaker than that of the Korteweg--de Vries equation cannot be solved by an iteration scheme based on the Duhamel formula. As a consequence, the flow map fails to be smooth.

[1]  Ronald R. Coifman,et al.  The scattering transform for the Benjamin-Ono equation , 1990 .

[2]  David J. Kaup,et al.  The Inverse Scattering Transform for the Benjamin–Ono Equation , 1998 .

[3]  J. Ginibre,et al.  Propriétés de lissage et existence de solutions pour l'équation de Benjamin-Ono généralisée , 1989 .

[4]  Athanassios S. Fokas,et al.  The Inverse Scattering Transform for the Benjamin‐Ono Equation—A Pivot to Multidimensional Problems , 1983 .

[5]  N. Tzvetkov,et al.  Remark on the local ill-posedness for KdV equation , 1999 .

[6]  Luis Vega,et al.  On the generalized Benjamin-Ono equation , 1994 .

[7]  Ronald Smith,et al.  Nonlinear Kelvin and continental-shelf waves , 1972, Journal of Fluid Mechanics.

[8]  Jean Bourgain,et al.  Periodic Korteweg de Vries equation with measures as initial data , 1997 .

[9]  V. Shrira,et al.  Nonlinear dynamics of vorticity waves in the coastal zone , 1996, Journal of Fluid Mechanics.

[10]  Michael M. Tom.  Smoothing properties of some weak solutions of the Benjamin-Ono equation , 1990, Differential and Integral Equations.

[11]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[12]  J. Ginibre,et al.  Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation , 1991 .

[13]  T. Kubota,et al.  Weakly-Nonlinear, Long Internal Gravity Waves in Stratified Fluids of Finite Depth , 1978 .

[14]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[15]  Gustavo Ponce,et al.  On the global well-posedness of the Benjamin-Ono equation , 1991, Differential and Integral Equations.

[16]  T. Benjamin.  Internal waves of permanent form in fluids of great depth , 1967, Journal of Fluid Mechanics.

[17]  Luis Vega,et al.  Well-posedness of the initial value problem for the Korteweg-de Vries equation , 1991 .

[18]  Jean-Claude Saut,et al.  Well-posedness and ill-posedness results for the Kadomtsev-Petviashvili-I equation , 2002 .

[19]  Felipe Linares,et al.  Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations , 2001 .

[20]  Jr. Rafael José Iório,et al.  On the cauchy problem for the benjamin-ono equation , 1986 .

[21]  J. Saut.  Quelques généralisations de l'équation de Korteweg-de Vries, II , 1979 .

[22]  J. Bona,et al.  Model equations for waves in stratified fluids , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[23]  J. Bona,et al.  Nonlocal models for nonlinear, dispersive waves , 1990 .