WELL-POSEDNESS FOR THE BENJAMIN EQUATIONS

  • Kozono, Hideo ;
  • Ogawa, Takayoshi ;
  • Tanisaka, Hirooki
  • Published : 2001.11.01

Abstract

We consider the time local well-posedness of the Benjamin equation. Like the result due to Keing-Ponce-Vega [10], [12], we show that the initial value problem is time locally well posed in the Sobolev space H$^{s}$ (R) for s>-3/4.

Keywords

Benjamin equation;local well-posedness;initial value problem;KdV equation

References

  1. Physica D v.40 Nonlocal models for nonlinear, dispersive waves L. Abdelouhab;J. L. Boca;M. Felland;J.-C. Saut
  2. J. Amer. Math. Soc. v.9 A bilinear estimate with applications to the KdV equation C. E. Kenig;G. Ponce;L. Vega
  3. J. Funct. Anal. v.158 no.2 Interaction Equations for Short and Long Dispersive D. Bekiranov;T. Ogawa;G. Ponce
  4. GAFA. v.3 Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear equations;Ⅱ. The KdV equation J. Bourgain
  5. J. Funct. Anal. (to appear) On the Cauchy problem for the Zakharov system J. Ginibre;Y. Tsutsumi;G. Velo
  6. Duke Math. J. v.71 The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices C. E. Kenig;G. Ponce;L. Vega
  7. Trans. Ameri. Math. Soc. v.342 On the generalized Benjamin-Ono equation C. E. Kenig;G. Ponce;L. Vega
  8. Invent. Math. v.134 no.3 Smoothing effects and local existence theory for the generalized nonlinear Schrodinger equations C. E. Kenig;G. Ponce;L. Vega
  9. Advances. Math. Sci. Appl. v.10 Well-posedness for the higher order nonlinear Schrodinger equation H. Takaoka
  10. Ill-posedness issues for the Benjamin-Ono and related equations, preprint L. Molinet;J. C. Saut;N. Tzvetkov
  11. J. Fluid Mech. v.245 A new kind of solitary waves T. B. Benjamin
  12. J. Differential Equations v.152 Existence and stability of solitary wave solutions of the Benjamin equation J. Angulo
  13. Comm. Appl. Math. v.46 Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction mapping principle C. E. Kenig;G. Ponce;L. Vega
  14. Diff. Integral Equations v.4 On the global well-posedness of the Benjamin-Ono equation G. Ponce
  15. SIAM J. Math. Anal v.20 The Cauchy problem for the Kortewed-de Vries equation with measure as initial data H. Takaoka
  16. Comm. Partial Differential Equations v.11 no.10 On the Cauchy problem for the Benjamin-Ono equation R. J. Iorio Jr.
  17. J. Differential Equations v.152 L² global well-posedness of the initial value problem associated to the Benjamin equation F. Linares
  18. Counter examples to bilinear estimates related with the KdV equation and the nonilnear Schrodinger equation, preprint K. Nakanishi;H. Takaoka;Y. Tsutsumi
  19. J. Fluid. Mech. v.29 no.2 Internal waves of permanent form in fluids of great depth T. B. Benjamin