Duke Mathematical Journal

On the ill-posedness of some canonical dispersive equations

Carlos E. Kenig, Gustavo Ponce, and Luis Vega

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the initial value problem (IVP) associated to some canonical dispersive equations. Our main concern is to establish the minimal regularity property required in the data which guarantees the local well-posedness of the problem. Measuring this regularity in the classical Sobolev spaces, we show ill-posedness results for Sobolev index above the value suggested by the scaling argument.

Article information

Source
Duke Math. J. Volume 106, Number 3 (2001), 617-633.

Dates
First available in Project Euclid: 13 August 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1092403945

Digital Object Identifier
doi:10.1215/S0012-7094-01-10638-8

Mathematical Reviews number (MathSciNet)
MR1813239

Zentralblatt MATH identifier
1034.35145

Subjects
Primary: 35R25: Improperly posed problems
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B65: Smoothness and regularity of solutions 35Q51: Soliton-like equations [See also 37K40] 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]

Citation

Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis. On the ill-posedness of some canonical dispersive equations. Duke Math. J. 106 (2001), no. 3, 617--633. doi:10.1215/S0012-7094-01-10638-8. http://projecteuclid.org/euclid.dmj/1092403945.


Export citation

References

  • M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974), 249--315.
  • H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I, II, Arch. Rational Mech. Anal. 82 (1983), 313--345., 347--375.
  • B. Birnir, C. E. Kenig, G. Ponce, N. Svanstedt, and L. Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, J. London Math. Soc. (2) 53 (1996), 551--559.
  • B. Birnir, G. Ponce, and N. Svanstedt, The local ill-posedness of the modified KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 529--535.
  • J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, II, Geom. Funct. Anal. 3 (1993), 107--156., 209--262.
  • --. --. --. --., Periodic Korteweg de Vries equation with measures as initial data, Selecta Math. (N.S.) 3 (1997), 115--159.
  • T. Cazenave and F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. 14 (1990), 807--836.
  • J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (1985), 363--401.
  • C. E. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), 527--620.
  • --. --. --. --., A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc. 9 (1996), 573--603.
  • --. --. --. --., Quadratic forms for the 1-D semilinear Schrödinger equation, Trans. Amer. Math. Soc. 348 (1996), 3323--3353.
  • C. E. Kenig and A. Ruiz, A strong type (2,2) estimate for a maximal operator associated to the Schrödinger equation, Trans. Amer. Math. Soc. 280 (1983), 239--246.
  • G. L. Lamb JR., Elements of Soliton Theory, Pure Appl. Math., Wiley, New York, 1980.
  • H. Lindblad, A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations, Duke Math. J. 72 (1993), 503--539.
  • R. M. Miura, Korteweg-de Vries equation and generalizations, I: A remarkable explicit nonlinear transformation, J. Math. Phys. 9 (1968), 1202--1204.
  • E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, 1993.
  • R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705--714.
  • Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), 115--125.
  • N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris. Sér. I Math. 329 (1999), 1043--1047. \enlargethispage20pt
  • M. Wadati, The modified Korteweg-de Vries equation, J. Phys. Soc. Japan 34 (1973), 1289--1296.