A bilinear estimate with applications to the KdV equation
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- by Carlos E. Kenig, Gustavo Ponce and Luis Vega PDF
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References
- 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, to appear J. London Math. Soc..
- Jerry Bona and Ridgway Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J. 43 (1976), no. 1, 87–99. MR 393887
- J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), no. 1287, 555–601. MR 385355, DOI 10.1098/rsta.1975.0035
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
- Amy Cohen Murray, Solutions of the Korteweg-de Vries equation from irregular data, Duke Math. J. 45 (1978), no. 1, 149–181. MR 470533
- Amy Cohen and Thomas Kappeler, Solutions to the Korteweg-de Vries equation with initial profile in $L^1_1(\textbf {R})\cap L^1_N(\textbf {R}^+)$, SIAM J. Math. Anal. 18 (1987), no. 4, 991–1025. MR 892486, DOI 10.1137/0518076
- Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. MR 257819, DOI 10.1007/BF02394567
- Charles Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44–52. MR 320624, DOI 10.1007/BF02771772
- C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, A method for solving the Korteweg-de Vries equation, Phys. Rev. Letters 19 (1967), 1095–1097.
- Clifford S. Gardner, John M. Greene, Martin D. Kruskal, and Robert M. Miura, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math. 27 (1974), 97–133. MR 336122, DOI 10.1002/cpa.3160270108
- Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, pp. 25–70. MR 0407477
- Tosio Kato, On the Korteweg-de Vries equation, Manuscripta Math. 28 (1979), no. 1-3, 89–99. MR 535697, DOI 10.1007/BF01647967
- Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR 759907
- Thomas Kappeler, Solutions to the Korteweg-de Vries equation with irregular initial profile, Comm. Partial Differential Equations 11 (1986), no. 9, 927–945. MR 844170, DOI 10.1080/03605308608820451
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33–69. MR 1101221, DOI 10.1512/iumj.1991.40.40003
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc. 4 (1991), no. 2, 323–347. MR 1086966, DOI 10.1090/S0894-0347-1991-1086966-0
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620. MR 1211741, DOI 10.1002/cpa.3160460405
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), no. 1, 1–21. MR 1230283, DOI 10.1215/S0012-7094-93-07101-3
- S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221–1268. MR 1231427, DOI 10.1002/cpa.3160460902
- —, Smoothing estimates for null forms and applications, to appear Duke Math. J.
- 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), 422–443.
- S. N. Kruzhkov and A. V. Faminskiĭ, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Mat. Sb. (N.S.) 120(162) (1983), no. 3, 396–425 (Russian). MR 691986
- Hans Lindblad, A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations, Duke Math. J. 72 (1993), no. 2, 503–539. MR 1248683, DOI 10.1215/S0012-7094-93-07219-5
- H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, preprint.
- Robert M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202–1204. MR 252825, DOI 10.1063/1.1664700
- Hartmut Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), no. 2, 261–270. MR 731347, DOI 10.1007/BF01181697
- Gustavo Ponce and Thomas C. Sideris, Local regularity of nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations 18 (1993), no. 1-2, 169–177. MR 1211729, DOI 10.1080/03605309308820925
- Robert L. Sachs, Classical solutions of the Korteweg-de Vries equation for nonsmooth initial data via inverse scattering, Comm. Partial Differential Equations 10 (1985), no. 1, 29–98. MR 773211, DOI 10.1080/03605308508820371
- J.-C. Saut, Sur quelques généralisations de l’équation de Korteweg-de Vries, J. Math. Pures Appl. (9) 58 (1979), no. 1, 21–61 (French). MR 533234
- J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation, Israel J. Math. 24 (1976), no. 1, 78–87. MR 454425, DOI 10.1007/BF02761431
- Anders Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569–579. MR 410135, DOI 10.1016/0022-247X(70)90068-5
- Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
- Shunichi Tanaka, Korteweg-de Vries equation: construction of solutions in terms of scattering data, Osaka Math. J. 11 (1974), 49–59. MR 352744
- R. Temam, Sur un problème non linéaire, J. Math. Pures Appl. (9) 48 (1969), 159–172 (French). MR 261183
- Yoshio Tsutsumi, The Cauchy problem for the Korteweg-de Vries equation with measures as initial data, SIAM J. Math. Anal. 20 (1989), no. 3, 582–588. MR 990865, DOI 10.1137/0520041
- A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189–201. MR 387950, DOI 10.4064/sm-50-2-189-201
Additional Information
- Carlos E. Kenig
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- MR Author ID: 100230
- Email: cek@math.uchicago.edu
- Gustavo Ponce
- Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
- MR Author ID: 204988
- Email: ponce@math.ucsb.edu
- Luis Vega
- Affiliation: Departamento de Matematicas, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao, Spain
- MR Author ID: 237776
- Email: MTPVEGOL@lg.ehu.es
- Received by editor(s): July 13, 1994
- Received by editor(s) in revised form: May 11, 1995
- Additional Notes: C. E. Kenig and G. Ponce were supported by NSF grants. L. Vega was supported by a DGICYT grant.
- © Copyright 1996 American Mathematical Society
- Journal: J. Amer. Math. Soc. 9 (1996), 573-603
- MSC (1991): Primary 35Q53; Secondary 35G25, 35D99
- DOI: https://doi.org/10.1090/S0894-0347-96-00200-7
- MathSciNet review: 1329387