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A bilinear estimate with applications to the $\text{KdV}$ equation

Author(s): Carlos E. Kenig; Gustavo Ponce; Luis Vega
Journal: J. Amer. Math. Soc. 9 (1996), 573-603.
MSC (1991): Primary 35Q53; Secondary 35G25, 35D99
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References:

1.
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..

2.
J. L. Bona and R. Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J 43 (1976), 87--99. MR 52:14694

3.
J. L. Bona and R. Smith, The initial value problem for the Korteweg-de Vries equation, Roy. Soc. London Ser A 278 (1975), 555--601. MR 52:6219

4.
J. Bourgain, Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geometric and Functional Anal. 3 (1993), 107-156, 209-262. MR 95d:35160a,b

5.
A. Cohen, Solutions of the Korteweg-de Vries equation from irregular data, Duke Math. J. 45 (1978), 149--181. MR 57:10283

6.
A. Cohen and T. Kappeler, Solution to the Korteweg-de Vries equation with initial profile in $ L^{1}_{1}(\mathbb{R}) \cap L^{1}_{N}(\mathbb{R}^{+})$, SIAM J. Math. Anal. 18 (1987), 991--1025. MR 88j:35138

7.
C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9--36. MR 41:2468

8.
------, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44--52. MR 47:9160

9.
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.

10.
------, The Korteweg-de Vries equation and generalizations. VI. Method for exact solutions, Comm. Pure Appl. Math. 27 (1974), 97--133. MR 49:898

11.
T. Kato, Quasilinear equations of evolutions, with applications to partial differential equation, Lecture Notes in Math. 448, Springer (1975), 27--50. MR 53:11252

12.
------, On the Korteweg-de Vries equation, Manuscripta Math 29 (1979), 89--99. MR 80d:35128

13.
------, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math. 8 (1983), 93--128. MR 86f:35160

14.
T. Kappeler, Solutions to the Korteweg-de Vries equation with irregular initial data, Comm. P.D.E. 11 (1986), 927--945. MR 87j:35326

15.
C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana U. Math. J. 40 (1991), 33-69. MR 92d:35081

16.
------, Well-posedness of the initial value problem for the Korteweg-de Vries, J. Amer. Math. Soc 4 (1991), 323--347. MR 92c:35106

17.
------, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), 527-620. MR 94h:35229

18.
------, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J. 71 (1993), 1-21. MR 94g:35196

19.
S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221--1268. MR 94h:35137

20.
------, Smoothing estimates for null forms and applications, to appear Duke Math. J.

21.
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.

22.
S. N. Kruzhkov and A. V. Faminskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Math. U.S.S.R. Sbornik 48 (1984), 93--138. MR 85c:35079

23.
H. Lindblad, A sharp counter example to local existence of low regularity solutions to nonlinear wave equations, Duke Math. J. 72 (1993), 503-539. MR 94h:35165

24.
H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, preprint.

25.
R. M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9 (1968), 1202-1209. MR 40:6042a

26.
H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), 261--270. MR 85h:35165

27.
G. Ponce and T. C. Sideris, Local regularity of nonlinear wave equations in three space dimensions, Comm. P.D.E. 18 (1993), 169-177. MR 95a:35092

28.
R. L. Sachs, Classical solutions of the Korteweg-de Vries equation for non-smooth initial data via inverse scattering, Comm. P.D.E. 10 (1985), 29--89. MR 86h:35126

29.
J.-C. Saut, Sur quelques généralisations de l' équations de Korteweg-de Vries, J. Math. Pures Appl. 58 (1979), 21--61. MR 82m:35133

30.
J.-C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation, Israel J. Math. 24 (1976), 78--87. MR 56:12676

31.
A. Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569--579. MR 53:13885

32.
R. S. Strichartz, Restriction of Fourier transforms to quadratic surface and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705-714. MR 58:23577

33.
S. Tanaka, Korteweg-de Vries equation: construction of solutions in terms of scattering data, Osaka J. Math. 11 (1974), 49--59. MR 50:5231

34.
R. Temam, Sur un problème non linéaire, J. Math. Pures Appl. 48 (1969), 159--172. MR 41:5799

35.
Y. Tsutsumi, The Cauchy problem for the Korteweg-de Vries equation with measure as initial data, SIAM J. Math. Anal. 20 (1989), 582--588. MR 90g:35153

36.
A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189-201. MR 52:8788


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Additional Information:

Carlos E. Kenig
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
Email: cek@math.uchicago.edu

Gustavo Ponce
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: ponce@math.ucsb.edu

Luis Vega
Affiliation: Departamento de Matematicas, Universidad del Pais Vasco, Apartado 644, 48080 Bilbao, Spain
Email: MTPVEGOL@lg.ehu.es

DOI: 10.1090/S0894-0347-96-00200-7
PII: S 0894-0347(96)00200-7
Keywords: Schr\"{o}dinger equation, initial value problem, well-posedness
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 of article: Copyright 1996, American Mathematical Society


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