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A bilinear estimate with applications to the equation
Authors:
Carlos E. Kenig, Gustavo Ponce and Luis Vega
Journal:
J. Amer. Math. Soc. 9 (1996), 573-603
MSC (1991):
Primary 35Q53; Secondary 35G25, 35D99
MathSciNet review:
1329387
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References |
Similar Articles |
Additional Information
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- 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
, 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:
http://dx.doi.org/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.
Article copyright:
© Copyright 1996 American Mathematical Society
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