Well-posedness of the initial value problem for the Korteweg-de Vries equation
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- by Carlos E. Kenig, Gustavo Ponce and Luis Vega
- J. Amer. Math. Soc. 4 (1991), 323-347
- DOI: https://doi.org/10.1090/S0894-0347-1991-1086966-0
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References
- T. B. Benjamin, Internal waves of permanent form in fluid of great depth, J. Fluid Mech. 29 (1967), 559-592.
- 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
- A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 177312, DOI 10.1073/pnas.53.5.1092
- Lennart Carleson, Some analytic problems related to statistical mechanics, Euclidean harmonic analysis (Proc. Sem., Univ. Maryland, College Park, Md., 1979) Lecture Notes in Math., vol. 779, Springer, Berlin, 1980, pp. 5–45. MR 576038 R. R. Coifman and Y. Meyer, Au delá des opérateurs pseudodifféntieles, Astérisque 57 (1978).
- R. R. Coifman and Yves Meyer, Nonlinear harmonic analysis, operator theory and P.D.E, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 3–45. MR 864370
- P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), no. 2, 413–439. MR 928265, DOI 10.1090/S0894-0347-1988-0928265-0
- Björn E. J. Dahlberg and Carlos E. Kenig, A note on the almost everywhere behavior of solutions to the Schrödinger equation, Harmonic analysis (Minneapolis, Minn., 1981) Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 205–209. MR 654188
- 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), no. 4, 363–401. MR 839728
- J. Ginibre and G. Velo, Commutator expansions and smoothing properties of generalized Benjamin-Ono equations, Ann. Inst. H. Poincaré Phys. Théor. 51 (1989), no. 2, 221–229 (English, with French summary). MR 1033618 —, Smoothing properties and existence of solutions for the generalized Benjamin-One equation, preprint.
- J. Ginibre and Y. Tsutsumi, Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 20 (1989), no. 6, 1388–1425. MR 1019307, DOI 10.1137/0520091 T. Kato, Quasilinear equations of evolutions, with applications to partial differential equations, Lectures Notes in Math., vol. 448, Springer-Verlag, Berlin and New York, 1975, pp. 27-50. —, On the Korteweg-de Vries equation, Manuscripta Math. 29 (1979), 89-99.
- 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
- Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. MR 951744, DOI 10.1002/cpa.3160410704
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J. 59 (1989), no. 3, 585–610. MR 1046740, DOI 10.1215/S0012-7094-89-05927-9
- 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 and Alberto Ruiz, A strong type $(2,\,2)$ estimate for a maximal operator associated to the Schrödinger equation, Trans. Amer. Math. Soc. 280 (1983), no. 1, 239–246. MR 712258, DOI 10.1090/S0002-9947-1983-0712258-4 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. 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
- Bernard Marshall, Mixed norm estimates for the Klein-Gordon equation, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 638–649. MR 730098
- Robert M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976), no. 3, 412–459. MR 404890, DOI 10.1137/1018076
- Hiroaki Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan 39 (1975), no. 4, 1082–1091. MR 398275, DOI 10.1143/JPSJ.39.1082
- 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, On the global well-posedness of the Benjamin-Ono equation, Differential Integral Equations 4 (1991), no. 3, 527–542. MR 1097916
- Gustavo Ponce and Luis Vega, Nonlinear small data scattering for the generalized Korteweg-de Vries equation, J. Funct. Anal. 90 (1990), no. 2, 445–457. MR 1052343, DOI 10.1016/0022-1236(90)90092-Y
- 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 P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J. 55 (1987), 699-715.
- E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 307–355. MR 864375
- Elias M. Stein and Rami Shakarchi, Fourier analysis, Princeton Lectures in Analysis, vol. 1, Princeton University Press, Princeton, NJ, 2003. An introduction. MR 1970295
- 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
- Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 358216, DOI 10.1090/S0002-9904-1975-13790-6
- Luis Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), no. 4, 874–878. MR 934859, DOI 10.1090/S0002-9939-1988-0934859-0 —, Doctoral thesis, Universidad Autonoma, Madrid, Spain, 1987.
Bibliographic Information
- © Copyright 1991 American Mathematical Society
- Journal: J. Amer. Math. Soc. 4 (1991), 323-347
- MSC: Primary 35Q53
- DOI: https://doi.org/10.1090/S0894-0347-1991-1086966-0
- MathSciNet review: 1086966