A bilinear estimate with applications to the KdV equation

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..

Bona, Jerry, 1976, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J., 43, 87

Bona, J. L., 1975, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278, 555, 10.1098/rsta.1975.0035

Kantorovitch, L., 1939, The method of successive approximations for functional equations, Acta Math., 71, 63, 10.1007/BF02547750

Murray, Amy Cohen, 1978, Solutions of the Korteweg-de Vries equation from irregular data, Duke Math. J., 45, 149

Cohen, Amy, 1987, Solutions to the Korteweg-de Vries equation with initial profile in 𝐿¹₁(𝑅)∩𝐿¹_{𝑁}(𝑅⁺), SIAM J. Math. Anal., 18, 991, 10.1137/0518076

Fefferman, Charles, 1970, Inequalities for strongly singular convolution operators, Acta Math., 124, 9, 10.1007/BF02394567

Fefferman, Charles, 1973, A note on spherical summation multipliers, Israel J. Math., 15, 44, 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.

Gardner, Clifford S., 1974, Korteweg-deVries equation and generalization. VI. Methods for exact solution, Comm. Pure Appl. Math., 27, 97, 10.1002/cpa.3160270108

Kato, Tosio, 1975, Quasi-linear equations of evolution, with applications to partial differential equations, 25

Kato, Tosio, 1979, On the Korteweg-de Vries equation, Manuscripta Math., 28, 89, 10.1007/BF01647967

Kato, Tosio, 1983, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, 93

Kappeler, Thomas, 1986, Solutions to the Korteweg-de Vries equation with irregular initial profile, Comm. Partial Differential Equations, 11, 927, 10.1080/03605308608820451

Kenig, Carlos E., 1991, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40, 33, 10.1512/iumj.1991.40.40003

Kenig, Carlos E., 1991, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4, 323, 10.2307/2939277

Kenig, Carlos E., 1993, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46, 527, 10.1002/cpa.3160460405

Kenig, Carlos E., 1993, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71, 1, 10.1215/S0012-7094-93-07101-3

Klainerman, S., 1993, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math., 46, 1221, 10.1002/cpa.3160460902

\bysame, 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.

Kruzhkov, S. N., 1983, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Mat. Sb. (N.S.), 120(162), 396

Lindblad, Hans, 1993, A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations, Duke Math. J., 72, 503, 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.

Miura, Robert M., 1968, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys., 9, 1202, 10.1063/1.1664700

Pecher, Hartmut, 1984, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185, 261, 10.1007/BF01181697

Ponce, Gustavo, 1993, Local regularity of nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations, 18, 169, 10.1080/03605309308820925

Sachs, Robert L., 1985, Classical solutions of the Korteweg-de Vries equation for nonsmooth initial data via inverse scattering, Comm. Partial Differential Equations, 10, 29, 10.1080/03605308508820371

Saut, J.-C., 1979, Sur quelques généralisations de l’équation de Korteweg-de Vries, J. Math. Pures Appl. (9), 58, 21

Saut, J. C., 1976, Remarks on the Korteweg-de Vries equation, Israel J. Math., 24, 78, 10.1007/BF02761431

Sjöberg, Anders, 1970, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl., 29, 569, 10.1016/0022-247X(70)90068-5

Strichartz, Robert S., 1977, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44, 705

Tanaka, Shunichi, 1974, Korteweg-de Vries equation: construction of solutions in terms of scattering data, Osaka Math. J., 11, 49

Temam, R., 1969, Sur un problème non linéaire, J. Math. Pures Appl. (9), 48, 159

Tsutsumi, Yoshio, 1989, The Cauchy problem for the Korteweg-de Vries equation with measures as initial data, SIAM J. Math. Anal., 20, 582, 10.1137/0520041

Zygmund, A., 1974, On Fourier coefficients and transforms of functions of two variables, Studia Math., 50, 189, 10.4064/sm-50-2-189-201