×

zbMATH — the first resource for mathematics

A bilinear estimate with applications to the KdV equation. (English) Zbl 0848.35114
The authors start the work with a wide panoramic view of the results related to the initial value problem for the Korteweg-de Vries equation. The main result of the paper is to have found the lower best index \(s\), such that we have local well posedness in \(H^s(\mathbb{R})\): Existence, uniqueness, persistence and continuous dependence on the data, for a finite time interval, whose size depends on \(|u_0|_{H^s}\). The value \(s> -3/4\), is the optimal one provided by the used method. The result improves a previous one of the same authors. The method combines oscillatory integral estimates with bilinear estimates for \(\partial_x(u^2/2)\) in the Bourgain function spaces associated with the index \(s\). The estimates extend to the periodic case.
Reviewer: L.Vazquez (Madrid)

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35G25 Initial value problems for nonlinear higher-order PDEs
35D99 Generalized solutions to partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
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.. · Zbl 0855.35112
[2] 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. · Zbl 0335.35032
[3] 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. · Zbl 0306.35027
[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. · Zbl 0787.35097
[5] Amy Cohen Murray, Solutions of the Korteweg-de Vries equation from irregular data, Duke Math. J. 45 (1978), no. 1, 149 – 181. · Zbl 0372.35022
[6] Amy Cohen and Thomas Kappeler, Solutions to the Korteweg-de Vries equation with initial profile in \?\textonesuperior \(_{1}\)(\?)\cap \?\textonesuperior _{\?}(\?\(^{+}\)), SIAM J. Math. Anal. 18 (1987), no. 4, 991 – 1025. · Zbl 0651.35075
[7] Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9 – 36. · Zbl 0188.42601
[8] Charles Fefferman, A note on spherical summation multipliers, Israel J. Math. 15 (1973), 44 – 52. · Zbl 0262.42007
[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. · Zbl 1061.35520
[10] 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. · Zbl 0291.35012
[11] 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), Springer, Berlin, 1975, pp. 25 – 70. Lecture Notes in Math., Vol. 448.
[12] Tosio Kato, On the Korteweg-de Vries equation, Manuscripta Math. 28 (1979), no. 1-3, 89 – 99. · Zbl 0415.35070
[13] 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. · Zbl 0549.34001
[14] Thomas Kappeler, Solutions to the Korteweg-de Vries equation with irregular initial profile, Comm. Partial Differential Equations 11 (1986), no. 9, 927 – 945. · Zbl 0606.35079
[15] 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. · Zbl 0738.35022
[16] 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. · Zbl 0737.35102
[17] 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. · Zbl 0808.35128
[18] 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. · Zbl 0787.35090
[19] 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. · Zbl 0803.35095
[20] ——, Smoothing estimates for null forms and applications, to appear Duke Math. J. · Zbl 0909.35094
[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. · JFM 26.0881.02
[22] 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). · Zbl 0537.35068
[23] 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. · Zbl 0797.35123
[24] H. Lindblad and C. D. Sogge, On existence and scattering with minimal regularity for semilinear wave equations, preprint. · Zbl 0846.35085
[25] Robert M. Miura, Korteweg-de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation, J. Mathematical Phys. 9 (1968), 1202 – 1204. , https://doi.org/10.1063/1.1664700 Robert M. Miura, Clifford S. Gardner, and Martin D. Kruskal, Korteweg-de Vries equation and generalizations. II. Existence of conservation laws and constants of motion, J. Mathematical Phys. 9 (1968), 1204 – 1209. · Zbl 0283.35019
[26] Hartmut Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), no. 2, 261 – 270. · Zbl 0538.35063
[27] 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. · Zbl 0803.35096
[28] 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. · Zbl 0562.35085
[29] 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). · Zbl 0449.35083
[30] J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation, Israel J. Math. 24 (1976), no. 1, 78 – 87. · Zbl 0334.35062
[31] Anders Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569 – 579. · Zbl 0179.43101
[32] 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. · Zbl 0372.35001
[33] Shunichi Tanaka, Korteweg-de Vries equation: construction of solutions in terms of scattering data, Osaka J. Math. 11 (1974), 49 – 59. · Zbl 0283.35063
[34] R. Temam, Sur un problème non linéaire, J. Math. Pures Appl. (9) 48 (1969), 159 – 172 (French). · Zbl 0187.03902
[35] 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. · Zbl 0679.35078
[36] A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189 – 201. · Zbl 0278.42005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.