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Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II: The KdV-equation. (English) Zbl 0787.35098
[For part I, cf. the review above.] The author studies the initial value problem for the periodic, in space, Korteweg-de Vries equation and some of its variants $$\partial\sb tu+\partial\sb{xxx}u+u\sp k \partial\sb xu=0,\ u(x,0)=\varphi(x)\quad (k \ge 1),\ u\text{ periodic in } x.$$ It is established a local result by Picard’s theorem, verifying a contracting property for the associated integral equation in a suitable space, mainly using Fourier analysis estimates. Combining the existence of local solutions with the $L\sp 2$- conservation law, it is obtained the global solution (after regularization).

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
42B10Fourier type transforms, several variables
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References:
[1] [BP]E. Bombieri, V. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), 337--357. · Zbl 0718.11048 · doi:10.1215/S0012-7094-89-05915-2
[2] [Bo1]J. Bourgain, On {$\Lambda$}(p)-subsets of squares, Israel J. Math. 67:3 (1989), 291-311. · Zbl 0692.43005 · doi:10.1007/BF02764948
[3] [Bo2]J. Bourgain, Exponential sums and nonlinear Schrödinger equations, GAFA 3 (1993), 157--178. · Zbl 0787.35096 · doi:10.1007/BF01896021
[4] [Bo3]J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I: Schrödinger Equations, GAFA 3 (1993), 107--156. · Zbl 0787.35097 · doi:10.1007/BF01896020
[5] [CW]T. Cazenave, F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation inH s , Nonlinear Analysis, Theory Methods and Applications 14:10 (1990), 807--836. · Zbl 0706.35127 · doi:10.1016/0362-546X(90)90023-A
[6] [GiV1]J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation, H. Poincaré Analyse Non Linéaire 2 (1985), 309--327. · Zbl 0586.35042
[7] [GiV2]J. Ginibre, G. Velo, The global Cauchy problem for the nonlinear Klein-Gorobon equation, Math. Z 189 (1985), 487--505. · Zbl 0566.35084 · doi:10.1007/BF01168155
[8] [Gr]E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag (1985). · Zbl 0574.10045
[9] [K1]T. Kato, StrongL p -solutions of the Navier-Stokes equations in $\mathbb{R}$m with applications to weak solutions, Math. Z 187 (1984), 471--480. · Zbl 0545.35073 · doi:10.1007/BF01174182
[10] [K2]T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Physique Théorique 46 (1987), 113--129. · Zbl 0632.35038
[11] [K3]T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Math. Suppl. Studies, Studies in Applied Math. 8 (1983), 93--128.
[12] [KePoVe1]C. Kenig, G. Ponce, L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. AMS 4 (1991), 323--347. · Zbl 0737.35102
[13] [KePoVe2]C. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction princple, preprint. · Zbl 0808.35128
[14] [KePoVe3]C. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Kortewegde Vries equation in Sobolov spaces of negative indices, to appear in Duke Math. J.
[15] [KruF]S. Kruzhkov, A. Faminskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Math. USSR Sbornik 48 (1984), 93--138. · Zbl 0549.35104 · doi:10.1070/SM1984v048n02ABEH002682
[16] [L]P. Lax, Periodic solutions of the KDV equations, Comm. Pure and Applied Math. 26, 141--188 (1975). · Zbl 0295.35004
[17] [LeRSp]J. Lebowitz, H. Rose, E. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Statistical Physics 50:3/4 (1988), 657--687. · Zbl 1084.82506 · doi:10.1007/BF01026495
[18] [MTr]H. P. McKean, E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), 143--226. · Zbl 0339.34024 · doi:10.1002/cpa.3160290203
[19] [MiGKr]R. Miura, M. Gardner, K. Kruskal, Korteweg-de Vries equation and generalizations II. Existence of conservation laws and constant of motion. J. Math. Physics 9:8 (1968), 1204--1209. · Zbl 0283.35019 · doi:10.1063/1.1664701
[20] [PosTr]J. Poschel, E. Trubowitz, Inverse Spectral Theory, Academic Press 37 (1987).
[21] [Sj]A. Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569--579. · Zbl 0185.34602 · doi:10.1016/0022-247X(70)90068-5
[22] [St]R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705--714. · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1
[23] [T]P. Tomas, A restriction for the Fourier transform, Bull. AMS 81 (1975), 477--478. · Zbl 0298.42011 · doi:10.1090/S0002-9904-1975-13790-6
[24] [Tr]E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), 325--341. · Zbl 0403.34022 · doi:10.1002/cpa.3160300305
[25] [Ts]Y. Tsutsumi,L 2-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcialaj Ekvacioj 30 (1987), 115--125. · Zbl 0638.35021
[26] [Vi]I.M. Vinogradov, The Method of Trigonometric Sums in the Theory of Numbers, Intersciences, NY (1954).
[27] [ZS]V. Zakharov, A. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34:1, 62--69 (1972).