Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II: The KdV-equation. (English) Zbl 0787.35098

Bourgain, J.

Geom. Funct. Anal. 3, No. 3, 209-262 (1993).

[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 \[ \partial_ tu+\partial_{xxx}u+u^ k \partial_ xu=0,\;u(x,0)=\varphi(x)\quad (k \geq 1),\;u\text{ periodic in } x. \] 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 \(L^ 2\)- conservation law, it is obtained the global solution (after regularization).

Reviewer: L.Vazquez (Madrid)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
42B10	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Keywords:

periodic solution; Fourier analysis; integral equation; Korteweg-de Vries equation; Picard’s theorem; local solutions; global solution

Citations:

Zbl 0787.35097

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
[2]	[Bo1]J. Bourgain, On {\(\Lambda\)}(p)-subsets of squares, Israel J. Math. 67:3 (1989), 291-311. · Zbl 0692.43005
[3]	[Bo2]J. Bourgain, Exponential sums and nonlinear Schrödinger equations, GAFA 3 (1993), 157–178. · Zbl 0787.35096
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[23]	[T]P. Tomas, A restriction for the Fourier transform, Bull. AMS 81 (1975), 477–478. · Zbl 0298.42011
[24]	[Tr]E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), 325–341. · Zbl 0403.34022
[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).