×

zbMATH — the first resource for mathematics

Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I: Schrödinger equations. (English) Zbl 0787.35097
In this first part the author develops a harmonic analysis method to solve the nonlinear periodic (in the space variables) Schrödinger equation (NLSE) \[ i\partial_ tu+\Delta_ xu+u | u |^{p- 2}=0,\quad (p \geq 3),\quad u(x,0)=\varphi(x). \] The method is based on solving the equivalent integral equation by Picard’s fixed point method, where the nonlinearity is controlled in the iteration process by the Strichartz’s inequality.
The main idea here is to try to adjust this approach to the periodic case. In such a case, the estimates are local in time and hence, at this stage, only a local solution is obtained. In order to obtain the global solution, the local results must be combined with the conservation laws. The main feature of the approach followed here is an analysis on multiple Fourier series.
As an illustration of the results, we have that, if \(n=3\), the NLSE has a global unique solution for \(4<p<6\) and sufficiently small \(H^ 1\)-data. For dimensions \(n \geq 4\), only local wellposedness statements are obtained.
[For part II, see the review below].
Reviewer: L.Vazquez (Madrid)

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [BP]E. Bombieri, J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59, 337–357 (1989). · 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, in this issue.
[4] [CW]T. Cazenave, F. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation inH 3, Nonlinear Analysis, Theory Methods and Applications 14:10 (1990), 807–836. · Zbl 0706.35127
[5] [GiV]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
[6] [Gr]E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag (1985). · Zbl 0574.10045
[7] [K1]T. Kato, StrongL p -solutions of the Navier-Stokes equations inR m with applications to weak solutions, Math. Z 187 (1984), 471–480. · Zbl 0545.35073
[8] [K2]T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Physique Théorique 46 (1987), 113–129. · Zbl 0632.35038
[9] [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.
[10] [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
[11] [KePoVe2]C. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, preprint. · Zbl 0808.35128
[12] [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
[13] [L]P. Lax, Periodic solutions of the KDV equations, Comm. Pure and Applied Math. 28, 141–188 (1975). · Zbl 0302.35008
[14] [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
[15] [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
[16] [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
[17] [PosTr]J. Poschel, E. Trubowitz, Inverse Spectral Theory, Academic Press 1987.
[18] [Sj]A. Sjöberg, On the Korteweg-de Vries equation: existence and uniqueness, J. Math. Anal. Appl. 29 (1970), 569–579. · Zbl 0185.34602
[19] [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
[20] [T]P. Tomas, A restriction for the Fourier transform, Bull. AMS 81 (1975), 477–478. · Zbl 0298.42011
[21] [Tr]E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math. 30 (1977), 325–341. · Zbl 0403.34022
[22] [Ts]Y. Tsutsumi,L 2 -solutions for nonlinear Schrödinger Equations and nonlinear groups, Funkcialaj Ekvacioj 30 (1987), 115–125. · Zbl 0638.35021
[23] [Vi]I.M. Vinogradov, The Method of Trigonometric Sums in the Theory of Numbers, Intersciences, NY (1954).
[24] [ZS]V. Zakharov, A. Shabat, Exact theory of two-dimenional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34:1, 62–69 (1972).
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.