Staffilani, Gigliola; Tataru, Daniel Strichartz estimates for a Schrödinger operator with nonsmooth coefficients. (English) Zbl 1010.35015 Commun. Partial Differ. Equations 27, No. 7-8, 1337-1372 (2002). Consequences of the stability analysis of the Cauchy problem for partial differential equations of Schrödinger type are numerous; besides their immediate dispersive interpretation and relevance they may concern, via inverse scattering method, the study (e.g., existence and uniqueness) of solitons, nonlinear KdV waves etc. A few mathematical challenges emerge in this context, one being the loss of an easy form of the localization argument which works easily for the hyperbolic wave equations. This is settled here via the assumption of an asymptotically constant form of the coefficients. Another technical difficulty emerges in connection with the non-smoothness of the coefficients in quasilinear problems. In this context, the main subject of this paper is the re-derivation of the fundamental (so-called Strichartz) inequalities (which, rougly speaking, inter-relate the \(q\)- and \(r\)-norms with different “admissible” \(q\) and \(r\), and provide an estimate for an “improved” norm of the solutions in terms of the “standard” \(r=2\) norm of the initial state) under the properly weakened assumptions. The proof starts from the “smooth plus non-smooth” split of the solution, with emphasis on the compact spatial support of the (mutually cancelled) inhomogeneous terms. Its main idea lies in a construction (by a suitable integral transformation) of the so-called microlocal parametrix, followed by the estimate obtained by the method of the stationary phase. Reviewer: Miloslav Znojil (Řež) Cited in 2 ReviewsCited in 91 Documents MSC: 35B45 A priori estimates in context of PDEs 81U30 Dispersion theory, dispersion relations arising in quantum theory Keywords:Schrödinger equation; Cauchy problem; microlocal parametrix PDFBibTeX XMLCite \textit{G. Staffilani} and \textit{D. Tataru}, Commun. Partial Differ. Equations 27, No. 7--8, 1337--1372 (2002; Zbl 1010.35015) Full Text: DOI References: [1] DOI: 10.1155/S107379289900063X · Zbl 0938.35106 · doi:10.1155/S107379289900063X [2] DOI: 10.1353/ajm.1999.0038 · Zbl 0952.35073 · doi:10.1353/ajm.1999.0038 [3] DOI: 10.1002/cpa.3160480802 · Zbl 0856.35106 · doi:10.1002/cpa.3160480802 [4] Shin-ichi Doi, J. Math. Kyoto Univ. 34 (2) pp 319– (1994) · Zbl 0807.35026 · doi:10.1215/kjm/1250519013 [5] DOI: 10.1080/03605309608821178 · Zbl 0853.35025 · doi:10.1080/03605309608821178 [6] Ginibre J., Sapporo, 1995, in: Nonlinear Waves pp pp. 85– (1997) [7] Hörmander L., The Analysis of Linear Partial Differential Operators. I–IV (1985) [8] Kapitanski L., Math. Res. Letters 3 pp 77– (1996) · Zbl 0860.35016 · doi:10.4310/MRL.1996.v3.n1.a8 [9] Kapitanski L., Differential Operators and Spectral Theory pp pp. 139– (1999) · doi:10.1090/trans2/189/11 [10] DOI: 10.1353/ajm.1998.0039 · Zbl 0922.35028 · doi:10.1353/ajm.1998.0039 [11] DOI: 10.1007/s002220050272 · Zbl 0928.35158 · doi:10.1007/s002220050272 [12] Klainerman S., C. R. Math. Acad. Sci. Paris. 334 (2) pp 125– (2002) · Zbl 1008.35079 · doi:10.1016/S1631-073X(02)02214-8 [13] Rolvung C, 1998. Ph. D. Thesis, in: Nonisotropic Schrödinger equations [14] DOI: 10.5802/aif.1640 · Zbl 0974.35068 · doi:10.5802/aif.1640 [15] DOI: 10.1080/03605300008821581 · Zbl 0972.35014 · doi:10.1080/03605300008821581 [16] Stein E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (1993) · Zbl 0821.42001 [17] Tataru D., AMS 15 (2) pp 419– (2002) [18] DOI: 10.1353/ajm.2000.0042 · doi:10.1353/ajm.2000.0042 [19] DOI: 10.1353/ajm.2001.0021 · Zbl 0988.35037 · doi:10.1353/ajm.2001.0021 [20] Taylor M. E., Partial Differential Equations. III (1997) [21] DOI: 10.1006/jfan.2000.3687 · Zbl 0974.47025 · doi:10.1006/jfan.2000.3687 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.