Inhomogeneous Strichartz estimates for the Schrödinger equation
HTML articles powered by AMS MathViewer
- by M. C. Vilela PDF
- Trans. Amer. Math. Soc. 359 (2007), 2123-2136 Request permission
Abstract:
We study Strichartz estimates for the solution of the Cauchy problem associated with the inhomogeneous free Schrödinger equation in the case when the inital data is equal to zero, proving some new estimates for certain exponents and giving counterexamples for some others.References
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta Math. 61 (1988), no. 4, 477–494. MR 952091, DOI 10.1007/BF01258601
- Thierry Cazenave and Fred B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992), no. 1, 75–100. MR 1171761
- Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409–425. MR 1809116, DOI 10.1006/jfan.2000.3687
- J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309–327 (English, with French summary). MR 801582
- Tosio Kato, An $L^{q,r}$-theory for nonlinear Schrödinger equations, Spectral and scattering theory and applications, Adv. Stud. Pure Math., vol. 23, Math. Soc. Japan, Tokyo, 1994, pp. 223–238. MR 1275405, DOI 10.2969/aspm/02310223
- Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048
- S. J. Montgomery-Smith, Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations, Duke Math. J. 91 (1998), no. 2, 393–408. MR 1600602, DOI 10.1215/S0012-7094-98-09117-7
- Irving Segal, Space-time decay for solutions of wave equations, Advances in Math. 22 (1976), no. 3, 305–311. MR 492892, DOI 10.1016/0001-8708(76)90097-9
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- 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. MR 512086
- Terence Tao, Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation, Comm. Partial Differential Equations 25 (2000), no. 7-8, 1471–1485. MR 1765155, DOI 10.1080/03605300008821556
- Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 358216, DOI 10.1090/S0002-9904-1975-13790-6
- Peter A. Tomas, Restriction theorems for the Fourier transform, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–114. MR 545245
- Kenji Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), no. 3, 415–426. MR 891945
Additional Information
- M. C. Vilela
- Affiliation: Departamento de Matemática Aplicada, Escuela Universitaria de Informática, Campus de Segovia - Universidad de Valladolid, Plaza de Santa Eulalia 9 y 11, 40005 Segovia, Spain
- Email: maricruz@eis.uva.es
- Received by editor(s): December 12, 2003
- Received by editor(s) in revised form: March 1, 2005
- Published electronically: December 15, 2006
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 2123-2136
- MSC (2000): Primary 35J10, 46B70
- DOI: https://doi.org/10.1090/S0002-9947-06-04099-2
- MathSciNet review: 2276614