On quadratic derivative Schrödinger equations in one space dimension
HTML articles powered by AMS MathViewer
- by Atanas Stefanov PDF
- Trans. Amer. Math. Soc. 359 (2007), 3589-3607 Request permission
Abstract:
We consider the Schrödinger equation with derivative perturbation terms in one space dimension. For the linear equation, we show that the standard Strichartz estimates hold under specific smallness requirements on the potential. As an application, we establish existence of local solutions for quadratic derivative Schrödinger equations in one space dimension with small and rough Cauchy data.References
- N. Burq, F. Planchon, personal communication.
- Hiroyuki Chihara, Local existence for semilinear Schrödinger equations, Math. Japon. 42 (1995), no. 1, 35–51. MR 1344627
- Hiroyuki Chihara, Gain of regularity for semilinear Schrödinger equations, Math. Ann. 315 (1999), no. 4, 529–567. MR 1731461, DOI 10.1007/s002080050328
- M. Christ, Ill-posedness of a Schrödinger equation with derivative nonlinearity, preprint.
- P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc. 1 (1988), no. 2, 413–439. MR 928265, DOI 10.1090/S0894-0347-1988-0928265-0
- Peter Constantin and Jean-Claude Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J. 38 (1989), no. 3, 791–810. MR 1017334, DOI 10.1512/iumj.1989.38.38037
- Shin-ichi Doi, On the Cauchy problem for Schrödinger type equations and the regularity of solutions, J. Math. Kyoto Univ. 34 (1994), no. 2, 319–328. MR 1284428, DOI 10.1215/kjm/1250519013
- Shin-ichi Doi, Remarks on the Cauchy problem for Schrödinger-type equations, Comm. Partial Differential Equations 21 (1996), no. 1-2, 163–178. MR 1373768, DOI 10.1080/03605309608821178
- J. Ginibre and G. Velo, Smoothing properties and retarded estimates for some dispersive evolution equations, Comm. Math. Phys. 144 (1992), no. 1, 163–188. MR 1151250
- Nakao Hayashi and Pavel I. Naumkin, A quadratic nonlinear Schrödinger equation in one space dimension, J. Differential Equations 186 (2002), no. 1, 165–185. MR 1941097, DOI 10.1016/S0022-0396(02)00010-4
- Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991), no. 1, 33–69. MR 1101221, DOI 10.1512/iumj.1991.40.40003
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 10 (1993), no. 3, 255–288 (English, with English and French summaries). MR 1230709, DOI 10.1016/S0294-1449(16)30213-X
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math. 134 (1998), no. 3, 489–545. MR 1660933, DOI 10.1007/s002220050272
- Carlos E. Kenig, Gustavo Ponce, and Luis Vega, The Cauchy problem for quasi-linear Schrödinger equations, Invent. Math. 158 (2004), no. 2, 343–388. MR 2096797, DOI 10.1007/s00222-004-0373-4
- H. Koch and N. Tzvetkov, Nonlinear wave interactions for the Benjamin-Ono equation, Int. Math. Res. Not. 30 (2005), 1833–1847. MR 2172940, DOI 10.1155/IMRN.2005.1833
- Igor Rodnianski and Terence Tao, Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions, Comm. Math. Phys. 251 (2004), no. 2, 377–426. MR 2100060, DOI 10.1007/s00220-004-1152-1
- Terence Tao, Global well-posedness of the Benjamin-Ono equation in $H^1(\textbf {R})$, J. Hyperbolic Differ. Equ. 1 (2004), no. 1, 27–49. MR 2052470, DOI 10.1142/S0219891604000032
- Daniel Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), no. 2, 349–376. MR 1749052
- Daniel Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc. 15 (2002), no. 2, 419–442. MR 1887639, DOI 10.1090/S0894-0347-01-00375-7
- Ph. Roux and D. Yafaev, The scattering matrix for the Schrödinger operator with a long-range electromagnetic potential, J. Math. Phys. 44 (2003), no. 7, 2762–2786. MR 1982788, DOI 10.1063/1.1576494
Additional Information
- Atanas Stefanov
- Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
- Email: stefanov@math.ku.edu
- Received by editor(s): February 25, 2005
- Published electronically: February 23, 2007
- Additional Notes: This research was supported in part by NSF-DMS 0300511.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 3589-3607
- MSC (2000): Primary 35Q55, 35J10
- DOI: https://doi.org/10.1090/S0002-9947-07-04207-9
- MathSciNet review: 2302508