Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space $W^{s,p}$ for $p<2$
HTML articles powered by AMS MathViewer
- by Yi Zhou PDF
- Trans. Amer. Math. Soc. 362 (2010), 4683-4694 Request permission
Abstract:
In this paper, we consider in $R^n$ the Cauchy problem for the nonlinear Schrödinger equation with initial data in the Sobolev space $W^{s,p}$ for $p<2$. It is well known that this problem is ill posed. However, we show that after a linear transformation by the linear semigroup the problem becomes locally well posed in $W^{s,p}$ for $\frac {2n}{n+1}<p<2$ and $s>n(1-\frac {1}{p})$. Moreover, we show that in one space dimension, the problem is locally well posed in $L^p$ for any $1<p<2$.References
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275, DOI 10.1007/978-3-642-66451-9
- J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 (1993), no. 2, 107–156. MR 1209299, DOI 10.1007/BF01896020
- Thierry Cazenave, Luis Vega, and Mari Cruz Vilela, A note on the nonlinear Schrödinger equation in weak $L^p$ spaces, Commun. Contemp. Math. 3 (2001), no. 1, 153–162. MR 1820017, DOI 10.1142/S0219199701000317
- Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal. 14 (1990), no. 10, 807–836. MR 1055532, DOI 10.1016/0362-546X(90)90023-A
- Michael Christ, James Colliander, and Terence Tao, A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order, J. Funct. Anal. 254 (2008), no. 2, 368–395. MR 2376575, DOI 10.1016/j.jfa.2007.09.005
- Axel Grünrock, Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS, Int. Math. Res. Not. 41 (2005), 2525–2558. MR 2181058, DOI 10.1155/IMRN.2005.2525
- S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space $\textbf {R}^{n+1}$, Comm. Pure Appl. Math. 40 (1987), no. 1, 111–117. MR 865359, DOI 10.1002/cpa.3160400105
- Herbert Koch and Daniel Tataru, A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN 16 (2007), Art. ID rnm053, 36. MR 2353092, DOI 10.1093/imrn/rnm053
- H. P. McKean and J. Shatah, The nonlinear Schrödinger equation and the nonlinear heat equation reduction to linear form, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 1067–1080. MR 1127050, DOI 10.1002/cpa.3160440817
- Yoshio Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115–125. MR 915266
- Ana Vargas and Luis Vega, Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite $L^2$ norm, J. Math. Pures Appl. (9) 80 (2001), no. 10, 1029–1044 (English, with English and French summaries). MR 1876762, DOI 10.1016/S0021-7824(01)01224-7
Additional Information
- Yi Zhou
- Affiliation: School of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China – and – Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, People’s Republic of China
- Email: yizhou@fudan.ac.cn
- Received by editor(s): May 28, 2008
- Received by editor(s) in revised form: September 2, 2008
- Published electronically: April 20, 2010
- Additional Notes: The author was supported by the National Natural Science Foundation of China under grant 10728101, the 973 Project of the Ministry of science and technology of China, the doctoral program foundation of the Ministry of education of China and the “111” Project and SGST 09DZ2272900
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4683-4694
- MSC (2010): Primary 35Q41
- DOI: https://doi.org/10.1090/S0002-9947-10-05055-5
- MathSciNet review: 2645046