Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation
HTML articles powered by AMS MathViewer
- by Thierry Cazenave, Daoyuan Fang and Zheng Han PDF
- Trans. Amer. Math. Soc. 368 (2016), 7911-7934 Request permission
Abstract:
In this paper, we consider the nonlinear Schrödinger equation $iu_t +\Delta u= \lambda |u|^{\frac {4} {N-4}} u$ in $\mathbb {R}^N$, $N\ge 5$, with $\lambda \in \mathbb {C}$. We prove local well-posedness (local existence, unconditional uniqueness, continuous dependence) in the critical space $\dot H^2 (\mathbb {R}^N )$.References
- Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976. MR 0482275
- Björn Birnir, Carlos E. Kenig, Gustavo Ponce, Nils Svanstedt, and Luis Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, J. London Math. Soc. (2) 53 (1996), no. 3, 551–559. MR 1396718, DOI 10.1112/jlms/53.3.551
- Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829
- Thierry Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 2002047, DOI 10.1090/cln/010
- Thierry Cazenave, Daoyuan Fang, and Zheng Han, Continuous dependence for NLS in fractional order spaces, Ann. Inst. H. Poincaré C Anal. Non Linéaire 28 (2011), no. 1, 135–147. MR 2765515, DOI 10.1016/j.anihpc.2010.11.005
- 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
- Wei Dai, Continuous dependence for $H^2$ critical nonlinear Schrödinger equations in high dimensions. arXiv:1204.0130 [math.AP]. http://arxiv.org/abs/1204.0130
- Wei Dai, Weihua Yang, and Daomin Cao, Continuous dependence of Cauchy problem for nonlinear Schrödinger equation in $H^s$, J. Differential Equations 255 (2013), no. 7, 2018–2064. MR 3072680, DOI 10.1016/j.jde.2013.06.005
- Daoyuan Fang and Zheng Han, On the well-posedness for NLS in $H^s$, J. Funct. Anal. 264 (2013), no. 6, 1438–1455. MR 3017270, DOI 10.1016/j.jfa.2013.01.005
- Giulia Furioli and Elide Terraneo, Besov spaces and unconditional well-posedness for the nonlinear Schrödinger equation in $\dot {H}{}^s(\Bbb R^n)$, Commun. Contemp. Math. 5 (2003), no. 3, 349–367. MR 1992354, DOI 10.1142/S0219199703001002
- Zheng Han and Daoyuan Fang, On the unconditional uniqueness for NLS in $\dot {H}{}^s$, SIAM J. Math. Anal. 45 (2013), no. 3, 1505–1526. MR 3056755, DOI 10.1137/120871808
- Tosio Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113–129 (English, with French summary). MR 877998
- Tosio Kato, Nonlinear Schrödinger equations, Schrödinger operators (Sønderborg, 1988) Lecture Notes in Phys., vol. 345, Springer, Berlin, 1989, pp. 218–263. MR 1037322, DOI 10.1007/3-540-51783-9_{2}2
- Tosio Kato, On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness, J. Anal. Math. 67 (1995), 281–306. MR 1383498, DOI 10.1007/BF02787794
- Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048
- Carlos E. Kenig and Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675. MR 2257393, DOI 10.1007/s00222-006-0011-4
- Rowan Killip and Monica Vişan, Nonlinear Schrödinger equations at critical regularity, Evolution equations, Clay Math. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 2013, pp. 325–437. MR 3098643
- Hartmut Pecher, Solutions of semilinear Schrödinger equations in $H^s$, Ann. Inst. H. Poincaré Phys. Théor. 67 (1997), no. 3, 259–296 (English, with English and French summaries). MR 1472820
- Keith M. Rogers, Unconditional well-posedness for subcritical NLS in $H^s$, C. R. Math. Acad. Sci. Paris 345 (2007), no. 7, 395–398 (English, with English and French summaries). MR 2361505, DOI 10.1016/j.crma.2007.09.003
- 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, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
- Terence Tao and Monica Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations (2005), No. 118, 28. MR 2174550
- Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540, DOI 10.1007/978-3-0346-0416-1
- Yoshio Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), no. 1, 115–125. MR 915266
- Yin Yin Su Win and Yoshio Tsutsumi, Unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrödinger equation, Hokkaido Math. J. 37 (2008), no. 4, 839–859. MR 2474179, DOI 10.14492/hokmj/1249046372
Additional Information
- Thierry Cazenave
- Affiliation: Université Pierre et Marie Curie and CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France
- MR Author ID: 46500
- Email: thierry.cazenave@upmc.fr
- Daoyuan Fang
- Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China
- Email: dyf@zju.edu.cn
- Zheng Han
- Affiliation: Department of Mathematics, Hangzhou Normal University, and Department of Mathematics, Zhejiang University, Hangzhou, 311121, People’s Republic of China
- MR Author ID: 924412
- ORCID: 0000-0002-9391-9352
- Email: hanzh_0102@163.com
- Received by editor(s): March 20, 2014
- Received by editor(s) in revised form: January 16, 2015
- Published electronically: March 2, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 7911-7934
- MSC (2010): Primary 35Q55; Secondary 35B30
- DOI: https://doi.org/10.1090/tran6683
- MathSciNet review: 3546788