Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case. (English) Zbl 0958.35126

Bourgain, Jean

J. Am. Math. Soc. 12, No. 1, 145-171 (1999).

Summary: We establish global wellposedness and scattering for the \(H^{1}\)-critical defocusing nonlinear Schrödinger equation in 3D \[ iu_{t}+\Delta u - u|u|^{4}=0 \] assuming radial data \(\phi \in H^{s}\), \(s\geq 1\). In particular, it proves global existence of classical solutions in the radial case. The same result is obtained in 4D for the equation \[ iu_{t}+\Delta u -u|u|^{2} =0 . \]

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35A05	General existence and uniqueness theorems (PDE) (MSC2000)
35L15	Initial value problems for second-order hyperbolic equations

Keywords:

nonlinear Schrödinger equation; global wellposedness

References:

[1]	Jeng Eng Lin and Walter A. Strauss, Decay and scattering of solutions of a nonlinear Schrödinger equation, J. Funct. Anal. 30 (1978), no. 2, 245 – 263. · Zbl 0395.35070 · doi:10.1016/0022-1236(78)90073-3
[2]	Michael Struwe, Globally regular solutions to the \?\(^{5}\) Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 3, 495 – 513 (1989). · Zbl 0728.35072
[3]	Manoussos G. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. (2) 132 (1990), no. 3, 485 – 509. · Zbl 0736.35067 · doi:10.2307/1971427
[4]	Jalal Shatah and Michael Struwe, Regularity results for nonlinear wave equations, Ann. of Math. (2) 138 (1993), no. 3, 503 – 518. · Zbl 0836.35096 · doi:10.2307/2946554
[5]	T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Metodes Matematicos 26 (Rio de Janeiro). · Zbl 0584.35022
[6]	Thierry Cazenave and Fred B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in \?^{\?}, Nonlinear Anal. 14 (1990), no. 10, 807 – 836. · Zbl 0706.35127 · doi:10.1016/0362-546X(90)90023-A
[7]	J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (1985), no. 4, 363 – 401. · Zbl 0535.35069
[8]	Jean Ginibre and Giorgio Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z. 170 (1980), no. 2, 109 – 136. · Zbl 0407.35063 · doi:10.1007/BF01214768
[9]	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. · Zbl 0372.35001