The Cauchy problem for the nonlinear Schrödinger equation on compact manifolds. (English) Zbl 1084.35086
Colombini, Ferruccio (ed.) et al., Phase space analysis of partial differential equations. Vol. I. Proceedings of the research trimester, Centro di Ricerca Matematica “Ennio de Giorgi”, Pisa, Italy, February 15–May 15, 2004. Pisa: Scuola Normale Superiore (ISBN 88-7642-150-5/pbk). Pubblicazioni del Centro di Ricerca Matematica Ennio de Giorgi, 21-52 (2004).
For the cubic nonlinear Schrödinger equation \(i\partial_{t}u+\Delta_{x} u=\pm| u|^2 u,\) on compact manifolds in \(\mathbb R^d\) a review of recent results on the well-posedness of the Cauchy problem is given. The authors emphasize the role played by bilinear Strichartz estimates, describing how they yield optimal results on spheres.
For the entire collection see [Zbl 1064.35001].
For the entire collection see [Zbl 1064.35001].
Reviewer: Boris V. Loginov (Ul’yanovsk)
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
