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Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem. (English) Zbl 1106.35096
Sanz-Solé, Marta (ed.) et al., Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume III: Invited lectures. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-022-7/hbk). 157-182 (2006).
The author deals with the nonlinear Schrödinger equations in inhomogeneous media and is mainly interested in the impact of the inhomogeneity on the dynamics of the equation, in particular, regarding the well-posedness theory of the Cauchy problem. The goal is to survey recent mathematical contributions in this direction. The author revisits the well-posedness problems by introducing multilinear Strichartz estimates and discusses in details the case of simple Riemannian compact manifolds, such as tori and spheres.
For the entire collection see [Zbl 1095.00006].

35Q55 NLS equations (nonlinear Schrödinger equations)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35R25 Ill-posed problems for PDEs