Two singular dynamics of the nonlinear Schrödinger equation on a plane domain

We study the cubic, focusing nonlinear Schrödinger equation (NLS) posed on a bounded domain of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathbb{R}^{2} $\end{document} with Dirichlet boundary conditions. We describe two types of nonlinear evolutions. First we obtain solutions which blow up with a minimal L2 norm in .nite time at a .xed point of the interior of the domain. The argument can be performed equally well for the cubic NLS posed on the .at torus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathbb{T}^{2} $\end{document}. In the case when the domain is a disc, we also prove that the Cauchy problem is ill posed in the following sense: the .ow map is not uniformly continuous on bounded sets of the Sobolev space Hs, s> 1/3, contrary to what is known on the square (recall that the scale invariant Sobolev space for the cubic NLS in 2D is L2).