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Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Lisa Onkes (University of Bonn)

Singularity formation for dispersive waves

We consider the (focusing) nonlinear Schrödinger equation \begin{align*} (\text{NLS})\qquad \left\lbrace\begin{array}{l} i u_t = - \Delta u - |u|^{p-1}u,\quad(t,x) \in \mathbb{R} \times \mathbb{R}^N \\ u(0,x)= u_0(x) \in H^1(\mathbb{R}^N, \mathbb{C}) \end{array} \right., \end{align*} which depending on the exponent is either subcritical, critical or supercritical. In the subcritical case all solutions are globally defined (J. Ginibre and G. Velo), while solutions in the critical case - assumed initial data mass comparable to the soliton mass and negative Galilean energy - present singularities with almost self-similar blow-up speed (F. Merle and P. Raphaël).
In the supercritical case physical experiments (V. Zakharov, E. Kuznetsov and S. Musher) suggest the existence of self-similar blow-up solutions. However this has only been proven in the slightly supercritical case (F. Merle, P. Raphaël, and J. Szeftel), by viewing this as a pertubation of the critical case. Our goal is to adopt an approach used by R. Donninger and B. Schörkhuber for the supercritical wave equation and obtain a proof for the existence of self-similar blow-up solutions of the mass supercritical Schrödinger equation, which does not depend on the closeness to the critical case. Thus (under numerically verifiable assumptions) providing a self-similar blow-up result for the whole supercritical range.