Nelson diffusions and blow-up phenomena in solutions of the nonlinear Schrödinger equation with critical power (Q2752232)

A Cauchy problem for the nonlinear Schrödinger equation with critical power NEWLINE\[NEWLINE 2i{\partial\psi\over\partial t} + \Delta\psi + |\psi|^{4/d} \psi = 0, \quad \psi(\cdot,0) = \psi_0, \quad (x,t)\in\mathbb R^{d} \times\mathbb R_{+}, \tag{1} NEWLINE\]NEWLINE is considered. It is known that if \(\psi_0\in H^1(\mathbb R^{d})\) then there exists a unique (maximal) solution \(\psi\in C(\mathopen[0, T_{\roman{max}}\mathclose[;H^1(\mathbb R^{d}))\) which, however, need not be global: one has \(T_{\roman{max}}<\infty\) for some initial data. The paper is a rather detailed overview, based mainly on the author's recent results, of the blow-up behaviour of solutions to (1), with a particular attention paid to tightness of the set \(M = \{|\psi (x,t)|^2 \roman dx; 0\leq t<T_{\roman{max}}\}\) of Radon measures defined by a blow-up solution \(\psi\) and to the structure of limiting measures, i.e. the weak\(^*\) cluster points of \(M\). Moreover, by using probabilistic methods coming from Nelson's stochastic mechanics the rate of blow-up and validity of the log log law \(\|\nabla\psi\|\sim [(T_{\roman{max}}-t)^{-1} \log\log(T_{\roman{max}}-t)^{-1}]^{1/2}\) are studied.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00038].