NLS blowup

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In the $L^2\,$ -supercritical focussing NLS one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality

$\partial^2_t \int x^2 |u|^2 dx \leq H(u)$ ;

see e.g. OgTs1991. By scaling this implies that we have instantaneous blowup in $H^s\,$ for $s < s_c\,$ in the focusing case. In the defocusing case blowup
is not known, but the $H^s\,$ norm can still get arbitrarily large arbitrarily quickly for $s < s_c\,$ CtCoTa-p2. In addition, the work about sharp criteria of blowup and global existence, see Zhj2002a, Zhj2002b.

NLS blowup

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