# NLS blowup

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

${\displaystyle \partial _{t}^{2}\int x^{2}|u|^{2}dx\leq H(u)}$;

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