NLS blowup: Difference between revisions
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<br /> In the <math>L^2\,</math>-supercritical focussing [[NLS]] one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality | <br /> In the <math>L^2\,</math>-supercritical focussing [[NLS]] one has blowup whenever the Hamiltonian is negative, thanks to Glassey's [[virial inequality]] | ||
<center><math>\partial^2_t \int x^2 |u|^2 dx \leq H(u)</math>;</center> | <center><math>\partial^2_t \int x^2 |u|^2 dx \leq H(u)</math>;</center> | ||
see e.g. [[OgTs1991]]. By scaling this implies that we have instantaneous blowup in <math>H^s\,</math> for <math>s < s_c\,</math> in the focusing case. In the defocusing case blowup <br /> is not known, but the <math>H^s\,</math> norm can still get arbitrarily large arbitrarily quickly for <math>s < s_c\,</math> [[CtCoTa-p2]]. In addition, the work about sharp criteria of blowup, see [[Zhj2002a]], [[Zhj2002b]]. | see e.g. [[OgTs1991]]. By scaling this implies that we have instantaneous blowup in <math>H^s\,</math> for <math>s < s_c\,</math> in the focusing case. In the defocusing case blowup <br /> is not known, but the <math>H^s\,</math> norm can still get arbitrarily large arbitrarily quickly for <math>s < s_c\,</math> [[CtCoTa-p2]]. In addition, the work about sharp criteria of blowup and global existence, see [[Zhj2002a]], [[Zhj2002b]]. | ||
[[Category:Schrodinger]] | [[Category:Schrodinger]] | ||
Latest revision as of 06:12, 21 July 2007
In the -supercritical focussing NLS one has blowup whenever the Hamiltonian is negative, thanks to Glassey's virial inequality
see e.g. OgTs1991. By scaling this implies that we have instantaneous blowup in for in the focusing case. In the defocusing case blowup
is not known, but the norm can still get arbitrarily large arbitrarily quickly for CtCoTa-p2. In addition, the work about sharp criteria of blowup and global existence, see Zhj2002a, Zhj2002b.
