Quintic NLS on R3

The theory of the quintic NLS on $\mathbb {R} ^{3}$ is as follows.

• Scaling is $s_{c}=1\,$.
• LWP is known for $s\geq 1\,$ CaWe1990.
  • For $s=1\,$ the time of existence depends on the profile of the data as well as the norm.
  • For $s<s_{c}\,$ we have ill-posedness, indeed the $H^{s}\,$ norm can get arbitrarily large arbitrarily quickly CtCoTa-p2. In the focusing case we have instantaneous blowup from the virial identity and scaling.
• GWP and scattering for $s\geq 1\,$ in the defocusing case CoKeStTkTa-p
  • For radial data this is in Bo-p, Bo1999.
  • Blowup can occur in the focussing case from Glassey's virial identity.