Quintic NLS on R2

The theory of the quintic NLS on $R 2$ is as follows.

Scaling is $s_c = 1/2\,$ .
LWP is known for CaWe1990.
- For $s=1/2\,$ 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 for by Hamiltonian conservation.
- This has been improved to $s > 1-\varepsilon$ in CoKeStTkTa2003b. This result can of course be improved further.
- Scattering in the energy space Na1999c
- One also has GWP and scattering for small $H^{1/2}\,$ data for any quintic non-linearity.