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 $s\geq 1/2\,$ 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 $s\geq 1\,$ 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.