Quintic NLS on R2

The theory of the quintic NLS on ${\displaystyle R^{2}}$ is as follows.

• Scaling is ${\displaystyle s_{c}=1/2\,}$.
• LWP is known for ${\displaystyle s\geq 1/2\,}$ CaWe1990.
• For ${\displaystyle s=1/2\,}$ the time of existence depends on the profile of the data as well as the norm.
• For ${\displaystyle s 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 ${\displaystyle s\geq 1\,}$ by Hamiltonian conservation.
• This has been improved to ${\displaystyle s>1-\varepsilon }$ in CoKeStTkTa2003b. This result can of course be improved further.
• Scattering in the energy space Na1999c
• One also has GWP and scattering for small ${\displaystyle H^{1/2}\,}$ data for any quintic non-linearity.