Cubic NLS on R4

The theory of the cubic NLS in R^4 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 $s<1\,$ 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 radial case Bo1999. A major obstacle is that the Morawetz estimate only gives $L^{4}\,$-type spacetime control rather than $L^{6}\,.$
  • For small non-radial $H^{1}\,$ data one has GWP and scattering. In fact one has scattering whenever the solution has a bounded $L^{6}\,$ norm in spacetime.

The large data non-radial case is still open, and very interesting. The main difficulty is infinite speed of propagation and the possibility that the $H^{1}\,$ norm could concentrate at several different places simultaneously.