Cubic NLS on R4

Cubic NLS on $\mathbb {R} ^{4}$

Description
Equation	$iu_{t}+\Delta u=\pm |u|^{2}u$
Fields	$u:\mathbb {R} \times \mathbb {R} ^{4}\to \mathbb {C}$
Data class	$u(0)\in H^{s}(\mathbb {R} ^{4})$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	${\dot {H}}^{1}(\mathbb {R} )$
Criticality	mass-supercritical; energy-critical; scattering-subcritical
Covariance	Galilean
Theoretical results
LWP	$H^{s}(\mathbb {R} )$ for $s\geq 1$
GWP	$H^{s}(\mathbb {R} )$ for $s\geq 1$ (+) or for $s\geq 1$, small norm (-)
Related equations
Parent class	cubic NLS
Special cases	-
Other related	quintic NLS on R3

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.