Cubic NLS on R4

**Cubic NLS on $\R^4$**
Description
Equation	$iu_t + \Delta u = \pm \|u\|^2 u$
Fields	$u: \R \times \R^4 \to \mathbb{C}$
Data class	$u(0) \in H^s(\R^4)$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	$\dot H^1(\R)$
Criticality	mass-supercritical; energy-critical; scattering-subcritical
Covariance	Galilean
Theoretical results
LWP	$H^s(\R)$ for $s \geq 1$
GWP	$H^s(\R)$ 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 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 (Ryckman-Visan)
- In the radial case this is in Bo1999.
- For small energy data this is in CaWe1990.