# Cubic NLS on R4

Cubic NLS on ${\displaystyle \mathbb {R} ^{4}}$
Description
Equation ${\displaystyle iu_{t}+\Delta u=\pm |u|^{2}u}$
Fields ${\displaystyle u:\mathbb {R} \times \mathbb {R} ^{4}\to \mathbb {C} }$
Data class ${\displaystyle u(0)\in H^{s}(\mathbb {R} ^{4})}$
Basic characteristics
Structure Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity ${\displaystyle {\dot {H}}^{1}(\mathbb {R} )}$
Criticality mass-supercritical;
energy-critical;
scattering-subcritical
Covariance Galilean
Theoretical results
LWP ${\displaystyle H^{s}(\mathbb {R} )}$ for ${\displaystyle s\geq 1}$
GWP ${\displaystyle H^{s}(\mathbb {R} )}$ for ${\displaystyle s\geq 1}$ (+)
or for ${\displaystyle s\geq 1}$, 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 ${\displaystyle s_{c}=1\,}$.
• LWP is known for ${\displaystyle s\geq 1\,}$ CaWe1990.
• For ${\displaystyle s=1\,}$ the time of existence depends on the profile of the data as well as the norm.
• For ${\displaystyle s<1\,}$ 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 ${\displaystyle s\geq 1\,}$ (Ryckman-Visan)
• In the radial case this is in Bo1999.
• For small energy data this is in CaWe1990.