Cubic NLS on R4
From DispersiveWiki
Description | |
---|---|
Equation | |
Fields | |
Data class | |
Basic characteristics | |
Structure | Hamiltonian |
Nonlinearity | semilinear |
Linear component | Schrodinger |
Critical regularity | |
Criticality | mass-supercritical; energy-critical; scattering-subcritical |
Covariance | Galilean |
Theoretical results | |
LWP | for |
GWP | for (+) or for , 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 .
- LWP is known for CaWe1990.
- For the time of existence depends on the profile of the data as well as the norm.
- For 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)