Energy critical NLS
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 or radial sub-ground state energy (-) |
| Related equations | |
| Parent class | NLS |
| Special cases | Energy-critical NLS on R^3, on R^4 |
| Other related | - |
The energy-critical NLS 
occurs when 
and 
. Note that the power non-linearity is smooth in dimensions 
(quintic NLS) and 
(cubic NLS).
LWP is known for all CaWe1990.
The GWP and scattering theory in the energy class is as follows.
- For small energy this is in GiVl1978, GiVl1979, Sr1981, Sr1981b.
- For radial focusing data with energy less than the ground state this is in [Kenig-Merle]
- For radial defocusing data in three and four dimensions this is in [Bourgain] (see also [Grillakis])
- For radial defocusing data in higher dimensions, see Ta2005, [Visan-Zhang]
- For defocusing data in three dimensions this is in [CKSTT]
- For four dimensions, this is in [Ryckman-Visan]
- For five and higher dimensions, this is in [Visan]
