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]