Energy critical NLS

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Energy-critical NLS
Data class
Basic characteristics
Structure Hamiltonian
Nonlinearity semilinear
Linear component Schrodinger
Critical regularity
Criticality mass-supercritical;
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]