Energy critical NLS

Energy-critical NLS

Description
Equation	$iu_{t}+\Delta u=\pm |u|^{4/(d-2)}u$
Fields	$u:\mathbb {R} \times \mathbb {R} ^{d}\to \mathbb {C}$
Data class	$u(0)\in H^{s}(\mathbb {R} ^{d})$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	${\dot {H}}^{1}(\mathbb {R} ^{3})$
Criticality	mass-supercritical; energy-critical; scattering-subcritical
Covariance	Galilean
Theoretical results
LWP	$H^{s}(\mathbb {R} )$ for $s\geq 1$
GWP	$H^{s}(\mathbb {R} )$ for $s\geq 1$ (+) 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 $s_{c}=1\,$ occurs when $d\geq 3$ and $p=1+4/(d-2)\,$. Note that the power non-linearity is smooth in dimensions $d=3\,$ (quintic NLS) and $d=4\,$ (cubic NLS).

LWP is known for all $s\geq 1$ 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]