Energy critical NLS

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

LWP is known for all ${\displaystyle s\geq 1}$ 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]