Cubic NLS on R4

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Cubic NLS on
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 norm (-)
Related equations
Parent class cubic NLS
Special cases -
Other related quintic NLS on R3


The theory of the cubic NLS in R^4 is as follows.

  • Scaling is .
  • LWP is known for CaWe1990.
    • For the time of existence depends on the profile of the data as well as the norm.
    • For we have ill-posedness, indeed the H^s norm can get arbitrarily large arbitrarily quickly CtCoTa-p2. In the focusing case we have instantaneous blowup from the virial identity and scaling.
  • GWP and scattering for (Ryckman-Visan)
    • In the radial case this is in Bo1999.
    • For small energy data this is in CaWe1990.