Cubic NLS on R3

Cubic NLS on $\mathbb {R} ^{3}$

Description
Equation	$iu_{t}+\Delta u=\pm |u|^{2}u$
Fields	$u:\mathbb {R} \times \mathbb {R} ^{3}\to \mathbb {C}$
Data class	$u(0)\in H^{s}(\mathbb {R} ^{3})$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	${\dot {H}}^{1/2}(\mathbb {R} ^{3})$
Criticality	mass-supercritical; energy-subcritical; scattering-subcritical
Covariance	Galilean
Theoretical results
LWP	$H^{s}(\mathbb {R} )$ for $s\geq 1/2$
GWP	$H^{s}(\mathbb {R} )$ for $s\geq 4/5$ (+) or for $s\geq 0$, small norm
Related equations
Parent class	cubic NLS
Special cases	-
Other related	-

The theory of the cubic NLS on $\mathbb {R} ^{3}$ is as follows.

• LWP for $s\geq 1/2\,$ CaWe1990.
  • For $s=1/2\,$ the time of existence depends on the profile of the data as well as the norm.
  • For $s<1/2\,$ 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.
  • For $s>1/2\,$ there is unconditional well-posedness FurPlTer2001
    • For $s>=2/3\,$ this is in Ka1995.
• GWP and scattering for $s>4/5\,$ in the defocussing case CoKeStTkTa-p8
  • For $s>5/6\,$ GWP is in CoKeStTkTa2002
  • For $s>11/13\,$ GWP is in Bo1999
  • For radial data and $s>5/7\,$ GWP and scattering is in $s>5/7\,$ Bo1998b, Bo1999.
  • For $s\geq 1\,$ this follows from Hamiltonian conservation. One also has scattering in this case GiVl1985.
  • For small $H^{1/2}\,$ data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More generally one has scattering whenever the solution is $L^{5}\,$ in spacetime.
  • In the focusing case one has blowup whenever the energy is negative Gs1977, OgTs1991, and in particular one has blowup arbitrarily close to the ground state BerCa1981, SaSr1985.If however the energy remains bounded (which is automatic in the defocusing case) then one has at most polynomial growth of high Sobolev norms, with the local higher Sobolev norms $H_{loc}^{s}\,$ remaining bounded for all time Bo1996c, Bo1998b. Also in the focusing radial case with bounded energy, the solution becomes asymptotically smooth and spatially decaying away from the origin, once one strips out the radiation component Ta2004b