Cubic NLS on R2

The theory of the cubic NLS on R^2 is as follows.

• Scaling is $s_{c}=0\,$, thus this is a [mass critical NLS].
• LWP for $s\geq 0\,$ CaWe1990.
  • For $s=0\,$ the time of existence depends on the profile of the data as well as the norm.
  • LWP has also been obtained in Besov spaces Pl2000, Pl-p and Fourier-Lorentz spaces CaVeVi-p at the scaling of $L^{2}\,$. This is also connected with the construction of self-similar solutions to NLS (which are generally not in the usual Sobolev spaces globally in space).
  • Below $L^{2}\,$ we have ill-posedness by Gallilean invariance considerations in both the focusing KnPoVe-p and defocusing CtCoTa-p2 cases.
• GWP for $s>4/7\,$ in the defocussing case CoKeStTkTa2002
  • For $s>3/5\,$ this was shown in Bo1998.
  • For $s>2/3\,$ this was shown in Bo1998, Bo1999.
  • For $s\geq 1\,$ this follows from Hamiltonian conservation.
  • For small $L^{2}\,$ data one has GWP and scattering for any cubic nonlinearity (not necessarily defocussing or Hamiltonian). More precisely, one has global well-posedness whenever the data has an $L^{2}\,$ norm strictly smaller than the ground state Q Me1993. If the $L^{2}\,$ norm is exactly equal to that of Q then one has blow-up if and only if the data is a pseudo-conformal transformation of the ground state Me1993, Me1992. In particular, the ground state is unstable.
    • Scattering is known whenever the solution is sufficiently small in $L^{2}\,$ norm, or more generally whenever the solution is $L^{4}\,$ in spacetime.Presumably one in fact has scattering whenever the mass is strictly smaller than the ground state, though this has not yet been established.
  • The $s>4/7\,$ result is probably improvable by correction term methods.
  • Remark: $s=1/2\,$ is the least regularity for which the non-linear part of the solution has finite energy (Bourgain, private communication).
  • For powers slightly higher than cubic, one has scattering for large mass data Na1999c, and indeed we have bounded $H^{k}\,$ norms in this case [Bourgain?].
  • If the data has sufficient decay then one has scattering. For instance if $xu(0)\,$ is in $L^{2}\,$ Ts1985. This was improved to $x^{2/3+}u(0)\in L^{2}\,$ in Bo1998, Bo1999; the above results on GWP will probably also extend to scattering.
• Remark: This equation is pseudo-conformally invariant. Heuristically, GWP results in $H^{s}\,$ transfer to GWP and scattering results in $L^{2}(|x|^{2s})\,$ thanks to the pseudo-conformal transformation. Thus for instance GWP and scattering occurs this weighted space for $s>2/3\,$ (the corresponding statement for, say, $s>4/7\,$ has not yet been checked).

Open question: large mass scattering

What happens for large $L^{2}\,$ data? It is known that the only way GWP can fail at $L^{2}\,$ is if the $L^{2}\,$ norm concentrates Bo1998. Blowup examples with multiple blowup points are known, either simultaneously Me1992 or non-simultaneously BoWg1997. It is conjectured that the amount of energy which can go into blowup points is quantized. The $H^{1}\,$ norm in these examples blows up like $|t|^{-1}\,.$ It is conjectured that slower blow-up examples exist, in particular numerics suggest a blowup rate of $|t|^{-1/2}(loglog|t|)^{1/2}\,$ LanPapSucSup1988; interestingly, however, if we perturb NLS to the Zakharov system then one can only have blowup rates of $|t|^{-1}\,.$