Cubic NLS on R2

Cubic NLS on $\mathbb {R} ^{2}$

Description
Equation	$iu_{t}+\Delta u=\pm |u|^{2}u$
Fields	$u:\mathbb {R} \times \mathbb {R} ^{2}\to \mathbb {C}$
Data class	$u(0)\in H^{s}(\mathbb {R} ^{2})$
Basic characteristics
Structure	Hamiltonian
Nonlinearity	semilinear
Linear component	Schrodinger
Critical regularity	$L^{2}(\mathbb {R} ^{2})$
Criticality	mass-critical; energy-subcritical; scattering-subcritical
Covariance	Galilean pseudoconformal
Theoretical results
LWP	$H^{s}(\mathbb {R} ^{2})$ for $s\geq 0$
GWP	$H^{s}(\mathbb {R} ^{2})$ for $s>4/7$
Related equations
Parent class	cubic NLS
Special cases	-
Other related	-

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

• 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 KnPoVe2001 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.
  • 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).
• The H^k norms grow like $O(t^{(k-1)+})\,$ as long as the H^1 norm stays bounded St1997, St1997b; this was improved to $t^{2/3(k-1)+}\,$ in CoDeKnSt-p, and also generalized to higher order multilinearity. A preliminary analysis suggests that the I-method can push the growth bounds down to $t^{(k-1)+/2}.$

Open question: large mass behavior

It is conjectured that global wellposedness, regularity, scattering and spacetime bounds occur for all large mass initial data in the defocusing case, and all data of mass less than that of the ground state in the focusing case.

• 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. In all known examples the mass has to be larger than that of the ground state.
  • 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 $|t|^{-1}\,.$