Mass critical NLS

$L^{2}$ critical NLS on $R^{d}$

The $L^{2}\,$ critical situation $s_{c}=0\,$ occurs when $p=1+4/d\,$. Note that the power non-linearity is smooth in dimensions $d=1\,$ (quintic NLS) and $d=2\,$ (cubic NLS). One always has GWP and scattering in $L^{2}\,$ for small data (see GiVl1978, GiVl1979, CaWe1990; the more precise statement in the focusing case that GWP holds when the mass is strictly less than the ground state mass is in Ws1983); in the large data defocusing case, GWP is known in $H^{1}\,$ (and slightly below) but is only conjectured in $L^{2}\,.$ No scattering result is known for large data, even in the radial smooth case.

In the focusing case, there is blowup for large $L^{2}\,$ data, as can be seen by applying the pseudoconformal transformation to the ground state solution. Up to the usual symmetries of the equation, this is the unique minimal mass blowup solution Me1993. This solution blows up in $H^{1}\,$ like $|t|^{-1}\,$ as $t\rightarrow 0-\,.$ However, numerics suggest that there should be solutions that exhibit the much slower blowup $|t|^{-1/2}(\log \log |t|)^{1/2}\,$ LanPapSucSup1988; furthermore, this blowup is stable under perturbations in the energy space MeRap-p, at least when the mass is close to the critical mass. Note that scaling shows that blowup cannot be any slower than $|t|^{-1/2}\,$.

The virial identity shows that blowup must occur when the energy is negative (which can only occur when the mass exceeds the ground state mass).Strictly speaking, the virial identity requires some decay on $u\,$ namely that $xu\,$ lies in $L^{2}\,$, however this restriction can be relaxed (OgTs1991, Nw1999, GgMe1995).

In one dimension d=1, the above blowup rate of $|t|^{-1/2}(loglog|t|)^{1/2}\,$ has in fact been achieved Per-p. Furthermore, one always this blowup behavior (or possibly slower, though one must still blow up by at least $|t|^{-1/2}\,$) whenever the energy is negative MeRap-p, MeRap-p2, and one either assumes that the mass is close to the critical mass or that $xu\,$ is in $L^{2}\,$. When the energy is zero, and one is not a ground state, then one has blowup like $|t|^{-1/2}(loglog|t|)^{1/2}\,$ in at least one direction of time ($t\rightarrow +\infty \,$ or $t\rightarrow -\infty \,$) MeRap-p, MeRap-p2.These results extend to higher dimensions as soon as a certain (plausible) spectral condition on the ground state is verified.

The exact nature of the blowup set is not yet fully understood, but there are some partial results.It appears that the generic rate of blowup is $|t|^{-1/2}(loglog|t|)^{1/2}\,$; the exceptional rate of $|t|^{-1}\,$ can occur for the self-similar solutions and also for larger solutions BoWg1997, but this seems to be very rare compared to the $|t|^{-1/2}(loglog|t|)^{1/2}\,$ blowup solutions (which are open in $H^{1}\,$ close to the critical mass MeRap-p).In fact close to the critical mass, there is a dichotomy, in that the blowup (if it occurs) is either $|t|^{-1}\,$ or faster, or $|t|^{-1/2}(loglog|t|)^{1/2}\,$ or slower MeRap-p, MeRap-p2.Also, near the blowup points the solution should have asymptotically zero energy Nw1999 and exhibit mass concentration Nw1992.

Conditions on the linearizability of this equation when the dispersion and nonlinearity are both sent to zero at controlled rates has been established in $d=1,2\,$ in CarKer-p (and in the $L^{2}\,$-supercritical case in CarFerGal-p.A key role is played by the size of the linear solution in the relevant Strichartz space.