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| ===<math>L^2</math> critical NLS on <math>R^d</math>===
| | #REDIRECT [[mass critical NLS]] |
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| The L^2 critical situation s<sub>c</sub> = 0 occurs when p = 1 + 4/d. Note that the power non-linearity is smooth in dimensions d=1 ([#Quintic_NLS_on_R quintic NLS]) and d=2 ([#Cubic_NLS_on_R^2 cubic NLS]). One always has GWP and scattering in L^2 for small data (see [[Bibliography#GiVl1978|GiVl1978]], [[Bibliography#GiVl1979|GiVl1979]], [[Bibliography#CaWe1990|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 [[Bibliography#Ws1983|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.
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| 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 [[Bibliography#Me1993|Me1993]]. This solution blows up in H^1 like |t|^{-1} as t -> 0-. However, numerics suggest that there should be solutions that exhibit the much slower blowup |t|^{-1/2} (log log|t|)^{1/2} [[Bibliography#LanPapSucSup1988|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}.
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| 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 \u2013 namely that x u lies in L^2, however this restriction can be relaxed ([[Bibliography#OgTs1991|OgTs1991]], [[Bibliography#Nw1999|Nw1999]], <br /> [[Bibliography#GgMe1995|GgMe1995]].
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| In [#Quintic_NLS_on_R one dimension d=1], the above blowup rate of |t|^{-1/2} (log log|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} (log log |t|)^{1/2} in at least one direction of time (t -> +infinity or t -> -infinity) [MeRap-p], [MeRap-p2].These results extend to higher dimensions as soon as a certain (plausible) spectral condition on the ground state is verified.
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| 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} (log log|t|)^{1/2}; the exceptional rate of |t|^{-1} can occur for the self-similar solutions and also for larger solutions [[Bibliography#BoWg1997|BoWg1997]], but this seems to be very rare compared to the |t|^{-1/2} (log log|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} (log log |t|)^{1/2} or slower [MeRap-p], [MeRap-p2].Also, near the blowup points the solution should have asymptotically zero energy [[Bibliography#Nw1999|Nw1999]] and exhibit mass concentration [[Bibliography#Nw1992|Nw1992]].
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| 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.
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