Mass critical NLS: Difference between revisions

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===<math>L^2</math> critical NLS on <math>R^d</math>===
===<math>L^2</math> critical NLS on <math>R^d</math>===


The <math>L^2\,</math> critical situation <math>s_c = 0\,</math> occurs when <math>p = 1 + 4/d\,</math>. Note that the power non-linearity is smooth in dimensions <math>d=1\,</math> ([#Quintic_NLS_on_R quintic NLS]) and <math>d=2\,</math> ([#Cubic_NLS_on_R^2 cubic NLS]). One always has GWP and scattering in <math>L^2\,</math> 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 <math>H^1\,</math> (and slightly below) but is only conjectured in <math>L^2\,.</math> No scattering result is known for large data, even in the radial smooth case.
The <math>L^2\,</math> critical situation <math>s_c = 0\,</math> occurs when <math>p = 1 + 4/d\,</math>. Note that the power non-linearity is smooth in dimensions <math>d=1\,</math> ([[Quintic NLS on R|quintic NLS]]) and <math>d=2\,</math> ([[Cubic NLS on R2|cubic NLS]]). One always has GWP and scattering in <math>L^2\,</math> 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 <math>H^1\,</math> (and slightly below) but is only conjectured in <math>L^2\,.</math> No scattering result is known for large data, even in the radial smooth case.


In the focusing case, there is blowup for large <math>L^2\,</math> 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 <math>H^1\,</math> like <math>|t|^{-1}\,</math> as <math>t \rightarrow 0-\,.</math> However, numerics suggest that there should be solutions that exhibit the much slower blowup <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math> [[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 <math>|t|^{-1/2}\,</math>.
In the focusing case, there is blowup for large <math>L^2\,</math> 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 <math>H^1\,</math> like <math>|t|^{-1}\,</math> as <math>t \rightarrow 0-\,.</math> However, numerics suggest that there should be solutions that exhibit the much slower blowup <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math> [[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 <math>|t|^{-1/2}\,</math>.
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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 <math>u \,</math> namely that <math>x u\,</math> lies in <math>L^2\,</math>, however this restriction can be relaxed ([[OgTs1991]], [[Nw1999]], [[GgMe1995]]).
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 <math>u \,</math> namely that <math>x u\,</math> lies in <math>L^2\,</math>, however this restriction can be relaxed ([[OgTs1991]], [[Nw1999]], [[GgMe1995]]).


In [#Quintic_NLS_on_R one dimension d=1], the above blowup rate of <math>|t|^{-1/2}(log log|t|)^{1/2}\,</math> 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 <math>|t|^{-1/2}\,</math>) whenever the energy is negative [[MeRap-p]], [[MeRap-p2]], and one either assumes that the mass is close to the critical mass or that <math>xu\,</math> is in <math>L^2\,</math>. When the energy is zero, and one is not a ground state, then one has blowup like <math>|t|^{-1/2} (log log |t|)^{1/2}\,</math> in at least one direction of time (<math>t \rightarrow +\infty\,</math> or <math>t \rightarrow -\infty\,</math>) [[MeRap-p]], [[MeRap-p2]].These results extend to higher dimensions as soon as a certain (plausible) spectral condition on the ground state is verified.
In [[Quintic NLS on R|one dimension d=1]], the above blowup rate of <math>|t|^{-1/2}(log log|t|)^{1/2}\,</math> 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 <math>|t|^{-1/2}\,</math>) whenever the energy is negative [[MeRap-p]], [[MeRap-p2]], and one either assumes that the mass is close to the critical mass or that <math>xu\,</math> is in <math>L^2\,</math>. When the energy is zero, and one is not a ground state, then one has blowup like <math>|t|^{-1/2} (log log |t|)^{1/2}\,</math> in at least one direction of time (<math>t \rightarrow +\infty\,</math> or <math>t \rightarrow -\infty\,</math>) [[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 <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math>; the exceptional rate of <math>|t|^{-1}\,</math> can occur for the self-similar solutions and also for larger solutions [[BoWg1997]], but this seems to be very rare compared to the <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math> blowup solutions (which are open in <math>H^1\,</math> 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 <math>|t|^{-1}\,</math> or faster, or <math>|t|^{-1/2} (log log |t|)^{1/2}\,</math> or slower [[MeRap-p]], [[MeRap-p2]].Also, near the blowup points the solution should have asymptotically zero energy [[Nw1999]] and exhibit mass concentration [[Nw1992]].
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 <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math>; the exceptional rate of <math>|t|^{-1}\,</math> can occur for the self-similar solutions and also for larger solutions [[BoWg1997]], but this seems to be very rare compared to the <math>|t|^{-1/2} (log log|t|)^{1/2}\,</math> blowup solutions (which are open in <math>H^1\,</math> 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 <math>|t|^{-1}\,</math> or faster, or <math>|t|^{-1/2} (log log |t|)^{1/2}\,</math> or slower [[MeRap-p]], [[MeRap-p2]].Also, near the blowup points the solution should have asymptotically zero energy [[Nw1999]] and exhibit mass concentration [[Nw1992]].

Revision as of 20:35, 4 March 2007

critical NLS on

The critical situation occurs when . Note that the power non-linearity is smooth in dimensions (quintic NLS) and (cubic NLS). One always has GWP and scattering in 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 (and slightly below) but is only conjectured in No scattering result is known for large data, even in the radial smooth case.

In the focusing case, there is blowup for large 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 like as However, numerics suggest that there should be solutions that exhibit the much slower blowup 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 .

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 namely that lies in , however this restriction can be relaxed (OgTs1991, Nw1999, GgMe1995).

In one dimension d=1, the above blowup rate of 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 ) whenever the energy is negative MeRap-p, MeRap-p2, and one either assumes that the mass is close to the critical mass or that is in . When the energy is zero, and one is not a ground state, then one has blowup like in at least one direction of time ( or ) 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 ; the exceptional rate of can occur for the self-similar solutions and also for larger solutions BoWg1997, but this seems to be very rare compared to the blowup solutions (which are open in 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 or faster, or 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 in CarKer-p (and in the -supercritical case in CarFerGal-p.A key role is played by the size of the linear solution in the relevant Strichartz space.