AN INFORMAL SUMMARY OF BOURGAIN’S RADIAL CRITICAL NLS RESULT

J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO

Abstract. We try to summarize in broad detail Bourgain’s argument for global well-posedness of H^{1}-critical NLS in the radial case. (Note added, Oct 2004: These are notes from 2002, developed in order to attack the non-radial GWP problem for the same equation. With our resolution of this problem in [5], some of the material here is obsolete; nevertheless these notes may still be of interest to give some context to our result, and as such I have chosen to only edit these notes minimally. Those of you who are interested in these problems may also read [7], [8]. I thank Monica Visan for corrections and helpful comments.).

1. Introduction

We consider the quintic defocussing Schrödinger equation in R^{3+1}:

iu_{t} + Δu = |u|^{4}u.  (1)

This equation has a conserved Hamiltonian

H(u) = ∫ \frac{1}{2}|∇u|^{2} + \frac{1}{6}|u|^{6}.

We take data which is radial and in H^{1}, so that H(u) is bounded. Bourgain’s result is that there is a global H^{1} solution, and furthermore there is a spacetime bound

∫ ∫ |u|^{10} dxdt ≲ C(H(u)).

In particular this yields scattering.

Here I’ll try to summarize the basic points of the argument.

J.E.C. is supported in part by an N.S.F. Postdoctoral Research Fellowship. M.K. is supported in part by N.S.F. Grant DMS 9801558. G.S. is supported in part by N.S.F. Grant DMS 9800879 and by a grant from Hewlett and

Packard and the Sloan Foundation.

H.T. is supported in part by J.S.P.S. Grant No. 13740087. T.T. is a Clay Prize Fellow and is supported in part by grants from the Packard Foundation.

2. Step 1: Push Strichartz to the limit

The first step is to review the local theory, and in particular see exactly what the obstruction is in getting global existence just from Strichartz alone. (The obstruction turns out to be “bubbles” of the field u).

Let X be all the stuff that Strichartz and Sobolev lets you control from H^{1} data; so that if

iu_{t} + Δu = F; u(0) = u_{0}

then

∥u∥_{X} ≲ ∥u(0)∥Ḣ^{1} + ∥∇F ∥_{L_{x,t}^{10/7}}.

Thus

∥u∥_{X} = ∥u∥_{L_{t}^{∞}L_{x}^{2}} + ∥u∥_{L_{x,t}^{10}} + ∥∇u∥_{L_{x,t}^{10/3}} + …

Inserting this into (1) we obtain

∥u∥_{X} ≲ data + ∥∇(|u|^{4}u)∥_{L_{x,t}^{10/7}}
≲ data + ∥|u|^{4}∇u∥_{L_{x,t}^{10/7}}
≲ data + ∥u∥_{L_{x,t}^{10}}^{4}∥∇u∥_{L^{10/3}}
≲ data + ∥u∥_{L_{x,t}^{10}}^{4}∥u∥_{X}.

So we conclude that we can close the argument, and bound ∥u∥_{X} by the initial data, whenever the L_{x,t}^{10} norm is small. Of course, since the L_{x,t}^{10} norm is part of X this means for instance that we have global well-posedness when the data is small. If the data is large, we still get local well-posedness if we restrict to a time slab short enough so that the L_{x,t}^{10} norm (of either the nonlinear solution or the linear solution, it doesn’t really matter) is small. But of course the length of this slab depends on the profile of the data and not just on the norm.

So, the natural thing to do is to divide up spacetime into time slabs R^{3} × I, such that the L_{x,t}^{10} norm is, say, equal to η on each slab - where η is a fixed constant which is small enough that we can close the local theory argument on each slab. (In particular, we expect on each slab that the nonlinear flow looks pretty much like the linear flow). Our mission is now to show that there can only be a bounded number of such slabs (in particular they don’t concentrate before some blowup time T_{∗}). This will also give a global L_{x,t}^{10} bound which leads to scattering by the usual arguments.

Now we look at a single slab of spacetime R^{3} ×I, where the L_{x,t}^{10} is somewhat large, and see what is actually causing this norm to be large. First we observe that this can only happen if u is localized to a single frequency. To see this let us go back to the expression

∥|u|^{4}∇u∥_{L_{x,t}^{10/7}}

in the local theory, and suppose everything is frequency localized:

∥u_{1}u_{2}u_{3}u_{4}∇u_{5}∥_{L_{x,t}^{10/7}},

where u_{j} has frequency ∼ N_{j}, and I am ignoring the conjugations. The worst case is when the derivative falls on the highest frequency, so suppose that N_{1} ≤ N_{2} ≤ N_{3} ≤ N_{4} ≤ N_{5}. Then the cleverest thing to do is to estimate the lowest frequencies in L^{∞}, e.g.

∥u_{1}∥_{L_{x,t}^{∞}}∥u_{2}∥_{L_{x,t}^{∞}}∥u_{3}∥_{L_{x,t}^{10}}∥u_{4}∥_{L_{x,t}^{10/3}}∥∇u_{5}∥_{L_{x,t}^{10/3}},

which after some Sobolev etc. is bounded by

N_{1}^{1/2}N_{2}^{1/2}N_{4}^{−1}.

So if N_{1} and N_{4} (for instance) are quite far apart, this expression becomes quite small. The moral of this is that if u has several distinct frequency components, then ∥|u|^{4}∇u∥_{L^{10/7}} will be much smaller than one might think, which ultimately would make ∥u∥_{X} small. In particular, in order for the L_{x,t}^{10} norm to be large, u has to be concentrated in a single frequency, say ∼ 2^{k} (or perhaps just a bounded number of such frequency ranges).

From Strichartz we know the L_{x,t}^{10} norm is bounded by the Hamiltonian, indeed we have from spatial Sobolev and Strichartz

∥u∥_{L_{x,t}^{10}} ≲ ∥∇u∥_{L_{t}^{10}L_{x}^{30/13}} ≲ ∥∇u∥_{L^{2}}.

On the other hand, the Sobolev inequality is really only sharp at frequency ∼ 2^{k} when the function u is supported on a ball of radius ∼ 2^{−k}. (The Sobolev inequality is closely related to the isoperimetric inequality, which is only sharp on balls. You can amuse yourself seeing how the Sobolev inequality becomes inefficient for a function which has frequency |ξ| ∼ 2^{k} but is spread out equally over a number of balls of radius ∼ 2^{−k}. Since time has twice the dimension of space in Schrodinger equations, we thus expect u to also have time frequency |τ| ∼ 2^{2k} and to be concentrated in time on an interval of length 2^{−2k}). (One can also get the L_{x,t}^{10} bound from temporal Sobolev instead of spatial Sobolev by changing the exponents a bit and using the equation i∂_{t}+Δ = 0 to trade spatial derivatives for temporal derivatives).

The conclusion of all this is that in order for a slab R^{3} × I to claim a large L_{x,t}^{10} norm (i.e. at least η), it must contain a “bubble” - a region of spacetime of spatial width 2^{−k} and temporal width 2^{−2k} where the function u has frequency 2^{k} and has large L_{x,t}^{10} norm (∼ η^{C}) on this bubble. A little calculation shows that this means that u must have size about ≳ η^{C}2^{k/2} on this bubble - note that this also means that this bubble absorbs a fairly large portion of the Hamiltonian (about η^{C} or so). Henceforth I’m ignoring all powers of η.

Thus, if there were only a bounded number of such bubbles in spacetime, we’d be done. The enemy is if these bubbles keep re-forming over and over again. (A soliton can be regarded as a permanently recurring bubble, so in particular we have to disprove the existence of solitons if we want to get scattering.).

Note that different bubbles can be of vastly different sizes. This is a feature of the scale-invariance of the problem. (For, say, the cubic scattering problem, bubbles can only be of width ∼ 1; this is a consequence of the critical norm Ḣ^{1/2} lying between a subcritical conserved norm (the Hamiltonian, which is essentially Ḣ^{1}) and a supercritical conserved norm (the L^{2} norm)).

In the radial case, bubbles must be centered at the spatial origin; in the non-radial case they can be located at arbitrary locations in space.

Bourgain now argues by contradiction. Suppose we have a huge (unbounded) number of bubbles in spacetime. Then he uses combinatorial arguments (pigeonhole principle, that kind of thing) to say that if the number of bubbles is sufficiently big, at least one of the following possibilities must occur:

• (Future-Isolated bubble) There exists a bubble B with time interval I which is somewhere in the middle of the bubble sequence (e.g. in the middle third of the series of slabs), and such that no bubble follows this bubble B for a time interval ≫ |I|. (In other words, there is no bubble B′ with time interval I′ such that I′ lies to the future of I and dist(I′, I) ≲ |I|).

• (Past-Isolated bubble) There exists a bubble B with time interval I which is somewhere in the middle of the bubble sequence (e.g. in the middle third of the series of slabs), and such that no bubble precedes this bubble B for a time interval ≫ |I|. (In other words, there is no bubble B′ with time interval I′ such that I′ lies to the past of I and dist(I′, I) ≲ |I|).

• (Cluster of similarly-sized bubbles) There exists a sequence B_{1}, …, B_{N} of bubbles which are roughly the same size and are roughly adjacent to each other. A more precise statement of this is that if I_{j} denotes the temporal interval of the bubble B_{j}, then the I_{j} are all contained in a single interval I and

∑_{j} |I_{j}|^{1/2} ≫ |I|^{1/2}.

(Note that this is the case, for instance, if the I_{j} are all equally sized and partition |I|, if the number N of bubbles is large.)

• (Cluster of differently-sized bubbles) There exists a sequence B_{1}, …, B_{N} of bubbles with N ≫ 1, not necessarily in chronological order, such that each bubble B_{j} is much smaller than the previous bubble B_{j−1}, but is still nearby in the sense that dist(I_{j−1}, I_{j}) ≲ |I_{j}|.

It’s actually rather intuitive as to why at least one of the above three scenarios must happen, if there are sufficiently many bubbles to play with. If there is at least one future-isolated or past-isolated bubble in the middle third of the sequence, we are done, so suppose that no bubble in the middle third is isolated. This means that every bubble is close to at least one other bubble in the near future, and close to at least one other bubble in the near past.

Now pick the biggest bubble B_{1} in the middle of the sequence. Of all the bubbles nearby to B_{1}, let B_{2} be the biggest of the nearby bubbles. Of all the bubbles nearby to B_{2}, let B_{3} be the biggest of the remaining bubbles which has not already been selected. And so on. Either these bubbles are rapidly decreasing in size or they are relatively stable. In the first case we have a cluster of differently sized bubbles. In the second case it is a relatively easy matter to show we have a cluster of similarly-sized bubbles, and we are done.

Now we analyze each of the above possibilities in turn. In capsule summary, the argument is as follows:

• A cluster of similarly sized bubbles cannot occur, because this would contradict (a localized version of) Morawetz’s inequality.

• A cluster of differently sized bubbles cannot occur, because each of these bubbles uses up a certain amount of L^{6} norm, and finite speed of propagation prohibits these bubbles from sharing much of their L^{6} norm with the other bubbles (which are of radically different size). (Actually one needs to use weighted L^{2} norm as a proxy for L^{6}). Since the L^{6} norm is finite, a contradiction ensues.

• If there is a future-isolated bubble, we split the evolution into the evolution of the bubble and the evolution of everything else. Because of the isolation, these two evolutions will evolve independently into the future for quite a while. Thanks to the pseudoconformal identity, the bubble will begin to disperse quite a bit, and in particular its L_{x,t}^{10} norm will decay to become very small by the time we start seeing other bubbles appear in the future. Meanwhile, the stuff outside the bubble has smaller Hamiltonian and so by an induction hypothesis, that stuff has globally bounded L_{x,t}^{10} norm. One can then use Strichartz analysis to show that one can in fact piece the two solutions together indefinitely into the future. This shows that the future portion of the L_{x,t}^{10} norm of u is bounded; since the bubble was initially in the middle third of the sequence of slabs, this shows that the L_{x,t}^{10} norm is globally bounded.

• The past-isolated case is a time-reversed version of the future-isolated case.

The good news is that everything in the above argument extends to the non-radial case except the use of localized-Morawetz in order to prevent a cluster of similarly sized bubbles occurring. This is where our super-Morawetz should come in handy, except that it needs to be localized just as the original Morawetz estimate was localized by Bourgain. (Note added, Oct 2004: we are referring here to the interaction Morawetz inequality, developed in [3], [4]; see [7] for more discussion).

3. The bubble chain: using Morawetz to disallow a cluster of similarly sized bubbles

Morawetz’s inequality in its original form states that

∫_{I} ∫ |u|^{6}/|x| dxdt ≲ sup_{t∈I} ∥u(t)∥2 ; Ḣ^{1/2}

for any time interval I. This is not directly useful because we don’t control the Ḣ^{1/2} norm (we do control Ḣ^{1}, but the problem is with very low frequencies). However, if time is localized to an interval of size |I|, then we expect from the uncertainty principle that τ should not be localized to anything less than |I|^{−1}, which suggests heuristically that ξ should not be localized to anything less than |I|^{−1/2}. Thus we expect that spatial frequencies less than |I|^{−1/2} are insignificant, which means that we should morally be able to estimate the Ḣ^{1/2} norm by |I|^{1/2} times the Ḣ^{1} norm. This gives some background for Bourgain’s localized Morawetz estimate

∫_{I} ∫_{|x|≲|I|^{1/2}} |u|^{6}/|x| dxdt ≲ |I|^{1/2} sup_{t∈I} ∥u(t)∥2 ; Ḣ^{1}

To prove this estimate, recall how Morawetz’s original inequality was proven: we take the Morawetz action

M_{0}(t) := Im ∫ \overline{u}u_{r} dx

and differentiate it in time to obtain (after much integration by parts)

∂_{t}M_{0}(t) = C ∫ |u|^{6}/|x| dx + positive terms.

Since M_{0} is bounded by the square of the Ḣ^{1/2} norm, we’re done.

What Bourgain does is simply to insert a radial spatial cutoff χ(x) adapted to the ball of radius |I|^{1/2} around the origin:

M_{0}^{χ}(t) := Im ∫ χ\overline{u}u_{r} dx

From Hölder and Sobolev we see that M_{0}^{χ}(t) = O(|I|^{1/2}∥u∥3 ; Ḣ^{1}) (take χ in L^{3}, u in L^{6} and u_{r} in L^{2}). A computation (e.g. by taking our super-Morawetz notes and inserting an χ) then shows

∂_{t}M_{0}^{χ}(t) = C ∫ χ|u|^{6}/|x| dx + positive terms + O(∫ |∇χ||∇u|^{2} + |∇χ||u|^{6} dx).

The error term can be controlled simply by estimating ∇χ = O(|I|^{−1/2}) and using Sobolev. Since we are integrating over a time interval of length O(|I|), the error term is acceptable. This basically finishes the argument.

The point of Bourgain’s localized estimate is that (a) the right-hand side depends only on the H^{1} norm and not on other weird norms such as the H^{1/2} norm, and (b) the power of I is strictly less than 1 on the right-hand side. To see how it is useful, suppose we wanted to count how many bubbles of frequency 1 (i.e. they have size 1 on a bubble of spatial radius 1 and time length 1) can fit in an interval of length R ≫ 1, assuming (as we may in the radial case) that each bubble is centered at the origin. Each bubble gives a contribution of ∼ 1 to the right-hand side, so the total number of such bubbles is at most R^{1/2}. In particular, as R gets larger, one cannot pack the time interval of length R completely with bubbles of length 1. More generally, if a bunch of bubbles B_{j} with time intervals I_{j} are contained inside a time interval I, then we must have

∑_{j} |I_{j}|^{1/2} ≲ |I|^{1/2}.

Now, we need to do the same in the case when the bubbles are not necessarily centered at the origin. We do have an interaction Morawetz estimate

∫_{I} ∫ |u|^{4} dxdt ≲ ∥u_{0}∥_{2}^{2} sup_{t∈I} ∥u(t)∥2 ; Ḣ^{1/2}.

In principle, this tells us that a time interval of length R ≫ 1 cannot contain too many bubbles of frequency 1 (and arbitrary spatial location), except for the fact that the right-hand side has a very screwy norm. If we were to replace everything by the H^{1} norm (again using the heuristic that only spatial frequencies ≳ |I|^{−1/2} are significant we might hope to end up with an estimate such as

∫_{I} ∫ |u|^{4} dxdt ≲ |I|^{3/2} sup_{t} ∥u∥4 ; Ḣ^{1}.

(possibly with some spatial localization to the natural scale of |I|^{1/2} thrown in). One can check easily that the power of 3/2 on the RHS is the only one which is consistent with scaling. Actually, such an estimate is not too hard to obtain; indeed just from Hölder and Sobolev we have

∫_{|x|≤|I|^{1/2}} |u|^{4} ≲ |I|^{1/2}∥u∥4 ; Ḣ^{1}

and hence

∫_{I} ∫_{|x|≤|I|^{1/2}} |u|^{4} dxdt ≲ |I|^{3/2} sup_{t} ∥u∥4 ; Ḣ^{1}.

Of course, the fact that such an estimate was so easily proven suggests that it is useless, and indeed it is - because the power 3/2 on the right-hand side is larger than 1. For instance, if one asks how many bubbles of frequency 1 can fit inside a time interval of length R, the above estimate gives the useless bound of O(R^{3/2}), which is even worse than the trivial bound of O(R).

So, we need to show tweak the scaling on our interaction Morawetz estimate so that we get a more favorable power of |I| on the right-hand side. A model problem is the following:

Problem: Let I be a time interval of length R ≫ 1. Show that the slab I × R^{3} cannot contain ∼ R bubbles of frequency ∼ 1, assuming only that u has bounded Hamiltonian.

If the bubbles were all centered at the origin then the ordinary Morawetz would be able to solve this problem (indeed it would say that only O(R^{1/2}) bubbles are allowed to exist. Also, if u somehow had no low frequencies in it - it only had frequencies ∼ 1 and higher - then our interaction Morawetz would be able to handle this problem easily. (In fact we would be OK even if u had spatial frequencies down to ≫ R^{−1/3}, or equivalently temporal frequencies down to ≫ R^{−2/3}). The problem occurs when u contains big, flat, low-frequency components (with spatial frequency ≲ R^{−1/3}); these are unable to directly cause a bubble, but they mess up the Morawetz estimate by means of having a huge L^{2} norm. I don’t know how to fix this problem yet (perhaps try to project away the low frequencies and just work with the high? Use the energy density instead of the L^{2} density?) (Note, Oct 2004: The former approach turns out, combined with some other tricks, to be what is needed; interaction Morawetz can be salvaged to give a useful inequality in the energy-critical case once one projects away the low frequencies.)

4. The bubble pyramid: using finite speed of propagation to disallow a cluster of differently sized bubbles

Now we look at another of Bourgain’s possibilities, namely that of a cluster of bubbles B_{1}, B_{2}, … which are rapidly decreasing in size, such that each B_{i} is close to the previous one. This forces there to exist a “blowup time t_{0}” such that each bubble B_{i}, with a time interval I_{i} (say), to lie within O(|I_{i}|) of t_{0}. Note that the bubbles can lie either to the future or the past of t_{0}, so perhaps “blowup time” is a misnomer. Perhaps “concentration time” is better.

A fixed bubble B_{i} has spatial frequency ∼ |I_{i}|^{−1/2}, and u has size about ∼ |I_{i}|^{−1/4} on the bubble. In particular, for a generic time t_{i} in the bubble, we have

∫_{B(x_{i},|I_{i}|^{1/2})} |u(x, t_{i})|^{6} dx ≳ 1

where x_{i} is the spatial center of the bubble (of course we have x_{i} = 0 in the radial case). Thus at time t_{i}, a rather large portion of the L^{6} norm (which is bounded, because the Hamiltonian is bounded) is stored in this bubble. The key thing is that this is also true at the concentration time t_{0} (with a slight growth in the radius of the ball):

∫_{B(x_{i},C|I_{i}|^{1/2})} |u(x, t_{0})|^{6} dx ≳ 1

Indeed, we even have

∫_{B(x_{i},C|I_{i}|^{1/2})\ ⋃_{j>i} B(x_{j},C|I_{j}|^{1/2})} |u(x, t_{0})|^{6} dx ≳ 1.

In other words, at time t_{0}, each bubble uses up a separate chunk of the L^{6} norm. Since there is only so much L^{6} norm to go around, this limits the number of bubbles which can form a cluster of this type.

How do we get this estimate? By Hölder it will suffice to show the L^{2} estimate

∫_{B(x_{i},C|I_{i}|^{1/2})} |u(x, t_{0})|^{2} dx ≳ |I_{i}|,

since one can use the a priori L^{6} bound and the rapid decrease of the I_{j} to show that the L^{2} norm on the balls B(x_{j}, C|I_{j}|^{1/2}) are ≪ |I_{i}|.

To get this L^{2} bound, we first go back to time t_{i}, and observe from the bubble bounds that

∫_{B(x_{i},|I_{i}|^{1/2})} |u(x, t_{i})|^{2} dx ≳ |I_{i}|.

On the other hand, the L^{2} norm is a conserved quantity. So if we can show some finite speed of propagation - in particular, that the L^{2} mass propagates at speed O(|I_{i}|^{−1/2}) - then we’re in business.

In principle, this finite speed of propagation is reasonable because the bubble has frequency ∼ |I_{i}|^{−1/2}, and in Schrödinger equations the propagation speed is proportional to the frequency. But it is difficult making this heuristic rigorous. Nevertheless, Bourgain has an ingenious pigeonholing argument to overcome this difficulty.

Let χ(x) be a spatial cutoff function to a ball B(x_{i}, C′|I_{i}|^{1/2}), where 1 ≤ C′ ≤ C is a radius to be chosen later, and consider the localized mass

E^{χ}(t) := ∫ χ(x)|u(x, t)|^{2} dx.

Thus we have E^{χ}(t_{i}) ≳ |I_{i}|, and our objective is to show that E^{χ}(t_{0}) ≳ |I_{i}|.

A standard integration by parts computation gives

∂_{t}E^{χ}(t) ≤ C ∫ |∇χ||u||∇u|.

The obvious thing to do at this bound is apply Hölder and estimate u, ∇u by the Hamiltonian. If one does this, we get a bound of ∂_{t}E^{χ}(t) = O(1). This is almost enough to conclude E^{χ}(t_{0}) ≳ |I_{i}|, if we had |t_{i} − t_{0}| ≪ |I_{i}|. Unfortunately we only have |t_{i} −t_{0}| ∼ |I_{i}| - the concentration time t_{0} is close to, but not actually contained in, the bubble.

To fix this problem Bourgain does something clever, of course. He picks C′ to be a power of two selected at random between 1 and C - there are about log C possible choices here. The function ∇η is then supported on a dyadic annulus |x − x_{i}| ∼ C′. For a randomly chosen C′, such an annulus will contain about 1/(log C) of the Hamiltonian, on the average. This is enough to squeeze an additional 1/(log C) or so in the bound for ∂_{t}E^{η}(t), which is enough to do what we want provided that C is large enough (roughly, C should be the exponential of the ratio |t_{i} − t_{0}|/|I_{i}|.

This entire argument works nicely in the non-radial case and so we don’t have to lift a finger to say that the cluster of differently-sized bubbles is not possible.

5. A bubble pops: How pseudoconformal decay and perturbation theory can inductively strip away an isolated bubble

We’ve seen that no clusters are possible, either of equally sized or differently sized bubbles. The only other option is that there is a bubble B which is isolated, either in the future or the past. By translation and scale invariance we can assume the bubble is centered at the spacetime origin with frequency 1, and that it is isolated in the future.

So the situation is as follows. At time 0 the solution u(0) can be written as a bubble v(0), which is localized to a unit ball and is essentially a bump function, plus a remainder term w(0), which has slightly smaller Hamiltonian. Also, for a long time interval 0 ≤ t ≤ T the solution u has a small L_{x,t}^{10} norm. (More precisely, the L^{10} norm is O(η), while H(w(0)) < H(u(0)) − Cη^{3}. It turns out that keeping track of the powers of η is slightly important, and this step is a little trickier in the 4D case.)

Now we try to decouple the v and w evolution. More precisely, we let the bump v(0) evolve by NLS

iv_{t} + Δv = |v|^{4}v

while w then evolves by the difference equation

iw_{t} + Δw = |(w + v)|^{4}(w + v) − |v|^{4}v

so that we have u = v + w throughout.

Since v is localized, we can take advantage of the pseudoconformal decay law

∥(x + 2it∇)v(t)∥_{2}^{2} + \frac{4}{3}t^{2}∥v(t)∥_{6}^{6} ≤ ∥xv(0)∥_{2}^{2}

to obtain a decay

∥v(t)∥_{6}^{6} ≲ t^{−2}.

Interpolating this with what one can get from Strichartz (in particular, using some Strichartz estimate in L_{x}^{p} for some p > 10), we see that the L^{10} norm of v gets small as one goes into the future:

∫_{t>T} ∫ |v|^{10} dxdt ≲ T^{−c}

for some c > 0.

Now look at w. We know that H(w(0)) < H(u(0)) − Cη^{3}. Now let’s try to get control on w up to time T. Since u has small L_{x,t}^{10} norm (about O(η)) on the slab [0, T], it is relatively easy to show that v and w also have norm O(η). A naive perturbation theory argument would then give

∥w(T) − w(0)∥_{H^{1}} ≲ η

but this is not good enough because the error of η will swamp the gain of η^{3} mentioned earlier. However, one can do better by dealing directly with the Hamiltonian. If one computes

∂_{t}H(w(t))

by hand, the quadratic terms cancel and we can bound the above by sextic and higher order terms such as ∫ |∇u|^{2}|u|^{4} + |u|^{10}, plus similar things where some of the u are replaced by v and w. This stuff is very small - about O(η^{4}) - and so we still have

H(w(T)) < H(u(T)) − Cη^{3}.

Now look at the evolution for times t > T. If w evolved by the original NLS evolution, we could use an induction hypothesis and conclude that w had a global solution with bounded L^{10} norm. Meanwhile, in this region [T, ∞), v has a small L^{10} norm, about O(T^{−c}). Thus if T is sufficiently large, one can use perturbation theory (or just plain old global well-posedness) to show that w evolves close to the NLS flow. In other words, we have a global L^{10} bound for both v and w on [T, ∞), and hence for u. Since we assume this original bubble B to be in the middle third of the stack of slabs, this means that we have a global bound on the L_{x,t}^{10} norm of u, and we can close the induction.

References

[1] J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, JAMS 12 (1999), 145-171.

[2] J. Bourgain, New global well-posedness results for non-linear Schrödinger equations, AMS Publications, 1999.

[3] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Existence globale et diffusion pour l’équation de Schrödinger nonlinéaire répulsive cubique sur R^{3} en dessous l’espace d’énergie , Journées “Équations aux Dérivées Partielles” (Forges-les-Eaux, 2002), Exp. No. X, 14, 2002.

[4] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Scattering for the 3D cubic NLS below the energy norm, preprint, 2003. see http://arxiv.org/abs/math.AP/0301260 .

[5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering in the energy space for the critical nonlinear Schrodinger equation in R^{3}, preprint.

[6] M. Grillakis, On nonlinear Schrödinger equations. , Comm. Partial Differential Equations 25 (2000), no. 9-10, 1827–1844.

[7] T. Tao, Viriel, Morawetz, and interaction Morawetz inequalities, unpublished.

[8] T. Tao, Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrodinger equation for radial data, preprint.

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