Birth of a Theorem: A Mathematical Adventure (17 page)

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SEVENTEEN

Princeton

Afternoon of February 25, 2009

The children are back from school, merrily building a playhouse. The squirrels are prancing on the lawn outside.…

I’m on the phone with Clément. He’s feeling rather less merry.

“We can resolve some of the problems I mentioned using stratified estimates. But there are still loads of other problems.…”

“Well, at least we’re making progress.”

“I’ve been studying Alinhac–Gérard, and there’s a serious problem with the estimates: you’d need some leeway regarding regularity to get convergence to zero in the regularization term. And it gets worse: regularization could eliminate the bi-exponential convergence of the scheme.”

“Damn! I missed that! You’re
sure
that Newton’s rate of convergence is lost? Well, we’ll figure out something.”

“And the regularization constants in the analytic are monsters!”

“They’re a big problem, absolutely. But we’ll find a way to deal with them.”

“And then in any case the constants are going to blow up way too fast to be killed off by the convergence in Newton’s scheme! Since the background has got to be regularized in order to cope with the error created by the function
b
, it’s the inverse of the time, but there’s a constant, and this constant has got to make it possible to control the norms that come from the scattering.… Don’t forget, these norms grow in the course of the scheme, since we want the losses to be summable on lambda!”

“Okay, okay. I don’t know yet how we’re going to handle all of this. But I’m sure we’ll find a way!”

“Cédric, do you believe—
really
believe—that we’ll be able to handle all of this by means of regularization?”

“Yes, of course. These are all technical details. Look at what we’ve done overall. We’ve made a hell of a lot of progress. We’ve figured out how to deal with resonances and the plasma echo, we’ve worked out the principle of time cheating, we’ve got good scattering estimates, good norms—we’re almost there!”

Clément must think that I’m a pathological optimist, one of those people you’re crazy to have anything to do with, the sort of person who goes on hoping when there’s no hope left to be had. This latest difficulty does indeed look formidable, I admit. But still I feel sure we’ll find a solution. Already three times in the last three weeks we’ve found ourselves at an impasse, and each time we’ve found a way around it. It’s also true, however, that some obstacles we thought we’d put behind us have come back to block our way in another form.… Nonlinear Landau damping is a monster, no doubt about it—with as many heads as the Hydra! Nonetheless, I remain convinced that nothing can stop us.
My heart will conquer without striking a blow.

*   *   *

Date: Mon, 2 Feb 2009 12:40:04
+
0100

From: Clement Mouhot

To: Cedric Villani

Subject: Re: aggregate-10

Here are some comments off the top of my head:

– I’m confident for the moment about the norms with two shifts, I’m looking carefully at the scattering to see if the estimates I’ve got are sufficient to express it in terms of norms with two shifts,

– ok for section 5, it fits nicely with the transfer of regularity
+
gain in decay, that’s a really lovely trick! Am I right in supposing that the “gain in decay” part carries forward the “big” phase interval to only one of the functions (becoming an interval between both shifts of the norm with two shifts), in the hope that by applying it to the field created by the density everything will work out smoothly?

– as for section 6, ok for the general idea and the calculations, but (1) I’d rather not try to sum the series in k and l since the coeffs don’t seem to be summable (no big deal), (2) to be able to assume that epsilon is small in th. 6.3, it seems to me we’ve got to make c small as well--is that borne out by what comes later? More comments to follow … best, clement

Date: Sun, 8 Feb 2009 23:48:32 -0500

From: Cedric Villani

To: Clement Mouhot

Subject: news

So, two pieces of good news:

– after reading two articles on the plasma echo I realize that this phenomenon is caused by exactly the same “resonances” that were such a headache in section 6. It’s all the more astonishing when you consider that they use almost identical notations, with a \tau … This makes me even more convinced that the danger identified in section 6 is physically significant, in short, it is a question of determining whether the SELF-CONSISTENT ECHOES in the plasma are going to accumulate and eventually destroy the damping.

–I think I’ve found the right way to treat the term \ell
=
0 that I had “provisionally” set aside in section 5 (in \sigma_0 of Theorem 5.8: it is estimated like the others, but one keeps all the terms, and makes use of the fact that \
|
\int f(t,x,v) dx \
|=
O(1) in large time (or rather \
|
\int \nabla_v f(t,x,v) dx \
|=
O(1)). This IS NOT a consequence of our estimate over f(t,x,v) in the gliding norm, it’s an additional estimate. For a solution of the free transport, \int f(t,x,v) dx is preserved over time, therefore it’s perfectly reasonable. When scattering is added, it will no longer be O(1), but O(t-\tau) or something of the sort, and then this has to be killed by the exponential decay in t-\tau that I’ve kept in the present version of Theorem 5.8.

The modifications that I’ve made in the version attached here are:

* sections 1 and 2 modified to fully take into account these papers on the plasma echo (I hadn’t really understood what the experiment involved, and probably all the math hounds missed the crucial importance of this move, here I believe we’re miles ahead of everyone else)

* subsection added at the end of section 4 to make it clear which time norms we’re going to be working with; I mention this business of regularization by spatial means, which moreover is equally consistent with the sources cited by Kiessling

* section 5 modified to take into account the treatment of the term \ell
=
0

* reference added on the plasma experiment

An IMPORTANT CONSEQUENCE is that in section 8 it will be necessary not only to propagate the gliding regularity on f, but also to propagate the uniform (in t) regularity (in v) on \int f dx.

I haven’t made any modif in section 7 but as you must have guessed, what I’ve put in section 7.4 “Improvements” is outdated, I wrote it before realizing that it was the difference (\lambda \tau
+
\mu) - (\lambda’ \tau’
+
\mu’), or something of this ilk, that should really count.

I haven’t modified section 8 either but a lot of what I wrote there concerning the “zero mode” of f_\tau is likewise obsolete.

What news on your end? Everything now depends on section 7.

Best

Cedric

Date: Sat, 14 Feb 2009 17:35:28
+
0100

From: Clement Mouhot

To: Cedric Villani

Subject: Re: final aggregate-18

So here’s version 19 with a complete version of the statements of theorems 7.1 and 7.3 on scattering in a hybrid norm with one and two shifts. Apparently the composition theorem with two shifts from section 4 is all we need for the proof (phew!). It looks like it should work but you’ll have to check carefully, the version with two shifts is still a bit of a mess. I haven’t yet incorporated the Sobolev correction, but surely this point is less of a worry. Otherwise I’ve modified one detail (also in the theorem with one shift): the estimates of losses on the indices and on the amplitude are now not only uniform, but tend to zero in \tau \to
+
\infty, as required in section 8. And these losses are in O(t-\tau) for small (t-\tau). I’ll get back to work tomorrow, adding the Sobolev correction and completing section 8 so it follows on from section 7.

Best regards, clement

Date: Fri, 20 Feb 2009 18:05:36
+
0100

From: Clement Mouhot

To: Cedric Villani

Subject: Re: Draft version 20

Here’s version 20 (still in draft), the complete stratified theorem with two shifts. There’s now a fundamental problem in connection with theorem 5.9: b can’t go to 0 during the Nash–Moser scheme since it corrects an error term due to the scattering itself, which doesn’t go to zero since it is associated with the field … I’m now scrutinizing theorem 5.9.

best regards,

clement

EIGHTEEN

Princeton

February 27, 2009

A bit of a party atmosphere at the Institute today now that the five-day workshop on geometric partial differential equations is coming to an end. Very fine casting, with many stars—all the invited speakers agreed to participate.

In the seminar room I found a place to stand all the way at the back, behind a large table. Sometimes an audiovisual control board is set up on it, but not today, so I could spread out my notes on top. There’s no better place to be. I was lucky to get there before Peter Sarnak, a permanent professor at the Institute who likes it as much as I do. You can always be sure of staying awake, for one thing. If you’re sitting in a chair you’re more likely to drift off—and you’ve also got to settle for writing on a small fold-down tablet.

I like to be able to pace back and forth in my stocking feet when I’m listening to a lecture, ideal for stimulating thought.