Birth of a Theorem: A Mathematical Adventure (14 page)

Date: Wed, 21 Jan 2009 23:44:49 -0500

From: Cedric Villani

To: Clement Mouhot

Subject:!!

There we go, finally, after hours of floundering miserably I’m pretty sure I’ve figured out why the O(t) I was complaining about on the phone today gets canceled. It’s a MONSTER!

Apparently the problem isn’t in the bilinear estimates or in Moser’s scheme, it occurs at the level of the “Gronwall” equation in which \rho is estimated as a function of itself … the point is that we’re dealing with something like

u(t) \leq source
+
\int_0^t a(s,t) u(s) ds

where u(t) is a bound on \
|
\rho(t)\
|
. If \int_0^t a(s,t) ds
=
O(1), everything’s fine. The problem is that \int_0^t a(s,t) can also be equal to O(t) (really not an obstacle in and of itself, I’ve chosen the most perfect possible cases and this can always happen). But when it does happen, it’s at a point strictly internal to [0,t], in the middle somewhere (as in the case where you’ve got k and \ell such that 0
=
(k
+
\ell)/2); or around 2/3 if you have 0
=
(2/3)k
+
\ell/3, etc. But then the recurrence equation on u(s) becomes

u(t) \leq source
+
epsilon t u(t/2)

and the solutions to this thing turn out not to be bounded, they slowly decay! (subexponential) But since the norm on \rho restricts exponential decay, you do in fact end up getting this decay...........

Whipping this thing into shape looks pretty daunting (resonances will all have to be catalogued, basically). That’s my job for tomorrow. In any case none of this relieves us of the obligation to verify the properties of the bihybrid norms.

Best

Cedric

Date: Wed, 21 Jan 2009 09:25:21
+
0100

From: Clement Mouhot

To: Cedric Villani

Subject: Re:!!

It really does look like we’ve got a monster on our hands! As for me, I’ve looked at the Nash–Moser part and I agree it seems unlikely that the factor t can be imported into it … On the other hand if I correctly understand the argument concerning the bound on u(t) it is absolutely necessary that the point s where a(s,t) is large remain uniformly at a strictly positive distance from t … Another thing is that we would therefore be dealing with a subexponential time bound in solving the nonlinear problem. And to have it eaten up by the norm on \rho we’d have to accept losing a bit on its index, which in my opinion must absolutely be avoided in the Nash–Moser part…?

best, clement

FOURTEEN

Princeton

January 28, 2009

Darkness! I’ve got to be in the dark. I have to be alone in the dark. Where? The children’s room, shutters closed!

Regularization. Newton’s scheme. Exponential constants. Everything was swirling around and around inside my head.

I’d brought the children back home from school and immediately locked myself away in their room. Had to kickstart my brain and put it into high gear. Tomorrow’s my talk at Rutgers, and the proof still hasn’t come together.

I’ve got to be by myself. I’ve got to walk around and think. Think hard. It’s
urgent
!

Claire has put up with worse than this without so much as a word. But here I was walking around in circles, alone in a dark room, while she was in the kitchen making dinner. It was a bit much.

“This is getting really weird!!”

I didn’t respond. Circuits overloaded, too many mathematical signals trying to get through. And the pressure of having to make something happen—fast. Even so, when dinner was served I came out to eat with the family, then went back to work for the entire evening. One calculation in particular that I thought for sure was correct turned out to be trouble; somehow I got something wrong. How serious an error was it? I had to find out.

Finally quit around two o’clock in the morning, now to bed. I have a feeling everything’s going to be fine after all.

*   *   *

Date: Thu, 29 Jan 2009 02:00:55 -0500

From: Cedric Villani

To: Clement Mouhot

Subject: aggregate-10

!!!! I think we’ve now got the missing pieces.

–First, I finally figured out (unless there’s a mistake) what needs to be done to lose an epsilon as small as one likes (even if it means losing a very large constant, either exponential or exponential squared in 1/epsilon). This drops out of a perfectly diabolical calculation that for right now I’ve simply sketched at the end of section 6. It seems totally miraculous but everything falls in place just the way it should, looks like it must be right.

– Next, I think I’ve also figured out exactly where we lose on the characteristics and the scattering. We’re going to have to redo all the calculations in this section, which won’t be any fun … I’ve inserted a few comments in a subsection at the end of this section.

With that, I think we’ve now got everything we need to feed the Nash–Moser. Tomorrow (Thursday) I’m not here. Here’s what I suggest we do after that: I go back over section 6 and subexponential growth again while you tackle the scattering estimates, which aren’t too depressing. Let’s aim to have everything except the last section revised and ready to go early next week sometime. Will that work for you?

Best

Cedric

FIFTEEN

New Brunswick

January 29, 2009

The dreaded day had finally come. The day of my talk to the mathematical physics seminar at Rutgers University, not quite twenty miles up the road from Princeton. I got a ride this morning with Eric Carlen and Joel Lebowitz, both of whom live in Princeton and work at Rutgers.

This was my second visit there. The first one was a couple of years ago for a gathering in memory of the late Martin Kruskal, the inventor of solitons, a great mind. I still vividly recall the amusing anecdotes recounted by the speakers that day. One in particular of Kruskal and two colleagues talking in an elevator, so deeply immersed in conversation that they went up and down for twenty minutes, oblivious to the people getting on and off.

Today was less entertaining. I was under a lot of pressure!

Usually a seminar talk is devoted to presenting a result that has been meticulously checked to be sure that everything is correct. That’s what I’ve always done in the past. Today was different: not only hadn’t the proof I was about to present been scrutinized down to the least detail, it wasn’t even complete.

Last night, of course, I had convinced myself that everything was in good shape, and that it remained only to write up the last part. But this morning my doubts returned. Before being dispelled once more. Then, in the car, I had to fight them off again.

When the time came to give my talk, I really was convinced that everything was fine. Self-delusion? I didn’t go into a lot of mathematical detail. Instead I spoke at some length about the significance of the problem and its mathematical interpretation, and unveiled my vaunted norm, the complexity of which made my listeners gasp—even though I had limited myself to presenting the version with five indices, not the one with seven.…

Afterward about a dozen of us sat down together for lunch. The conversation flowed easily. In the audience earlier one could not have failed to notice Michael Kiessling, a giant elf of a man, eyes sparkling with an air of impish exuberance. Now, at the table, Michael was telling me with his usual infectious enthusiasm about how as a young man he fell in love with plasma physics, screening, the plasma wave echo, quasi-linear theory, and so on.

Michael Kiessling

His mention of the plasma echo immediately concentrated my full attention. What a lovely experiment! You begin by preparing a plasma, which is to say a gas in which the electrons have been separated from the nuclei, so that the plasma is at rest. Then you disturb this motionless state by briefly applying an electric field, which is to say a “pulse” (actually a train of pulses) that excites the gas. Once the current that has been propagated in this manner has faded away, a second field is applied. You wait for the current to fade away a second time. It is at this point that the miracle occurs: if the two pulses have been well modulated, you will observe a spontaneous response, at a precise instant. This response is called the
echo.
…

Spooky, huh? One utters an (electrical) cry in the plasma wilderness, then a second cry (in a different pitch), and a few moments later the plasma responds with a cry of its own (in yet another pitch)!

All this brought to mind some calculations I did a few days ago: a temporal resonance … the plasma reacting at certain quite specific moments … I thought that I’d lost my mind—but perhaps it’s the same thing as the echo phenomenon that plasma physicists have known about for years?

I made a mental note to think about this some more. For the moment, however, I was content to make small talk with my hosts. Who have you got in your department right now? Recruited any good people lately? Why yes, things are going very well, there’s So-and-So and So-and-So, also So-and-So, and So-and-So—

This last name gave me a start.

“What!? Vladimir Scheffer works here!!”

“Yes—though it’s been ages since anyone’s seen him. Why do you ask, Cédric? Do you know his work?”

“Yes, of course! I gave a Bourbaki seminar just last year on his famous existence theorem for paradoxical solutions of Euler’s equations.… I’ve got to meet him!”

“He’s not around much, no one’s talked to him in a long time. I can try to track him down after lunch if you’d like.”

In the event, Joel did manage to get in touch with Scheffer. He agreed to join us later that afternoon.

I won’t forget my encounter with him anytime soon.

Scheffer apologized at length for not having been able to come earlier, something to do with administrative duties that seemed to involve forestalling threats of legal action against the university by disgruntled students. It wasn’t very clear.

After a while Scheffer and I excused ourselves to talk privately in a small room with a blackboard next door to Joel’s office.

“I gave a Bourbaki seminar on your work. Here, I’ve printed out the text for you! It’s in French, but perhaps you’ll be able to get something out of it. I explain in great detail how your existence theorem for paradoxical solutions was improved and simplified by De Lellis and Székelyhidi.”

“Ah, this is very interesting, thank you.”

“I wanted to ask you, how did you ever come up with the idea of constructing these incredible solutions?”

“Well, it’s really very simple. In my thesis I had shown that there exist impossible objects, things that shouldn’t exist in our world. Here’s the method.”

He drew a few humps on the board, and a sort of four-pointed star. I recognized the figure.

“Yes, of course, that’s Tartar’s T
₄
configuration!”