Best
Cedric
Date: Sun, 18 Jan 2009 17:28:12 -0500
From: Cedric Villani
To: Clement Mouhot
Subject: Re: transfer
Here’s the revised file. For it all to hold together (I haven’t mentioned Newton’s scheme yet), we’ll have to (i) check to see that the “bihybrid” norms I’ve introduced at the end of section 4 exhibit the same properties as the “simple” hybrid norms, and that therefore one gets similar estimates for the characteristics in these norms(!); (ii) find a way of combining the two distinct effects that are described in the new section 5; (iii) put all that at the end of chapter 7 and fill in the missing details to give an estimate for the total density; (iv) check everything! In other words, our plates are full. For the moment I suggest that you go over what I’ve written and tell me if you see anything that doesn’t look right. I’ll let you know later if I see things that we can clearly work on together at the same time …
A few more points:
I think your estimates for transfer of regularity were flawed, the result was too strong, I wasn’t able to reproduce them in the usual norms; on the other hand I did use your transfer strategy in section 5. But when you try to apply it in large time (ttoinfty, tau remaining small) the transfer seems to crash, the permissible exponents don’t allow the integral to converge over time. I concocted (don’t ask me how) a method for getting more integration over time, but this time without getting more regularity. We’ve got to find a way to combine the two.
More later, best
Cedric
Date: Mon, 19 Jan 2009 00:50:44 -0500
From: Cedric Villani
To: Clement Mouhot
Subject: Re: transfer
I’ve gone through the file again and cleaned it up a bit, so the attached version should be considered definitive. Going forward I suggest the following division of labor:
– you see to it that Proposition 4.17 and Theorem 6.3 are in good shape, that’s asking a lot, I know, but it will have the advantage of forcing you to go over my estimates in sections 4 and 6 again line by line:-) which has to be done, because if we make a mistake calculating the conditions that the exponents must satisfy we’re sunk. For the moment in both these sections I’ve entered some pretty much arbitrary statements as placeholders, with guesstimates that for all I know may turn out to be right but it may also turn out that reality is more complicated. No need to recheck the proofs, but we do have to be absolutely sure about the bounds we end up with, everything else depends on them.
– in the meantime I’ll get to work finishing sections 5 and 7 (modulo the input that will come from Theorem 6.3).
– also I’m going to talk to Tremaine tomorrow about the physics part of the intro.
– if you have time to flesh out your comment following, you can incorporate it in the introduction to section 5, where I’ve already mentioned the connection with averaging lemmas. (Be careful, since we’re working in the analytic class it’s not wholly obvious that this is an L^1/L^infty phenomenon??)
If you can get started right away, and assuming all goes well, it ought to be possible to get all this done 2-3 days from now, and only Newton/Nash–Moser will be left to work out in detail. (But I think our priority has to be getting the statements in 4.17 and 6.3 right so we’re sure we’re not building castles in the air.)
Best
Cedric
Date: Mon, 19 Jan 2009 13:42:27 +0100
From: Clement Mouhot
To: Cedric Villani
Subject: Re: transfer
Hi Cedric,
it’s turning into a real monster;)!!
* * *
Excerpts from aggregate file-3 (January 18, 2009)
4.7 Bihybrid norms
We shall be led to use the following more complicated norms:
Definition 4.15. We define the space by
(…)
After trial and error, the best we could do was to recover this decay in the “bihybrid” norms described in subsection 4.7:
Proposition 5.6 (regularity-to-decay estimate in hybrid spaces).
Let f = ft(x, v), g = gt(x, v), and
Then
THIRTEEN
Princeton
January 21, 2009
Thanks to the rabbit I pulled out of my hat on my returning from the museum the other evening, I’ve been able to get back on track. But today I’m filled with a strange mixture of optimism and dread. Got around one major roadblock: made a few explicit calculations and eventually figured out how to manage a term that had gotten too big—that much gives me hope. At the same time, the complexity of the mathematical landscape that’s now opened up makes my head spin if I think about it for more than a few moments.
Could it really be that Vlasov’s splendid equation, which I thought I was beginning to get a handle on, operates only by fits and starts? On paper, at least, it looks as though sometimes the response to external perturbations suddenly occurs very, very quickly. I’ve never heard of such a thing; it’s not in any of the articles and books that I’ve read. But in any case we’re making progress.
* * *
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. I
n 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 T4 configuration!”
Luc Tartar
“Really? Well, maybe, I don’t know. In any case I did that in order to construct impossible solutions to certain elliptical equations. And then I realized that there was a general formula.”
He explained the formula.
“Yes, of course, that’s Gromov’s method of convex integration!”
“Really? No, I don’t think so. What I was doing is much simpler. The construction is very straightforward, it works because you’re in the convex envelope and you can express the approximate solution each time as a convex combination, and then.…”
But these are the main elements of convex integration theory! Did this guy really rediscover everything all by himself, without any idea what other people had already done? Where was he living? On Mars?
“So what about fluid mechanics?”
“Oh, right! I’d heard Mandelbrot give a talk, and I said to myself, I’d like to do something similar. So I began to study Euler’s equations from the fractal point of view, and I realized that I could reinterpret the same kind of things I’d done in my thesis. But it was complicated.”
Birth of a Theorem: A Mathematical Adventure Page 7
