Birth of a Theorem: A Mathematical Adventure
Page 9
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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.
At the break I rushed outside without bothering to put my shoes back on and ran upstairs to my office. Quick phone call to Clément.
“Clément, did you see my message from yesterday with the new file?”
“You mean the new scheme you got by writing down the characteristic equations? Yeah, I looked at it and I began to do the calculations. Looks like a bear to me.”
Monster, beast, bear—these words occur over and over again in our conversations.…
“I have a feeling we’re going to run into problems with convergence,” Clément continued. “I’m also worried about Newton’s scheme and the linearization error terms. There’s another technical detail, too—you’re always going to have scattering from the previous step, and it won’t be trivial!”
I was a bit annoyed that my brilliant idea hadn’t convinced him.
“Well, we’ll see. If it doesn’t work, too bad, we’ll stick with the present scheme.”
“It’s pretty wild—we’ve got more than a hundred pages of proofs by this point and we’re still not done yet!! Do you really think we’ll ever finish?”
“Patience, patience. We’re almost there.…”
The intermission in the seminar room was over. I hurried back downstai
rs to hear the concluding talks.
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Partial differential equations express relations between the rates of variation of certain quantities as a function of different parameters. PDEs constitute one of the most dynamic and varied domains of the mathematical sciences, defying all attempts at unification. They are found in every phenomenon studied by the physics of continuous systems, involving all states of matter (gases, fluids, solids, plasmas) and all physical theories (classical, relativistic, quantum, and so on).
But partial differential equations also lurk behind many geometric problems. Geometric PDEs, as they are called, make it possible to deform geometric objects in accordance with well-established laws. The application of methods of analysis to problems in other fields of mathematics is an example of the sort of cross-fertilization that became increasingly common in the course of the twentieth century.
The February 2009 workshop at the IAS addressed three principal themes: conformal geometries (involving transformations on a space that distort distances but preserve angles), optimal transport (the movement of mass from an initial defined configuration to a final defined configuration while expending the least energy possible), and free-boundary problems (the form of the boundary that separates two states of matter or two materials). Three topics that touch on geometry and analysis in addition to physics.
In the 1950s, John Nash disrupted the balance between geometry and analysis when he discovered that the abstract geometric problem of isometric embedding could be solved by the fine “peeling” of partial differential equations.
A few years ago, Grigori Perelman solved Poincaré’s conjecture by using a geometric PDE known as the Ricci flow, invented by Richard Hamilton. Once again the analytic solution of an emblematic problem in geometry shattered the status quo and created an unprecedented interest in exploring other applications of geometric PDEs. The shock waves from Perelman’s bomb were felt throughout the world of mathematics—an echo of the one Nash had set off fifty years earlier.
NINETEEN
Princeton
March 1, 2009
I read the message that had just appeared on my computer screen, and then read it again. Couldn’t believe my eyes.
Clément’s come up with a new plan? He wants to give up on regularization? Wants to forget about making up for the loss of regularity encoded in the time interval?
Where did all this come from? For several months now we’ve been trying to make a Newton scheme work with regularization, as in Nash–Moser—and now Clément is telling me that we need to do a Newton scheme without regularization? And that we’ve got to estimate along the trajectories, while preserving the initial time and the final time, with two different times??
Well, maybe he’s right, who knows? Cédric, you’ve got to start paying attention, the young guys are brilliant. If you don’t watch out, they’re going to leave you in the dust!
Okay, there’s nothing you can do about it, the next generation always ends up winning … but … already?
Save the sniveling for later. First thing, you’ve got to try to understand what he’s getting at. What does this whole business of estimating really amount to, when you get right down to it? Why should it be necessary to preserve the memory of the initial time?
In the end, Clément and I will be able to share the credit for the major innovations of our work more or less equally: I came up with the norms, the deflection estimates, the decay in large time, and the echoes; he came up with the time cheating, the stratification of errors, the dual time estimates, and now the idea of dispensing with regularization. And then there’s the idea of gliding norms, a product of one of our joint working sessions; not really sure whose idea that was. To say nothing, of course, of hundreds of little tricks …
Perhaps it wasn’t such a bad thing after all that an ocean and a few time zones came between us in the middle of the project: for a couple of months each of us has been forced to concentrate on his own strategy without having to listen to any opposing arguments. It’s now become clear, however, that our separate points of view will have to be reconciled somehow.
If Clément is right, the last great conceptual obstacle has just been overcome. On this first day of March our undertaking has entered into a new phase, less fun, but also more secure. The overall plan is in place, the period of free-ranging, open-ended exploration is over. Now we’ve got to consolidate, reinforce, verify, verify, verify.… The moment has come for us to deploy the full firepower of our analytical skills!
Tomorrow I’m taking care of the kids; there’s no school on account of the snowstorm. But come Tuesday, the final push begins. One way or another the Problem simply has got to be tamed, even if it means going without sleep. I’m going to take Landau with me everywhere—in the woods, on the beach, even to bed. Time now for him to watch out!
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Long afterward, Clément confessed to me that he had decided to bail earlier that weekend. On Saturday morning, February 28, he began to compose an ominous message: “All hope is lost … the technical hurdles are insurmountable … can’t see any way forward … I give up.” But just as he was about to send it, he hesitated. He wanted to find the right words to convince me, but also to console me. So he saved his message in the draft folder. Going back to it that evening, armed with pencil and paper in order to make a list of all the paths we had explored and all the dead ends they had led us into, he saw, to his amazement, the right way to proceed opening up before him. The next morning, having gotten up at six o’clock after a few hours of fitful sleep, he wrote out everything again so he’d have a clean copy of the key idea that might just save us, and then finished his message to me.
That day, Sunday, March 1, we came within a whisker of abandoning our dream. Several months of work very nearly disappeared—at best, filed away in a drawer; at worst, gone up in smoke. On the other side of the Atlantic, however, I had no idea that we had come within an inch of catastrophe. All I sensed was the enthusiasm emanating from Clément’s message.
During the month of February I exchanged a good one hundred emails with Clément. In March, more than two hundred!
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Date: Sun, 1 Mar 2009 19:28:25 +0100
From: Clement Mouhot
To: Cedric Villani
Subject: Re: aggregate-27
Might be some hope if we go at it from another angle: not to regularize but to try to propagate the norm to a shift that’s needed at each step of the scheme, only along the characteristics of the preceding step. In order, then, we would estimate at rank n (I don’t write down the summable losses on lambda and mu each time):
1) F norm of the density rho_n with lambda index t + mu
2) Z norm of the distrib h_n with lambda coefficient, mu and t
3) C norm of the spatial average
4) Z norm at time tau with a shift -bt/(1+b) along the (complete) characteristics S_{t,tau} of order n-1. Differentiating with respect to tau we obtain an equation on
H_tau:= h^n _tau circ S_{t,tau} ^{n-1}
of the type (I don’t include any possible minus signs)
partial_tau H = (F[h^n] cdot nabla f^{n-1}) circ S_{t,tau} ^{n-1} + (F[h^{n-1}] cdot nabla h^{n-1})
circ S_{t,tau} ^{n-1}
So basically in this equation there’s no longer any field at all and we treat the whole right-hand term as a source term, by using the bounds on density in point 1) above: The Z norm is estimated with the b shift: as for the density we treat the error committed on account of the characteristics by means of this shift (since the norm is projected on x) and for the other terms we use the recurrence assumption from the preceding point to bound the existing norms.
5) Now we need to have a bound (in shifted norm) on f^n circ S_{t,tau} ^n (with well-defined characteristics n), by using the bound of the recurrence assumption (in shifted norm) on f^{n-1} circ S_{t,tau} ^{n-1}. Tha
nks to 4) above, by addition we get a limit on f^n circ S_{t,tau} ^{n-1}. Then we’ve got to exploit the possibility of bounding f^n circ S_{t,tau} ^n (characteristics of step n) as a function of f^n circ S_{t,tau} ^{n-1} (characteristics of step n-1) modulo a loss, summable as n goes to infinity.
The general idea may be summarized as follows:
– To estimate the density there isn’t any choice, we’ve got to have characteristics and a shifted norm (with a shift of order 1) on the distribution of the preceding step, along the characteristics of the preceding step,
– But once you’ve got the bound on the characteristics, you can work along the characteristics and in shifted norm, since when they’re projected onto the density, the two phenomena cancel each other.
There’s one thing I omitted to mention, the gradient in v on the background, which doesn’t commute with the composition by the characteristics, but one might hope to have something like shifted norm of (nabla_v f^{n-1}) circ S_{t,tau} ^{n-1} smaller than constant times shifted norm of nabla_v (f^{n-1} circ S_{t,tau} ^{n-1}) …
If you’re around we can talk about it on the phone I’m at home for another hour: I think that this accords with your scheme for the most part, with the difference that it fundamentally distinguishes between two steps and only later looks at things along characteristics.
Best regards, Clement
Date: Mon, 2 Mar 2009 12:34:51 +0100
From: Clement Mouhot
To: Cedric Villani
Subject: Version 29
So here’s version 29, in which I’ve really tried to implement the strategy I spoke to you about yesterday: it’s in section 9 on linear stability which I’ve entirely rewritten, and subsections 11.5 and 11.6 of the section on the Newton scheme where I’ve sketched the convergence study. Unless I’m hugely mistaken, I really have the impression that we’re nearing our goal!!