A pyramid-shaped blow-up set for the 2d semilinear wave equation
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00:00
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Transcript: English(auto-generated)
00:15
So first of all, I would like to thank the organizers for giving me the opportunity to talk in this nice conference.
00:24
Today I will talk about a very recent work done jointly with Franck Merle. Okay, so let me first start with the equation. We have the semi-linear wave equation in two space dimensions.
00:41
So here, the nonlinearity is u to the p, where p is subconformal. And the conforming exponent is less than the sub exponent. So here, p is less than 5, because most of the time I will be in two dimensions. Of course, I will talk first about the one-dimensional case, which is very well understood.
01:04
But then I will focus on the 2D case. We had, as far as blowup is concerned, we had many, many works before. So many people in this conference room have already worked on this equation.
01:22
So I cite some of them, and I apologize for those who I might have forgotten. Okay, now I will not consider global in-time solutions, but only solutions which are not, which exist up to some time, here t bar, okay.
01:47
But then, from the finite speed of propagation, the solution may stop of existing at this minimal existing time, and then it continues to exist somewhere else, up to some surface, which is a graph.
02:04
X gives capital T of x, okay, which is a kind of local blowup time. And from the finite speed of propagation, this graph is already one Lipschitz. Why? Because my domain of definition is simply a union of backward light cones.
02:25
So a union of backward light cones is either the whole half space T positive, or the subgraph of one Lipschitz function. All this comes directly from the Cauchy theory, which we do in H1 times L2,
02:40
by the way, because we are subaleft subcritical anyway. Okay, so you see, as I said, if you take any point on this surface, the backward light cone with vertex x, T of x is included in the domain of definition.
03:00
Then we will give a geometric definition. If you can change the slope of your light cone, which is in blue here, and make a green cone with a slope which is strictly less than one, and still stay in the domain of definition, we will say that we have a non-characteristic point.
03:24
Their set will be called script R. All the other points will be called characteristic points, and their set will be called S. S like singular, because in fact the solution blows up everywhere, you see? Okay, so you are either non-characteristic or characteristic,
03:43
and in fact the characteristic points will be the nasty ones when you will have a more complicated behavior. Now I start with the case n equals one. First thing, any blowup solution has a non-characteristic point.
04:02
Why? Because you always take the minimum, the minimal time, the place where you have the minimal time, and then you will see that, of course, you can have even a flat cone with slope zero if you want. Of course, I take initial data for this in h1 loc uniform times l2 loc uniform,
04:23
meaning that the l2 norm on every wall with radius one is uniformly bounded. So you have always your solution which is defined in a strip, at least, okay, your domain of definition contains a strip, so you have a minimal time, and there you have a non-characteristic point.
04:45
But then this is for free, if I can say. What is difficult to have is to have a solution with a characteristic point, and this was an open problem some years ago
05:00
when we solved that with Franck Merle with this small example. Take initial data which is odd, okay, and then with large plateaus here and here, so from the finite speed of propagation it will stay constant and dependent of x in a smaller interval of space.
05:23
So here you can solve it like the ODE. You know that it will blow up. You can see that it blows up. The ODE is explicit, okay, so this is not a global solution. It's a non-global solution which will blow up, okay,
05:42
and then because it's initially odd, it will stay odd all the time, so u of zero will be always equal to zero, and we will have a characteristic point at the origin. And by the way, having characteristic points is connected to sign changing,
06:00
in fact, as we will see this in a moment. An important property in one-space dimension, this solution is stable, meaning that you may perturb it and breaking the symmetry and still have a characteristic point in the middle close to zero,
06:21
not necessarily at zero but close to zero. Okay, so this is a robust property, having a characteristic point with one change of sign is stable with respect to initial data. Okay, so this is the picture in one-space dimension with the existence of non-characteristic points which is free,
06:44
and then having a characteristic point, this is something more difficult. Now I would like to move to the asymptotic behavior near blow-up points. So we will have two kinds of behaviors, near non-characteristic points and near characteristic points.
07:04
To see in a nice way the behavior, it's useful to introduce similarity variables. So here we have a new function, new space variable, new time variable. Let me start with time. Time is simply a slow time, okay,
07:21
s equals negative log of capital T of x0 minus t, going to infinity as t goes to capital T minus x0. y is a zoom near the singularity, and this zoom is in the wave style, meaning that we have x minus x0 to the power one and capital T minus t to the power one,
07:41
unlike the heat equation where, as you may know, we have a square root here because space and time do not play the same role. And then, for the new function, we simply divide u by the rate of the ODE. So the question we are asking, can we compare the growth of the solution of the PDE
08:02
to the growth of the ODE which is explicit, which is known. Okay, then I'm not writing the equation satisfied by w, but of course doing some algebra we can have it, and we are working mostly in the w variable, but here just to illustrate the blow-up behavior. I'm just saying that if you write this PDE,
08:24
you will find a class of trivial solutions. These trivial solutions in the u variable are exactly self-similar blow-up solutions. They give you exactly self-similar blow-up solutions. When d is equal to zero, you have the ODE solution.
08:42
When d is non-zero, it's just moving the ODE solution with the Lorentz transform. Okay, now if x0 is non-characteristic, then wx0 has a profile, and that profile is this guy, because this family is the only possibility
09:00
for having stationary solution in w. We have Lyapunov functional in w, decreasing energy, so this helps us to have a limit, a limit to the set of stationary solutions, and this set is completely characterized. You have zero or this family with plus and minus.
09:22
Now, if you have a characteristic point, as I have already told you, this is the nasty case. So here, and I'm making connection with the talk of Patrick, we have multisolutants. So look here, you have this kind, this family,
09:43
but the parameter is moving, and we will see that two neighbors have opposite signs. And this parameter which is moving is explicitly given by this formula. So just to give you the result,
10:04
you will see some parameters going to one, others going to negative one, and if you have an odd number of solitons, the middle solitons, soliton will not move. Okay, all these modalities, in particular the multisolutants do occur.
10:23
We were able with Raphael code to construct a solution for any k, we have a solution which behaves like that. But of course, it's better to have a picture, and let me here make a picture for you when k equals four. If you have four solitons,
10:41
it's nice to further change the variables because let me go back a little bit. Here, if x and t is in your light code with vertex x0, t of x0, then y is in the unit pool. Okay, now I will make further change of variable,
11:03
y equals hyperbolic tangent of xi, so xi is in r, y is still in the unit pool, okay, negative one, one, okay, and I multiply w by this rate, and here, miracle, you see the KdV solitons.
11:20
So your w bar is decomposing into a sum of alternate KdV solitons, okay, and you see that two neighbors have opposite signs, and the two of the left will go to the left, and the two on the right will go to the right. Okay, if you have an odd number of solitons,
11:43
then the middle soliton would stay in the middle. Okay, what else to be said here? Of course, the middle of the soliton is given completely sharply. We have an explicit formula for that. It moves like log of s,
12:01
which makes log of log of capital T minus T, in fact. This is the motion of the center of the soliton. And in fact, as Patrick said before, there are many, many connections between the two talks, the center of the solitons
12:21
obeys some kind of orthogonality condition, and that orthogonality condition gives us the ODE satisfied by the center of the solitons. And this ODE, as you see here, is maybe familiar to some of you. This is not the Toda system,
12:42
because the Toda system comes with double derivative here, second derivative for zeta i. Here we have only one. Somehow it's maybe easier to handle, but not that much. And of course, we have this kind of small terms,
13:02
which come here, simply because our equation is not linear equation. So if you have... Let me go back here. This guy is a solution, but the sum of solitons is not a solution. And because it's not a solution, we have some defect, and that defect is proportional to the small terms,
13:25
and this gives us the law. Then you see that we have an explicit family, which is given here, just to check that we have this family is completely easy. Now, we have already talked about the behavior of the solution.
13:44
If you are at a non-characteristic point, you have one soliton with a parameter that does not move. If you are at a characteristic point, then you have a sum of two or more solitons. Now, what about the regularity of the blow-up curve?
14:02
They are connected, in fact. You cannot have one before the other. Asymptotic behavior and the regularity of the blow-up set has to be somehow advanced side by side. But here I'm just giving you the result. So R, the set of non-characteristic points, is open.
14:23
And T is C1 there. And then, if you know that W approaches kappa of d of x0, then this d of x0 is the derivative of T. You see how asymptotic behavior and the regularity are linked.
14:42
Now, S, the set of characteristic points, is finite on compact sets. And near each a in S, your blow-up curve here in dashed line is tangent to the backward light cone.
15:02
So it has a corner with 90 degrees. And it's not differentiable. So we have half a derivative from the left, which is 1, and from the right, which is negative 1. And as I told you, every characteristic point is isolated.
15:24
Then we can have a further expansion of the difference between T and the equation of the backward light cone. It's given here. So it has a log correction. And in the log correction, you see the number of solitons.
15:42
Once again, the regularity is linked to the asymptotic behavior. You can see it. So then, usually, it's not symmetric. This is really something strange, because usually when we find blow-up profiles,
16:01
we will find a profile which is somehow symmetric. But here, the first part is symmetric. But the second part is by no means symmetric. Why? Because here in this solution, this zeta i, they're invariant with translation in zeta i. You can change zeta i, make their barycenter different.
16:24
The barycenter is conserved. The center of mass is conserved here. So if you take the center of mass, which is 0, then this guy is symmetric. If you take a center of mass here, which is not 0, then this correction is not symmetric with respect to x0.
16:42
I think that I said everything. So I can move to the next slide. Okay, I also see time running. We have some generalizations, which, well, I would say easy generalization, because of what will come later. But at that time, this was not that easy.
17:02
Well, anyway, you can add some lower order terms, f of u, where f of u is less than u to the power q with q less than p. And even some terms involving derivatives in time, in space, x and t, as far as their growth does not exceed this.
17:21
So here we cannot put other power. The maximum power here is one. Unfortunately, we could not fill the gap to what the scaling would give us. Okay, so here everything is true for this kind of equations. Second case, if you take radial equation, okay, but outside the origin.
17:45
If you are outside the origin, this term will be of lower order. Okay, so you can handle it as we did here. And everything is like the 1D case. For example, if you have, but of course, we have radial symmetry.
18:03
So if you have a characteristic point, the whole sphere will be characteristic. Okay, et cetera. Then you can maybe make a mixture of both cases, radial with perturbation. This is possible, but still outside the origin.
18:20
You can also take the complex case and even the, this is very recent contribution by my former student, Azaih. You can take u in r to the power m and have generalizations, at least for non-characteristic case. And even with strong perturbations, going up to u to the p,
18:42
divided by log of u to some power a, and this was done by Amza and Saeedi. Okay, then what happens when n is more than two? Which is, of course, the topic of my talk here. So it's high time for me to start talking about n more than two.
19:03
The first result is completely general. No symmetry, no hypothesis. It's about the blow-up rate. So you define w, the similarity variable version. As I told you, we define the similarity variable version like that
19:23
by dividing u by the rate of the ODE solution. So we are asking whether w will be bounded, which means that we will have ODE rate. Okay, so this is true, near non-characteristic points.
19:40
And of course, sharp also from the solution of the equation h1 times l2, and the finite speed of propagation. We have a lower bound. At characteristic points, we have the same bound from above, but with some weights.
20:00
So we remove the weights by considering only balls of radius one half. And then after that, as far as classification is concerned, we don't have a lot of results. But let me concentrate on characteristic points, because in fact, characteristic points is my aim in this talk,
20:25
although it doesn't appear in the title, but you will see it in this pyramid soon. So here we have no classification of characteristic points. In 1D, we said that every characteristic point is isolated.
20:40
Here, no result. And also for construction, for examples, if you ask me, can you construct a solution with characteristic points? Then I tell you, well, I can take a 1D solution and then make some truncations. And in some place where the solution will be rigorously 1D,
21:02
I will see, for example, a solution which is in 2D having a line of characteristic points, because in 1D, it's just one point. Or if I am making truncations with a radial solution, then I will see a characteristic set which is a sphere, for example.
21:23
But this is always rigorously 1D behavior. And that's it. We have no other examples. So the questions, when we started with Frank to do this, is can we find new blow-up solutions with characteristic points with a non-1D behavior?
21:42
That was the challenge. And let me here stop a little bit and talk about a general question, the geometry of the set of the singular points. So our dream with Frank two years ago, a year and a half ago, was to find a solution where S is cross-shaped.
22:02
For example, in 2D, you take the axis x1 equals 0, x2 equals 0. Can we have a solution where locally near the origin, the set of characteristic points is equal to the axis? That was our aim at first.
22:22
But I will tell you if this is true or not in a minute, so please wait a little bit. But more generally, the geometry of the singular set is, for me at least, the problem which is largely open. And for the case of the semi-linear heat equation, which is probably the easiest case of PDE having blow-up,
22:43
there are, the question is largely open, as you will see, but I'm not sure if many people are aware that this question is completely open. Look here. You can construct, you have examples where the blow-up set.
23:03
And let me just say something that for the heat equation, we have only one blow-up time, capital T. After that, the solution does not exist. So at capital T, we have two kinds of points. Blow-up points where the solution goes to infinity, and the regular points where the solution is bounded.
23:24
Okay, so I will concentrate on the set of blow-up points, in fact. I don't have characteristic or non-characteristic, only blow-up points. We can have solutions where the singular set is only one point,
23:41
or a finite number of points. You can choose them. Okay, or a sphere, or a finite number of concentric spheres, and you can choose them, and that's it. For example, in 2D, we don't have, we don't know if we have a solution which may blow up on an ellipse. Open problem.
24:01
Can we have a cross in 2D? Open problem. Can we have a segment? Open problem, etc. Can you imagine any other geometry? And we don't have an answer. And the same question, of course, somehow exists for the similar wave equation, semi-linear wave equation, but for the similar wave equation,
24:22
we have to change a little bit the definition of singular set. Because here, all points are blow-up points, but at later times. So here, the good notion for singular is the notion of characteristic points. Because near non-characteristic points, the situation is,
24:41
at least in one space dimension, completely easy. We have a profile where t is c1, etc. So here, the corresponding question would be, can we have, for example, a solution where the set of characteristic points is a cross? That was our initial thought.
25:01
And this is the theorem. Well, we could not answer the question, but at least we had, we came out with a new type of, a new solution. So we have a solution in 2D, which blows up on one Lipschitz graph, which is pyramid shaped. And let me show you here the picture.
25:23
Thank you, Frank. Okay, so you can see, yes, the picture is here. Okay, so Tx is T0 minus maximum of x1 minus x2. Maximum x1 minus x2 is like the pyramid of Egypt.
25:41
Okay, you see, so something like that. So if you see, in the direction of the axis x1, okay, if you restrict yourself only to this direction x1, you take x2 equals 0, you will see the corner with 90 degrees,
26:03
like in one space dimension. Same thing with x2, okay, when x1 is equal to 0. But along the bisectress, you will see another angle, okay, which, well, you will have a slope of 1 over square root of 2, in fact,
26:22
because this is the pyramid, okay. We have exactly the geometry of the pyramid. So this is local near 0. And let me make some remarks. First of all, Ux of t is, of course, non-radial, because we have a pyramid. Pyramid is not radial, okay. Second thing, my solution, we construct that
26:43
as being symmetric with respect to the axis, and anti-symmetric with respect to bisectresses, which means that U will remain always rigorously 0 on bisectresses, okay. And the surprising result is that only 0 is the characteristic point.
27:06
All other points are non-characteristic. And let me tell you about something which is counterintuitive in some sense, because in one space dimension, we have noticed that the change of sign,
27:22
or being 0, is linked to having a characteristic point. So here on the bisectress, our solution is always 0, but we have a non-characteristic point. So this is something which is completely different from the 1D situation. So only the origin is a characteristic point.
27:45
So here, in fact, this result we proved only locally, okay, because we were somehow lazy, but we are able to say that all the other points, even far in space, are non-characteristic. This is possible, though somehow technical.
28:03
Now let me talk about the regularity of the blowup graph. I will simply say, okay, that the regularity is the same as the regularity of the pyramid, okay. So the pyramid, you see here, sorry, what happens? Okay, I will, no, no, no problem, I can do it.
28:25
Okay, so you see here, your pyramid is regular everywhere except at the origin and on bisectresses.
28:42
Same for my blowup graph. We are C1 everywhere except on the bisectresses, okay. So if we are outside the bisectresses, from symmetries, it's enough to consider the case where zero is less than x2,
29:01
strictly less than x1. Here, look at your derivatives. Your derivative is like the derivative of the pyramid, negative one, but with logarithmic correction, like what we had in one space dimension, in fact. Okay, and this is almost zero. In fact, the derivative in the other direction.
29:22
Now, on the bisectresses outside the origin, you have directional derivatives in all directions, except in the direction of the bisectrices, bisectrics, and this is like the pyramid, in fact. But at the origin, you have directional derivatives
29:42
except once again along the bisectresses, okay. And let me insist on something here. Unlike the 1D, we have on the bisectresses the first example of non-characteristic points where t is non-differentiable.
30:00
Because in 1D, if you are at non-characteristic points, then you are differentiable. And at characteristic points, you have a corner of 90 degrees. Here, we still have a corner at the characteristic point, which is the origin, but on the bisectresses,
30:22
we have non-characteristic points and t, which is not differentiable. Okay, now what about the behavior? So, at the origin, you remember this change of variable, it's always centered at the point where you want to see the behavior.
30:44
Okay, so when I talk about w0, this means that I'm working in the backward light cone with vertex 0, t of 0. Here, I will see four solitons, two along the x1 direction,
31:04
one going to the right, this one, and one going to the left, they are symmetric, okay, and two along the x2 direction, they are also symmetric, and we see the anti-symmetry. So, this means that if you draw a circle,
31:23
which is the section of your backward light cone with vertex 0, t0, you will see, if you go clockwise, for example, you have positive, negative, positive, negative, okay, if you encounter all the axes, okay. Once again, we have this kind of sign change, in fact,
31:44
like in the 1D situation. Here, the solitons are always the same, okay, and then the parameter is, once again, completely explicit, and it satisfies the same kind of ODE, which is connected to the TODA system.
32:01
Here, because of symmetry, so we have only one center to deal with. If we had here d1, d2, d3, d4, we would have something involving zeta 1, zeta 2, and zeta 4. Okay, then, well, what I would say is that d bar is negative 1
32:23
plus some correction here, which is a logarithmic correction, because s is negative log of capital T minus t, and you see that d goes to negative 1, and negative d goes to 1, which gives you the slopes of your pyramid.
32:40
So this is for w sub zero. What happens outside the origin? So if x0 is outside the origin, as I told you, we have non-characteristic points, and here we have a convergence to stationary solutions.
33:01
Okay, and we have two cases. If we are outside the bisectresses, then we will converge like in the 1D situation to this special soliton, the same one. Okay, and its slope is given by negative 1 plus this logarithmic correction.
33:20
So we are not rigorously equal to the pyramid, but we have a correction to the pyramid, which is given like that. Of course, this is given just here, but if you change by symmetry, you can find the behavior everywhere outside the bisectresses. And now, if you are on one of the bisectresses,
33:43
then you will find a new blow-up, a new stationary solution, which is not radial, which is anti-symmetric, which is odd. Well, which is anti-symmetric with respect to the bisectress.
34:05
Okay, so yes, in higher space dimension, the kappa d is not the only possibility for the set of stationary solutions. We have a new set.
34:21
We are not able to characterize everything, but we found a new stationary solution. Okay, then the proof. This is maybe the most hard part of the proof. I should have started with the proof, because at first, maybe it's easier to follow,
34:42
but at the end, well, it's more difficult. But of course, the results are more important to state anyway. We have two major steps. First step, the construction in the light cone. You have finite speed of propagation. So it's completely meaningful to start with initial data
35:02
in the section of a backward light cone and to follow it only in the backward light cone. And after that comes step two, where we will find the behavior outside the origin. So here we make a construction with prescribed behavior
35:22
in the light cone, and then we find the regularity. Between the two steps, there is a small step, which is not very long, but we need to use the finite speed of propagation and extend our initial data outside the section of the light cone
35:44
in order to have something which is defined, hopefully, everywhere in space. So in fact, our initial data is defined in an initial, is compactly supported and defined in a square. And the square is suitable to the geometry of the pyramid, in fact.
36:04
And it is really supported in a set which is strictly larger than the section of the light cone. And as I told already for the 1D case, the asymptotic behavior and the regularity of blowup set
36:22
are completely linked and advanced side by side in the proof. Okay, let me now go to the first step. As I told you, we are working only in the light cone, which means that when I introduce W, I will work only in the unit ball.
36:40
Y is in the unit ball. Okay, so we are able, so this is the goal, find a solution in W which obeys this behavior. Okay, so we see our solitons, as I stated before in the theorem. Okay, so two solitons which are symmetric in the X1 behavior,
37:03
two solitons symmetric in the X2 behavior, and we have anti-symmetry with respect to bisectrices. Okay, and I give you all other characterization of the parameter, d of s is completely explicit. So this is my goal, okay.
37:20
The framework is the construction of a solution for PDE with prescribed behavior. Okay, the method. Well, you linearize the equation around the intended behavior, and we find three regions in the spectrum, negative spectrum. And this is controlled thanks to a linearized version of the Lyapunov functional,
37:43
because in W we have a Lyapunov functional for the whole system, the non-linear. So even for the linearized equation, you have a Lyapunov functional, and this helps you to control all the negative parts of the spectrum. We do not compute any eigenvalue explicitly in the negative range.
38:04
Then we have lambda equals zero, which is controlled thanks to modulation in the parameter d of the solitons. Remember that this d parameter is, in fact, the parameter of the Lorentz transform, which operates on the ODE solution in the UXT setting.
38:23
So we change it a little bit to kill the projection on lambda equals zero. Lambda equals one and not negative one. Sorry for the mistake. Okay, lambda equals one. It's controlled and killed, in fact, thanks to modulation with respect to this parameter, nu, in this family.
38:43
So this family, when nu is zero, you have your solitons. Take your soliton kappa of d and y, then go back to UXT, and then again to W, but with a different time. You find this family. Okay? And of course, this construction is inspired by the construction we did with Raphael Cote
39:06
for multi-solitons in one space dimension. Okay. Then let me give you a history of the construction with prescribed behavior. Well, again, sorry, I'm sure I forget some people and probably some recent work,
39:27
but it was possible to use these ideas, NLS, KDV, WaterWave, Schrodinger maps, Gensburg, Landau, Keller-Siegel, Wave, Heat, Schrodinger maps with many, many people, which I cite here.
39:46
Now I move to step two, the behavior of W x zero, okay? And the regularity of T of x zero when x zero is outside the origin. Let me suggest the following.
40:02
You take x zero, okay? And then if you want to know the behavior of W x zero, which is completely equivalent to knowing the behavior of UXT in the backward light cone with vertex x zero T of x zero,
40:21
in this case, you just remark when x zero is small, the sections of the cone with vertex x zero T of x zero, and the cone with vertex zero T of x zero are almost the same when you are far from the singularity, okay? So far from the singularity, you can start from W x zero,
40:44
go back to U, and from U, go to W zero, and you will find that W x zero and W zero are linked with this nice algebraic identity, okay? So this is completely explicit. Since W zero has four solitons,
41:03
W x zero far from the singularity will also have four solitons, but with a deformation, because you see, we are multiplying W zero with this factor, and the four solitons for W x zero, in fact, involve this kappa tilde, kappa star, this family, okay?
41:26
So we have these four generalized solitons with deformation. When I say deformation, because we have this factor, which means that here, we have a nu which is not zero, okay? And then, what we will do with some dynamics,
41:42
we will follow these four solitons, and we will see that two will disappear, because if I go here, when nu is positive, because nu is mu e to the power s, if mu is positive, s going to infinity, okay?
42:01
All this will go to zero. So two will disappear, and then I will have only two solitons, and in some cases, only one solid. And we need strong analysis here, and the strong analysis is the following. If you start with four solitons which are decoupled,
42:22
which do not see each other, you can follow them for a long time, okay? And you see if someone would go to zero, or would go to infinity, or stay close to kappa of D. Okay, then if we are not on the bisectancies,
42:41
say in this range, then we will be left with only one soliton, okay? At some point, s star, such that s star is like negative log of x1, so if x1 is close to zero, we will have to wait a long time
43:00
until making the three solitons disappear, and having only one. If you are far from x1, then well, this would happen rather quickly, okay? And then we have a trapping result, okay? Which we first proved in one space dimension, and then in any dimension for subconformal exponent.
43:22
If you are close to this family, then you would converge to a member of that family. And of course, when you converge, the parameter here is a gradient of t, and it is close to this parameter.
43:42
So we have some analysis to make our solution close to one soliton, and once it is close to one soliton, it's trapped. It would converge to one soliton, and then since the gradient is like this parameter, it gives us directly the gradient of t at this point.
44:05
Here I forgot something important. This is done only at non-characteristic points, because our trapping result works only at non-characteristic points. So here I'm outside the bisectrices. Okay, if I am non-characteristic,
44:20
then I know the gradient, and I have convergence. But then, at some point later in the proof, I will have to show that all points outside the bisectrices are indeed non-characteristic. And this is difficult. For the moment, I don't have it, okay? And this will be an important step.
44:43
Note once again the link between the asymptotic behavior of wx0 and the regularity. You see it here. We converge to something where we see the gradient. Okay, now case two, if we are on the bisectrices, this situation, the solution is anti-symmetric.
45:01
So you cannot have only one soliton. Always, all that you can do is have only two solitons. So you will have two solitons which will decrease to zero, and you are left with two solitons like that, one along x1 and the other along x2, and the parameter is almost the same, in fact.
45:22
Okay, and here, because we have a non-characteristic point, and this comes from the behavior of the neighbors, you are on a bisectrices, but you have neighbors which are not on the bisectrices,
45:40
and there you know the gradient. So from that, you can have the gradient, or at least directional derivatives, outside the direction of the bisectrices, and see that you are non-characteristic. Then, because we have a Lyapunov function in similarity variables, we will converge to a solution which will be close to this guy.
46:03
So we have a new kind of stationary solution, which are neither radial nor 1D. Okay, a new kind of stationary solution. Now, we are left with only one thing,
46:21
to show that outside the bisectrices, all the points are non-characteristic. Okay, so in fact, this is what I wanted to call the umbrella technique. Okay, maybe it's suitable for today, because it's going to rain.
46:41
Okay, but let me tell you what I call the umbrella. Look, here, in the domain of definition, in 1D, but in multi-D, you just take the cone. Okay, any blue cone with slope one, okay, a light cone,
47:02
is completely included in the domain of definition, with vertex x, t of x here. But then, if you take an umbrella which is green, and you start from the bottom, imagine your green umbrella is going up, okay, and then it touches your blow-up graph at some point,
47:25
then that point is for sure a non-characteristic point. Okay, take your umbrella and touch in every place, the first time when your umbrella or your cone,
47:41
non-characteristic with a slope which is strictly less than one, touches your graph, that point is by definition a non-characterist. You see it, okay? That's not more complicated than that. So here, this is what I will do. Sorry, I will get back to the same place.
48:04
Okay, take x outside the bisectresses. For example, here, x2 is strictly less than x1, and we'll show that x is non-characteristic. We take gamma, which is rather small, between zero and x1 squared,
48:20
smaller than the log, the log correction, because our slopes are all negative one or one, with a log correction. So x1 squared is much faster than one over log of x1. Okay, and we consider a family of cones, with vertex centered in x and height t.
48:43
t is a parameter, and the slope is always one minus gamma. So this is an umbrella. It's coming from the bottom, the green umbrella, and going up at some point, for some value of t, it will touch the graph.
49:00
At some point, x bar, t of x bar. Immediately, x bar is non-characteristic, because of this slope, which is strictly less than one. If x bar is equal to x, because maybe the umbrella will touch the graph at its top, if this happens, then we are done.
49:22
x is non-characteristic. If it does not happen, if it touches elsewhere, we will try to find a contradiction. Okay, so let's see. Imagine that the touching point is not the center of all these cones.
49:41
If x bar is only bisectresses, well, this is a little bit complicated. I'm sorry, I don't want to mention that at all. Okay, maybe later, if you are interested or in the paper. Okay, but now when x bar is not on the bisectresses,
50:00
the argument is in fact easy. Let's see. Imagine that x bar is also by symmetry here, so it has positive coordinates. Okay, so in this place, both the cone in x bar and the graph are differentiable.
50:22
The cone is differentiable because we are not at the center, but the graph is differentiable because we have a non-characteristic point. x bar by construction is non-characteristic. Imagine two surfaces which touch each other at some point, so they should share the same tangent plane.
50:46
Okay, so their gradient has to be the same. Okay, their slopes have to agree. Okay, so the slope of the cone is this guy, which is less than this one, and the slope of the graph is this guy. And you see, log of x1 bar is less than x1 squared,
51:05
because the parameter comes from the x, the point where I'm starting, and the slope comes from the touching point x bar. So all this means that x1 bar is negligible, with respect to x1.
51:21
Okay, this is on one hand. On the other hand, I have a series of three inequalities, which are easy to understand, hopefully. T of x is more than T of x bar, because the cone is under the graph, in fact.
51:47
Okay, T of x is more than the top of the cone, and the top of the cone is more than T of x bar. This is one thing. Second thing, T is one Lipschitz, so T x bar is more than T of zero minus x bar.
52:03
But since we are here, it's more than x1 bar square root of two. Then, because wx is bounded, thanks to our work in 2003-2005, we have the following upper bound,
52:22
which is not sharp. T of x is, in fact, always less than T of zero minus x1 over two. And now let's put all these three inequalities all together. If you put them, this is more than this, more than this, more than this. So you find that x1, okay,
52:43
is less than two x1 bar. But from the previous slide, x1 bar was negligible with respect to x1. A contradiction. Okay. Now, all points outside the bisectrices are non-characteristic,
53:00
and we have the gradient, which is the slope of the pyramid, negative one in this region, with a logarithmic correction. Then, when we integrate this estimate between zero and x, we will find that Tx minus T0 is like negative x1
53:21
with a logarithmic correction. And of course, by symmetry, we find all the pyramid shape like that. Okay. Thank you for your attention.
53:41
So if you take the bisectrix, and you get a Lorentz boost so it's horizontal, in what way does it differ from the 1D blow-up shape behavior? Sorry, can you say it again? It takes the bisectrix, and Lorentz boosts it so it's horizontal.
54:03
Ah, yes, okay. Then how does that deviate from the 1D behavior, which blows up also along a line? Yes, it's a different stationary solution. Because on the bisectrix, we don't have kappa of D,
54:22
we don't have the usual solitons. We have convergence to a new stationary solution. So even if you change with Lorentz transform, you will find, yes, this is good. Yes, we can make that like you are saying. We'll find a new type of non-characteristic point where the profile is not the soliton in 1D.
54:44
Very good remark. Yes, a new profile. And how do I see the deviate? I mean, what does it look like that makes it look different? Let me show you. It's here. It's here. You see? And Frank can do a picture, of course.
55:01
So if you see it like that, then it will be with a different scope. One of the delta, one minus delta of the delta. But it will be wider. Wider than the characteristic, I mean,
55:21
in 1D you either have slope which is rigorously one, or we have something which is differentiable. And here we don't have something which is differentiable, but two slopes which are far from one.