 Okay, there was, before I move on, I want to mention one further thing, which I should have mentioned the first time, which is that there's another important conservation law for the energy critical wave equation and that's the momentum. So this is another conservation law that the nonlinear wave equation in the energy critical case has. Notice that it makes sense because the gradient is in L2 and the derivative is in L2. Okay, and this is a constant in time and somehow this is fundamental in the nonradial case. In radial situations this doesn't play a role because it's zero. Right? If you take two radial functions, u0 and u1, this integral is zero. Okay, you don't need to know any equation. You can, maybe I leave a small exercise to check that. Okay, so I just wanted to mention this because it's an important fact in nonradial situations, even though the first thing I'm going to do today is completely radial, but anyway it comes with preliminaries. So basically what we regard is the first lecture was preliminaries and today we actually start. Maybe I erase to be with a clean slate. If you think about the statement that I made as to what soliton resolution means and remember that we know what the traveling waves are already for the nonlinear wave equation. They're precisely the Lorentz transformations of solutions of the elliptic equation and soliton resolution means that as you approach the final time of existence your solution can be decomposed into a finite sum of modulated traveling waves plus a linear solution plus a term that goes to zero. Okay, now if you think about this statement for a minute you see that if it held it would mean that the energy norm of the solution would have to remain bounded up to the final time. And so because of that we will consider here only bounded solutions of the energy critical wave equation. That is to say, so what are bounded solutions? Suppose that T plus is finite then and recall this type 2 blow up solutions. Okay, so the two statements are the same, but I'm just dividing it into the two cases. Okay, so we will now stick to this kind of solutions. And one thing that is important in the case when T plus is finite is something that we call the singular set. So the singular set is defined as follows. We will first define the regular set and the singular set will be its complement. Okay, so we say that a point x0 is regular if there's no uniform concentration at x0. So that means that for all epsilon positive there exists an r positive such that for all T up to T plus, so T plus is finite. Okay, the integral from x minus x0 less than r of the gradient xt u of xt squared the x is less than epsilon. Okay, so there's uniform absolute continuity of the gradient squared for all times. And we say that x0 belongs to s or s is just a complement of the critical of the regular points. And it's an important fact that s is never empty and finite. Okay, so if T plus is finite and you remain bounded there must be at least one singular point. If there were no singular points you could go forward more. And because the energy norm is uniformly bounded then that guarantees there's only finitely many singular points. So in particular suppose that u is radial. So suppose that we have a radial solution. And its values are the same on a whole sphere which is an infinite set. So what can be the only singular point? The origin just by counting. So we will now turn to a soliton resolution in the radial case. And I reemphasize that we're working on r3. And here in what I'm going to say this three has a special role. Okay, so soliton resolution in the radial case in r3 was first proved by Leukeyer, myself and Merle in 2012 along a well-chosen sequence of times. And then for any sequence of times. So first let me explain what this statement means. So for a well-chosen sequence of times means that we can find one sequence of times that we choose very carefully along which we can prove that for u in the sequence of times we can decompose it as a sum of modulated solitons plus radiation term plus error that goes to zero. Okay, but we have to select the sequence of times. And the second statement says that no matter what sequence of times we take going to the final time this holds. So clearly the second result is stronger than the first. The second thing I want to mention is what are the solitons in the radial case? So that means that we have to go back to radial solutions of the nonlinear elliptic equation. What are the nonlinear, the radial solutions of the nonlinear elliptic equation? And we saw last time that by work of Giede's, Nierberg and Bohorzhev that's exactly w and minus w and their scalings. So there's only one soliton. No other solution of the elliptic equation is radial. And by taking Lorentz transformations you cannot make something radial that wasn't radial. So this is it. So what is it that happens that produces soliton resolution? So there's a heuristics that says let's say we look at the finite time blow up case that what's happening is that because we're working in an infinite space domain energy is being shifted away from a truncated cone with vertex at the singular point towards infinity. And as you shift a little bit of energy you get to an exact amount of energy like some of solitons and you do this through the radiation. Then you take out some more energy moving to infinity and you pick up a soliton that does that and you continue like that. So that's the intuition that what you need to understand is how energy gets pushed out to spatial infinity as you approach the blow up time. So what we managed to do is find a way to measure this in math instead of in pictures with my hands or physical experiments. So the fundamental new tool which I'm going to explain now is a dynamical characterization of W that has to do with this ejection of energy. So fundamental new fact of course it's not new today it was new when we proved this. Let you be a radial bounded solution of NLW and exists for all positive and negative time. Suppose U is not zero which is not not interested and it's not W or one of its scaling or negative W. Okay then we have the following estimate that I'm going to call this star. So there's always energy for all t bigger than or equal to zero or all t less than or equal to zero. So in one time direction there's always a definite amount of energy outside unless you're W. So the first thing you need to convince yourself remember what W is. The first thing you have to do in your heads is that this doesn't happen for W. So the gradient of W would be 1 over x squared that's infinity and then you're in 3D you get 1 over r to the fourth you integrate r cubed and so anyway r squared I'm sorry so you're integrating 1 over r squared you get a 1 over r that goes to zero at the infinity you can never get something like that because you have the t here okay. So definitely for W this isn't true. What we're saying here is that unless we meet W this is true and this is a way to capture W and this is of course what's capturing the fact that there's energy going out. This is a lower bound on the outside part for t positive or t negative. Okay so let me just say a few words on the proof of this thing. Let me just say a few words about the proof. So our proof relied on a basic fact about radial solutions of the linear wave equation in 3D. Okay so this is a basic fact that could have been proved by D'Alembert. Okay so it's a completely elementary fact. Okay so I'll write what the estimate is. So V a radial solution of the linear wave maybe D I don't know and D equals 3. Okay let me just say put that here then t positive or for all t negative. So the difference between positive and negative time comes from the fact of whether you're considering incoming waves or outgoing waves and that's all that this is. You're splitting your your solutions into incoming or outgoing. If you are one you get one time direction if not you can get the other. Okay so you have this inequality that for all t positive or for all t negative we have a uniform lower bound on this outer part of the energy in terms of the initial one. Okay so let me first make one remark. Here I don't exactly have the initial the initial energy because I have the R inside instead of outside. Why do I do that? Because if I put the R outside this is false. Okay so I just make v1 v0 and v0 equal 1 over r or r bigger than r0 and 1 over r0 for r less than r0. Okay this is a perfectly acceptable choice. What is v? Let me draw a picture of the v. Here is r0. I'll make the cone like that. And here v is 1 over r. Why is that? Because 1 over r is the fundamental solution for the Laplacian. It's the Newtonian potential. So away from the origin it's harmonic. Okay since it's harmonic it will solve the wave equation and by finite speed of propagation it will have to be exactly this in this region. Okay and I don't care what it is inside because my estimate looks at things outside. Okay and now what happens with 1 over r this thing gets to go to 0 when t goes to infinity. If I had the r outside this would be 0. This would not be 0 I'm sorry. But if I have the r inside it is 0. No no no no no no no no no. Not if the r is outside. No no. No this is the exact point that if the r is outside that thing is not 0 if the r is inside it is 0. Because r times 1 over r is 1 and therefore the derivative of 1 is 0. That's why this is the correct formulation. So let me explain how one uses this and to do the proof of soliton resolution. Okay I will do it in the finite time blow up because it's slightly easier to draw the pictures. So the proof is just the proof. The proof of this is yes it's one line basically. It's just one line you say let me do f of rt which is rv of rt which solves the 1d nonlinear linear wave equation and you split the 1d linear wave equation as a composition of two transport equations and there you have it. Okay that's why as I was saying this is completely elementary. I mean you can I can give this as an exercise. No that's true I can give it as an exercise. From what I've said if you have the patience you don't need to know anything else. Okay that's why you know it's completely elementary. Okay all right. Okay so let me show you how you use this to prove the soliton resolution in the radial case in the finite blow up. Let me assume that t plus is 1 and I have u and I need I want to do the soliton resolution for you. Okay let me take a sequence t and tending to 1. Then the first observation is that this has a weak limit. I'm used to calling it v zero so I stick to that. It has a weak limit. Of course along any sequence it has a weak limit just by boundedness. On the other hand one can prove that the weak limit is unique. It doesn't depend on the sequence. Okay the next thing that you do is you call v be the solution of n l w with data v zero v one at t plus equal one. So that certainly exists and it's a very nice solution up to t plus equal to one for t near one. I mean this is small data theory if you wish. Okay now what happens with is that the support of u of t minus v of t is contained in the ball around zero of radius one minus t. This is where u minus v lives. And you can see that if this wasn't the case using finite speed of propagation you could prove that zero is not in the singular set. Okay so this gets completely localized. So everything that is happening here is inside this inverted light cone. Outside that inverted light cone my solution equals a regular solution so I don't have to do anything to it. And this will be the radiation term, this v. And because this is a regular solution v of t is like v l of t for t near one. That's from the small data theory. The time is very close to one, the nonlinear solution and the linear solution are very close together. So that's why this is the radiation term. Okay so what am I going to do next? What I do next is split u of t n minus v of t n into blocks. Okay so these are nonlinear blocks. What are these? In the technical language they are the nonlinear profiles and the linear profile decomposition of this difference along this sequence of times. Okay we just think of them as blocks. Blocks of energy. Okay now we're going to understand each one of these blocks of energy. Let's take one of these blocks of energy and assume it is not w lambda. Okay what we really want to show is that they're all w or minus w. Right scaled because that's what the Soliton resolution says. So suppose one of them is not. Okay then it will have this kind of dynamic character. It will have this kind of outer energy property. There's some energy outside and so if that property holds for all positive time there'll be a chunk of energy here right from this inequality. But there can't be any energy there because the support is in here. So that's a contradiction. So that had to be w. Now if the thing happens for negative time let me use another color. I'm going to use that going backwards like that and then there'll have to be energy here. A fixed amount though. It would have to be a fixed amount of energy there. And as I approach here this gets smaller and smaller and this is a fixed function. So it can't have a fixed amount of energy in a set of measure going to zero. So that's also a contradiction. So either way you reach a contradiction and so that block had to be w. Now we've killed all the all the profiles. All the profiles are now w. Now the error remember only went to zero in the dispersive sense. We now want it to go to zero in the energy sense okay. But then you run the same argument on the error because everything here but now you use this property. And the constants are uniform even though I have a sequence. And either it happens going forward or backward but in each case I reached a contradiction. So the energy has to go to zero okay. And so that's the proof. Okay there's some cheating involved. Obviously I haven't given a complete proof but this is the idea. All right. Now it yes. This is the proof of an every sequence of times you never say. No right. This is this is here. I will now explain the other one okay. It's weaker but it's it's weaker but more flexible in some way okay. So that's why I want to. Do you have another question? Yeah. So the proof that you give in the free case is just an heuristic argument to show that to convince us or you need this. You need this to do that. The point is that you can use the linear one in the nonlinear case because of this freedom in the R. Suppose that your solution didn't have compact support. So that there's always energy far out. Then I pick my R very very far out so that the part outside at time zero has small norm. And then by the finite speed I can reduce to that case and if it has small norm I can use a the fact that the linear solution and the nonlinear solution are close to each other. Now if it has compact support what I do I go to the very edge of the support and that's how I pick my R okay. So this is flexible enough that by always going out to outside of cones I can manage to pass from the linear to the nonlinear case without assuming smallness. So this is the the point of this idea okay. We have another question. So you said the D doesn't depend on the sequence. And the result depends on the sequences or the solids don't depend on the sequences? No the solids are just W in the radial case. There's nothing else. But that can be a few ones. Yes the number gets fixed by the energy. By energy okay. But the fact that the V does not depend on the sequences is just some extra information okay. Other questions? Please ask because this is a good time to ask. I mean this is really the idea of the proof I mean I haven't cheated. Okay. So let me explain this other proof. Oh but before I do that okay. I have to explain a few other things. First you saw that I was very careful here to say that this is D equal to 3. Okay. So the first thing is that for any even dimension this fails. Its corresponding statement is false. It's not that we don't know how to prove it. We know how to find the counter example. Okay. This is false for radial solutions in every even dimension. Okay first thing. Even with r0 equals 0. Even in that case. Okay first thing. Second thing I'd like to say is that in the non-radial case even with V equals 3 and r0 is positive such an inequality is also false. So you can see by going to higher order spherical harmonics. Okay. So there's not that much wiggle room here. For the radial case in higher odd dimensions there's a variant of this inequality. That's true. That was proved by Lori, Liu, Schlag and myself. You can ask Baoping who's here. Baoping Liu who's sitting here about the proof of that. Still the constant is one half. The expression in here becomes much more complicated as the dimension increases. So you can interpret this as being the difference between the initial data and the orthogonal complement in the Hilbert space of the one-dimensional line produced by this. Okay. And then in higher dimensions you take orthogonal projections to the complement of finite dimensional spaces with increasing size depending on the dimension. Okay. And there's some interesting combinatorics and some linear algebra in those proofs. Okay. So now let me go to this proof. I need some help for that. So for this proof the idea was different and I'll explain how we choose the sequence of times somehow. So the first point was to prove that there's no cell similar blow up and this problem is always fundamental to understand cell similar behavior. So for instance in the finite time blow up case that means that the integral for a x bigger than lambda times 1 minus t and t of grad xt u of xt square dx is always tends to 0 with t as t tends to 1 for all lambda less than 1. So in here there's no energy. So all the energy concentrates close to the axis. I mean there's a spike. Pardon me? 1 minus t here. Yeah I'm sorry. Thank you. Okay. So that was the first step. Now how did we prove this? Okay. Our original proof of this fact used double star. Now since then there's been a different proof that uses ideas developed in the theory of wave maps but I will not go to that. Okay. So the second step is that this implies the lack of cell similar blow up implies that in the Cesaro means the DDT derivative goes to zero inside the cone. Okay. So how does one prove from this that one uses virial identities and I need to mention virial identities. This seems to be a good time to do it. They play a role everywhere in this theory. In the radial case we don't use the last virial identity. We just use the two. This one is the one that you use in non-radial situations and you see that it connects with the momentum which is what I started with. Okay. So how do you use these two virial identities? We multiply this one by one-half and we add them and that isolates minus DDT squared plus the error and we do the integration in time and the time integral makes the things go to zero provided that we show we choose this cut-off function to be like that supported in that situation and the errors go to zero because there's no cell similar blow up. Okay. And that's how you prove this. So what does this mean? Somehow on average my solution is time independent because the DDT derivative is going to zero. Since it is time independent has to be resembling W which is the only time independent solution and that's how you prove that each profile has to be W in this case. Where do you pass from a sequence to a subsequence? Let me call this something. This is the classical Tauberian argument that if you have the Cesare means going to zero for a subsequence you're converging to zero. Basically this is the idea. Okay. And that's how you need a subsequence. This is the classical Tauberian kind of argument. So this kind of thing used to be thought in first year analysis. Not anymore. Not even when I was a student was that thought in first year analysis. Okay. So this gives you an idea of how this works. Okay. And let me just say that this idea allowed one to generalize the along a well-chosen sequence of times to all dimensions eventually. Okay. In the radial case. But now we will see how that works even in the non-radial case. So that's the next part of the course. Okay. So here the exact statement could be what? Like in these two theorems. So what are the assumptions? In this let me just go precisely. But I will give a precise statement of a more general result. Okay. But I'll say it in words quickly. Here it says that let's say that let's do the case of infinite time. There's this for any sequence Tn going to infinity. I can find scalings lambda sub n's and sine ij and such that the sum of ijw scaled by lambda j plus a linear solution minus my non-linear solution tends to zero along this sequence of times. And this can be done for any sequence of times. And this statement is the fact that I can choose a sequence of times going to infinity such that that happens. Okay. But there will be a time when the theorem will be stated very precisely. And it's coming very soon. Okay. Before the end of the day today. Okay. And yes. So when the dynamic categorization of W you explain ideal proof. I understand that there is a place the company is on for her. Which step do you use? It's not a problem. So which step do you use to condition the solution is not a problem? To show that there's these channels of energy going out that give me kill each profile. No. I mean how is the linear property to derive the non-linear property? Yes. I cheated a little bit. Yeah. No. There's a part that has to do with W is a solution up to the inside that I can't always cut that I haven't shown. Okay. But it is in the notes that are on the web. Okay. Not only in the paper but in the in those notes. Okay. And the arguments are reminiscent of elliptic theory. Okay. So those arguments resemble elliptic theory. Is there a way to state the result of dk and 13 without referring to sequence? Oh yes. Yeah. Oh yeah. Oh yes. Yeah. Yes. Yes. No. But I'm doing it with a sequence because I'm comparing it to the other to the other result. No. But that's how it's stated. Yes. Okay. Yeah. Other questions? Yeah. So this this theorem is not really a theorem about sequences but our continuous parameters. But anyway, but it's equivalent. No. Because I'm doing it only in the sequence case. I'll just say it. It is in some notes. Yes. It's in the notes on the web. Yeah. Okay. So I'm going to now go to the non-radial case. So a non-radial case. This is work of Gia 15, dkm 16, and djkm 16. So there's three papers comprising this work. Okay. And I'll explain how it goes. So but the first thing I want to say is what are the difficulties in the non-radial case? Let's try to see. First, as I explained earlier, this kind of inequality to star for all solutions of the linear equation is just not available. Okay. This is not that we can prove it. It's just not true. Okay. Second thing, in the case of radial solutions, there's only one possible soliton, right? It's just w. And we know it's formula. It's right over there. It's a very simple object. In the case of the non-radial case, we have a zoo of possible traveling waves. Each solution of a nonlinear elliptic equation gives rise to a whole family of traveling waves by Lorentz transformations. And we don't know who these nonlinear solutions are. They're just some objects that live out there. So a dynamical characterization along these lines is out of the question. So we have to abandon such a plan. And so we're going to do a plan which is more reminiscent of this one, one where we use this. Okay. So that's the approach. And if we're going to do that, then we're going to get a result for a well-chosen sequence of times, not for all sequences of times. And I'll say more about that tomorrow. Okay. So the first thing I want to say is I want to make a few more comments about the radiation term before I begin. In the case when t plus is less than infinity, we showed in this paper, as I mentioned in the other board, that, let me just do it this way, that u of t converges weakly to a v0, v1, which is in h1 cross l2. So there is a weak limit. And if we call v of t, again the solution of the nonlinear wave, then let me write down, so those are the singular points. Okay. And those are the singular points. Then the support of u of t minus v of t is contained in the union of these cones. And that's a similar argument using finite speed of propagation. And the fact that otherwise this would be not regular points, not singular points, but regular points. So the picture is that we have a whole bunch of cones here. But there's only finitely many. Okay. So if we go close enough to t equal to 1, they're all completely separated. Then we can just concentrate on each one singular point at a time. Okay. And that's what we will do. Now in the case of infinite time, the extraction of this radiation term, which is what we call the scattering profile, is much more difficult. So for t plus equals infinity, this is, in this paper that I mentioned, we showed how to extract the radiation. So this is the part of a nonlinear solution that has linear behavior. Okay. When it is scattered, when the solution scatters, it's the whole thing. But when it doesn't scatter, there's obstructions to it being the whole thing. And so what we did is the following. We look at the linear solution on this data and show that this has a weak limit as t goes to infinity. Then we constructed vL of t is simply the linear solution with data, this weak limit. And we proved that for all A and R, so that any finite distance from the light cone, the nonlinear solution behaves like this linear solution vL. So again, all of the action is inside, it's strictly inside the light cone. Otherwise, it's linear behavior. The key idea of the proof is to use very identities, the three identities that I either erased or hid. I hid. Use very real identities to show that there are no blocks close to the light cone. And this is coherent with the fact that the nonlinear objects, which are the traveling waves, travel at speeds strictly less than one. And so if we stick close to speed one, we have to avoid all those traveling waves, so they're only thing that can be linear behavior. And we proved this using this, well, the proof is, I would say, not simple, but the key tools are these three identities. And this one is used because we're in non-radial setting. And I will say no more about this. So now I'm going to show up, I need the transparencies now. So I'll show the theorem now. So I'm afraid that, oops, the theorem is lengthy. So we start with a bounded solution of a nonlinear wave equation. We look at this first defined blow-up case, and we fix one singular point. Then we can find a number of traveling waves, J star, a little radius R star that keeps us away from all the other possible singular points. V0, V1, which is the radiation term, a sequence of times, which tends to T plus scales, which go faster than the self-similar rate to zero. Positions, which are the center of the traveling waves, and which are strictly inside by a factor beta of the inverted light cone at x star. And with this LJ, which gives me the direction of the traveling waves, given by this limit between the position and the time, this is well defined. And traveling waves, that means Lorentz transformations in the direction LJ of solutions QJ to the nonlinear elliptic equation, such that inside this little ball that stays away from all the other singular points. U is the radiation term, the sum of the translated and rescaled solitary waves, plus an error that goes to zero. And moreover, the sum is decoupled in the sense that the parameters are orthogonal in this sense. So this is the precise statement, T plus finite. Now let me go to the T plus infinity case. There's a VL solving the linear wave equation with this property that I explained away from the light cone. The difference goes to zero. Then I have J traveling waves. There might be no traveling wave in case my solutions cutters. If my solutions cutters, I have no traveling wave. Then I have scales, again faster than the self-similar rate, positions strictly inside the light cone, directions of traveling waves LJ, strictly less than one. Travelling waves, such that the solution is the linear solution plus the sum of the rescaled traveling waves plus an error that goes to zero. And again, the traveling waves are decoupled. So this is a precise description of the asymptotics along this sequence of times. And what we're going to do the rest of the time is sketch a proof of this. So I want to make some comments. The T plus being a finite, that case, was proven by G.A. in the 15, but the error term went to zero in the weaker dispersive norm, in his case. Then in this joint paper, using the fact that we could extract the scattering profile, we were able to do this pose for finite and infinite time and show that the errors tend to zero in the energy space. So the plan is now to show this proof. And I have one lecture and 10 minutes to do that. I wonder if you can show the result in the radial case, because I think it's too unclear. Not in this one. I don't think, no. But I have it in another one. Are you telling me people are tired, Frank? Oh, but I did write it always for a sequence. I'm sorry. No, I don't have that in, but I will write it. But maybe we turn off there. So I'll give you the statement in the radial case. So in the radial case, let's say, let me give it in the case T plus is infinite. I mean, I don't want to do it in both cases. This is one. So now U is a solution of NLW. Then there exists a J star. J star could be zero. Functions lambda J of T that have the property IJ, which signs plus or minus one, such that U of T equals VL of T plus sum J equals zero to J star of IJ W, scale by lambda J of T of X plus little o of one and h1 cross L2. And notice one thing that I didn't do in this statement is I didn't assume that U was bounded. Because in fact, our proof shows that if it exists for all time, it has to be bounded in the radial case. Okay? So this is the precise statement in the radial case. So and then you can relate G star to the bound to the energy here. So which one gives what? I mean, which? Well, somehow the number J star and the energy are related. And they are related also to the energy of W. And I don't remember the exact relationship, but somehow the J stars is how many pieces of energy of W you have to put to get to the energy. What is true is that the energy of U has also to be positive, strictly positive. Otherwise, it cannot exist globally. Okay? So it's not an assumption, it's a consequence of existing globally. All right? Yeah. Did you have a question? No? Any other questions? For the neural radial case, if I remember well, only for a specific C. Yes. So far. Hope springs eternal. Okay? Okay, so maybe I don't punish you anymore. And we have five minutes of extra time of rest or questions. Be happy to answer questions. Anything that needs clarifying and be happy to clarify some more. I was wondering, can you say here what is the argument to say that the number of singularities finite? Yes, it has to do with the fact that let me see, this epsilon, you actually don't need it to be all epsilon, but a fixed amount that depends on the local well-posedness theory, suffices. And because of that, if there's infinitely many, if there's infinitely many, you accumulate too much norm and the norm is bounded. Okay? Other questions? Yes? It's interesting to notice that even in the non-radial case, your scattering profile is completely determined, does not depend on any sequence. No, no, no. That's correct. So that's one good feature. Yeah, absolutely. It's one step forward. Yes. So the proof is about, because even this sweet convergence for the whole time, does not seem very... No, it's non-trivial. It's actually a consequence of the whole proof. So what we do is we first start with the sequence and prove the weak limit and then show that this weak limit has to be real. Okay? The J-score, does that have to be finite? Yes. Again, for the energy bound. For the same reason. Maybe I make a comment before the question. And that's also the reason why the traveling waves, these parameters, Lj, have to be strictly less than 1, because as L goes to 1, the energy norm of the Lorentz transformation of an elliptic thing blows up. Okay? So in your statement, you had the ratios between the frequencies, such as the frequencies 10 to 0, is there a heuristic explanation for why this might be the case? What happens is that they are truly separated. They live in different scales. For each scale, there's one. Okay? That's it. It will be a different function. Yeah, you can... Yes? So this is more a philosophical question in general. So this soliton resolution conjecture should, I guess, not only be true for nonlinear waves, but also for nonlinear Schrodinger and other equations. It seems to me that somehow finite speed of propagation really played a huge role. Yes, this is completely correct. Can you somehow comment on how you view what should go on for sure? Give me that and a million dollars, right? No, I think it is too early to speculate about that. I mean, this, I think, are very solid and general arguments in the presence of finite speed of propagation. How to go to infinite speed is another big jump. What's the key argument behind the fact that if you're global, then you're bound to... Oh, the key is that you can't... Well, not exactly. I mean, what it is, is you run through the whole proof, basically. First, you prove that there is a sequence on which you are bounded, because otherwise you must blow up infinite time by an argument of the vine, an old argument. Then you use this sequence, which you are bounded, to do the decomposition into solitons. From that, once you have that, you can prove the bound. So the first sequence that you start is the one on which it is bounded and which you know exists, because otherwise it would blow up. In the dynamical characterization of the ground state, you said that you could use the linear equation by saying that if the support is infinite and you chosen are sufficiently large, and outside, it behaves like the linear equation. But I don't see in the argument where you use H or not W. Well, no, that's the cheating. Yeah, I mean, this is... There are arguments that are, I would say, of iterative elliptic type. Somehow, there is an ODE that W's verifies and it depends on the precise values of solutions of the ODE all the way up to zero, okay? This dynamic characterization of W resembles the, you will kind of... Exactly, because you turn from the feeling and the less you use the fact that you are closed and it will be on the case in this particular thing where you can prove it without the ground state. The fact that we are closed or... To a ground state. It's very close, but the norm was closed to the, the true norm was closed to the... Here, there is no assumption of size. So if you take a solution which you eventually allowed into the support and bubbles, and if you prove the initial data of this bit, would it possible to have all the other cases coming up? Yes. The short answer is yes. Take W, okay? In any neighborhood of W, you can find type one blow-up solutions, type two blow-up solutions, solutions that exist globally and scatter to a linear solution, solutions that exist globally and scatter to W, okay? In any neighborhood of W. So the thing is extremely instinctive. For example, if you do n-bubble, you're suspected that like a mind, one bubble and all the other possibilities. Yeah, yeah, but that's energetic considerations, yeah. Just stop early, like 15 minutes. Okay, thank you.