Bubbling blow - up in critical parabolic problems
Pino, Manuel Del
We construct solutions that blow up in the form of energy invariant, time dependent space scalings of steady states for some problems where such steady states exist.
 Well, I would like to thank Ivan for the introduction, for the invitations, for Frank and the other organizers. So it's great to be here. It's my first time here. So the topic of my talk is actually parabolic equations, so not really connected with wave phenomenon. However, what I will discuss is blow up phenomenon. So that, of course, as it has been deeply studied already, it has a lot of parallels with singularity formation in parabolic problems, except that here the mechanism that triggers it is diffusion. So I will mention next three problems that have rather similar features. Of course, with a lot of technical differences, one with respect to each other, but with a common flavor. I will only have time to speak about one of them. So I prefer to say a little bit more about one of them specifically. So one of the problems, one of the examples, actually, of the type of phenomenon I would like to see, where perhaps it is easier to explain what happens, is say the three problems that I'm going to mention are actually have been studied a lot. They are quite standard in the literature. So one of them is that reaction diffusion equation with this special power, which is the critical solar exponent. Say we can consider, for instance, the problem of finding positive solutions in the domain, maybe with boundary conditions, and define up to some time capital T that could even be infinity. So a second problem, which is the one I will really talk about, is the harmonic map flow in 2s2 from R2 into 2s2. This is the special case, of course, the flow of harmonic maps, the evolution problem for harmonic maps. But in the case of values in 2, it takes a particularly simple form, which is this semi-linear equation, where the non-linearity carries this gradient square term. And some special feature that is actually preserved along the flow is the fact that the absolute value of u is 1. So u really has values in 2s2. So we will consider this problem in a bounded domain, omega, although most of what I will mention is valid also in the space, or certain classes of bounded domains, but doesn't really matter, and up to finite time T. I will consider this problem with a given boundary condition and corresponding initial data. So a third problem of this type is the Kepler-Siegel equation. The Kepler-Siegel equation is an unlocal equation, if you want, or a system. So this is simply the heat equation, the standard heat equation, modified by a drift operator that depends on u and on this v. And what is this v? This v is inverse Laplacian of u, which by definition, for me, will be simply the convolution of u with the fundamental solution. This is a two-dimensional problem, which you can also write this problem in divergence form like this. Okay, what do these problems have in common? One of them, one very important that makes the problems somehow nice, is the fact that they all have Lyapunov functionals. Actually, this is valid not just for the critical power, but for any power you wish, formally. Say this functional defined like this, is a Lyapunov functional. In fact, it is strictly decreasing unless we are at steady states, at this where we are standing at steady states, and more precisely, it's just a computation, formal computation tells us that the derivative in time of this guy is going to be minus integral of u t square, so it's negative. So similar feature we have for the harmonium flow, harmonium flow by definition actually, is the two-gradient flow or the two-negative gradient flow for the Dirichlet integral. For here, I insist u has values into s2. And again, we have the same characteristic for e2. So e2 is strictly decreasing along solutions, along smooth solutions of it, non-trivial solutions, non-static solutions. And well, perhaps a little bit more complicated, but still similar, is the Kepler-Siegel system, the Kepler-Siegel equation, the Kepler-Siegel non-local equation which can be written in this form as I mentioned, and it turns out that the Lyapunov functional is given by this quantity, which is a little bit different. Now the derivative of this guy along solutions, along smooth solutions, is negative, but now the right-hand side is given by this. Okay, so this is one feature, yeah? This is one feature, the fact that they have this structure, this variational structure. Yeah, but another one that they share that is very special for these problems is the following. It's the presence of a special family of steady states, as a matter of fact, a continuum of steady states, but a continuum of steady states depending on scaling parameter. The three problems have special scaling variances, and that asymptotically becomes singular. Say, what do I mean? I mean, let us consider this, which is a special solution of this entire space, but in fact all solutions of this can be written as a translation and a scaling of this function, so they are all given by this formula, this is known, the positive solutions, I mean. And a characteristic they have is that they all have the same energy, the same energy that I wrote before, so it's actually constant along this scaling. So we have energy invariant family, an energy invariant continuum of steady states, which as the parameter takes to the limit to lambda going to zero, say this function becomes singular. Of course, away from the origin this goes to zero, but exactly at the origin, this is one over lambda raised to the power, it goes to infinity, so it has spike form, so like that. Okay, so what is the corresponding object that I will consider, because it's not the only one, unlike the critical exponent equation, say an object that seems to play a similar role is the following. This is a harmonic map, this function is a harmonic map, a harmonic map is by definition steady state of the harmonic map flow, and this function indeed solves this elliptic equation with absolute value one. We call it the one-corrotation or harmonic map or the standard one-corrotation or harmonic map. Again, we have a same type of translation invariant, we also have a dilation invariant, if you simply scale the argument of it, the energy is preserved, the energy meaning the Dirichlet integral this time, no lambdas in front of course, because you have values in each one, but there is an extra, there is an extra invariance that is this, say if you rotate, u is a vector, so it's an element in S2, if you rotate it by an orthogonal transformation of R3, then you still get a solution of the equation and they all have the same energy once more. As lambda goes to zero, this of course does not blow up itself, it has absolute value one, but what blows up here is its gradient, the gradient of u is actually going to infinity. Now what is the corresponding object in the Keller-Siegel system? In the Keller-Siegel system, say in the Keller-Siegel equation in R2, well, an actual object to refer here is this function and this function has again the characteristic that if you scale it now this way, you still get a solution and this scaling preserves again the energy, this funny energy, as I wrote at the beginning. They have an additional feature that they all have mass 8 pi. So what is the topic of this talk? The topic of this talk or actually the issue that I would like to analyze for the three problems, but in reality we will refer only to the second one now, is the construction of solutions that have blow up phenomenon, the exhibit blow up phenomenon as the approaches finite time or even infinite time. And what is the special pattern of blow up that these equations exhibit or that is already more or less known that they blow up a rise, that there is blow up arising in this way, is precisely like this, say, is precisely in the form of blow up around the finite number of points, maybe one or maybe more, where the profile of blow up near the, near each of the blow up points is scaling asymptotically singular or resembles the scaling and asymptotically singular scaling of a steady state of one of these energy preserving steady states. This is not so crazy because since the energy is preserved, say, the energy of an object like I am picturing here actually gets stabilized, because the scaling doesn't change the energy. So this is what is special in this problem, in these problems. That's feature sometimes called criticality as a matter of fact in this setting. Okay, so now I would like to refer to the harmonic map flow from R2 into S2. Then I consider this problem which from now on I will call HMF. So I consider this parabolic equation up to some given finite time capital T in a bounded domain or in entire R2, but if omega is not the whole space, I consider boundary conditions and an initial condition. So of course both the initial and the boundary condition have to go into S2. If that's the case, that property of having absolute value one is indeed preserving along the flow. Okay, so there is a lot of literature around this problem. In fact, I would like to make a sort of review of known resource. A lot of people have worked on this since long ago. Well, the harmonic map flow was introduced not just in the setting, the two dimensional setting, but as say as the L2 gradient flow or L2 negative gradient flow for the Dirichlet integral for maps between two more general remaining manifolds. And it was introduced as a way to build harmonic maps in reality. So these flows was introduced by Elson and Samson many years ago, 50 years ago. And concerning for instance the problem of existence, local existence in time of classical solutions, this was first seen by Elson and Samson themselves, then Struve, and then for the case of compact manifolds and for the case of the boundary value problem by Casey Chanck, more or less in 85. Okay, so I repeat what I said before. Say the Dirichlet integral is actually strictly decreasing along non-trivial trajectory, non-steady trajectories, non-stationary trajectories of the flow, and we have this identity. So the energy is decreasing. It's decreasing along the flow, along the flow as long as the solution remains smooth. But something that has become, that became known is that in general we don't expect global existence, global existence of a smooth solution. That's typically not the case, not typically, but it may not be the case. Say that the solutions remain smooth at all times, while this didn't really become known up to much later. However, Struve in 84, say this is an important result of the theory, actually, proved the following, proved that even though the solution may develop singularities, those singularities can more or less be understood. And it can be understood enough in the case, so that the solution can actually be continued in a natural way. In fact, it can be continued in a way, continued as a weak solution. It can be continued as an H1 weak solution, where the energy has discontinuities. The energy still decreases, but it has jumps. This is what happens with the definition of Struve. Well, as a matter of fact, it turns out that only at a finite number of points in space-time, so only up to only for finite number of times, and for each of those times at a finite number of points, we can have low up. And at those points, at those times, jumps down in the energy take place. Say, there is this resolve by Freyr in 2002. He proved that for a certain class of solutions where energy is decreasing even if it has jumps, this solution is unique. So that means that the definition of Struve is, in a way, a good definition. Okay. More recently, I don't remember exactly the reference, but very recently, a student of Struve proved that if the jumps, say, the solution may not have decreasing jumps. Say, even if you have a weak solution with jumps in the energy that could even go a little bit up, but not too much, the solution is still unique. What I mean is that the jumps, as I will try to argue a little bit later, the jumps in the energy are quantized. In fact, in all examples known, this jump cannot be less than 4 pi. And I think this is a general fact indeed. So this is Struve's solution, the energy in Struve's solution, when you reach a singularity. Okay. So as I mentioned, the, say, U itself does not blow up, has absolute value 1, but the gradient blows up when we have blow up. So in fact, if I call capital T the first time at which regularity is lost, which is strictly positive, as we know, we must have that the gradient goes to infinity as T goes from below to the blow up to the time capital T. Well, I will mention, although maybe the statement is not completely precise, I will make, I will state the property that is a summary of results by several people who work on this in the 80s and 90s, starting with Struve and several other authors, as I mentioned here. So the question is, how does the solution looks like when it blows up? And it turns out that quite a bit of it can be set. The solution actually does blow up, if it does, if it blows up, it blows up in the form that I mentioned before, namely as scaling around a finite number of points of steady states, steady states, steady states Ui, which are not necessarily the U that I wrote before. So more precisely, so the result, the statement is a little bit weaker, but still quite a strong assumption. If we approach capital T along a sequence, then I can extract a subsequence such that for a finite number of points Q1 up to Qk, the following holds. U along that sequence resembles an H1 function plus the sum of these bubbles. This is the subtraction of the value of infinity of the bubble, which makes sense. And say where this guy, say this is just a normalization, where this actually goes to zero, this difference, say pointwise, pointwise away from the points Qi, Qin is approaching each of the Qis, and this lambda In goes to zero. In fact, a little bit of important information can be obtained about the relation between the scaling parameters and the mutual distances of the points, which is this fact. This number is here, for I different from J, go to infinity. What does this mean? What happens is that if the points, for instance, Qi are all well separated, one from each other, say this is automatically satisfied, doesn't matter what this is. This goes to infinity because this goes to zero. But it could be the case at all points Qi or finding the number of them coincide. What this says is that if they coincide, if they coincide, for instance, then this goes to infinity, which means that the rates are not comparable. One of the rates is much smaller than the other. So if we scale around the point, that means that you see just a single object. Say, if you use lambda i or lambda gegeo, we'll just see one object. So this is what that property means. So the UIs, as I mentioned, are harmonic maps. So harmonic maps with finite energy in entire ones. So what I mean is the UIs add entire solutions of this equation with values in S2 and with finite energy, which is natural. I mean, we are starting with finite energy configuration. So in the limit, we should see something with finite energy. This is also a standard fact. Is that U is a function that goes from R2 into S2. But I could regard it after stereographic projection as a function from R2 into R2 or from the complex numbers into the complex numbers or with the composition in the other way from S2 into, say, I can regard it as a function from complex to complex or from S2 into S2. So this is known. This is a known fact. These maps are, say, conformal and in S2. And that makes, well, in fact, when it is lifted to S2, it becomes a smooth function. So that the, in particular, the value U at infinity, say infinity is one of the poles in S2, is well defined. Some fact that is known is that using correspondence in the complex numbers with a rational function. And in fact, these also holds. So if we compute, again by conformality in S2, say the integral of gradient U squared turns out to be a multiple of 4 pi. Say 4 pi is the area of S1, of S2. So it's an integer times the area of S2. And this fact holds for some M in the national numbers. So in particular, the least energy, non-trivial, harmonic maps, non-constant harmonic maps have energy 4 pi. So this is the reason of the 4 pi I mentioned before. So in particular, say what are we going to have? We are going to have that this U along Tn goes weekly to U star in H1. And the energy, say the defect with respect to the gradient of U star squared, are precisely asymptotic for the Dirichlet measure. It's given by this asymptotically. Simply integer multiples of 4 pi times Dirichlet delta set, the corresponding lower points. So this is a way to visualize quickly what the statement says. OK. So of course, there are many questions around it. This is a very interesting phenomenon. By the time of Struves results or many others that I mentioned, it was not really much known about existence, for instance, of such patterns or further properties of it. Stopping in 2004 proved this very interesting fact. Proved that the rates at which the lambda is, remember, are these blow-up rates around each of the bubbles in this decomposition. This is what I mean. So what Topping proved is, say, along subsequences, but I mean, it doesn't matter, say he proved that these blow-up rates have to be of this form. So they go to 0 faster than T minus T to power 1 half. In fact, the estimate he finds, I think, is a little bit better, say, with the log in the denominator. But what this says in the language of these type of geometric problems, or actually of many singularity formation phenomena, what this says is that the singularities developed must necessarily be of type 2. Type 2 means a singularity that cannot be captured with a subsimilar form. This is more or less what it means. And the right rate for that in the harmonium flow is this 1 half. So lambda i actually goes to 0 faster than that. This is also valid in more general targets, not just in S2. In fact, Topping finds examples that show that this estimate is, in fact, optimal or approximately optimal. There are examples of manifolds for which you can get blow-up rates as T minus T to 1 half plus delta for any arbitrarily small delta. So let me consider now the function U that I am calling from now 1 U0. This standard harmonic map, this standard harmonic map, actually when you is just lifting, say, this is a function from R2 into S2. But if you take it from R2 into R2, this is the identity. So this just comes from a stereographic projection. So these guys have exactly energy given by 4 pi, degree 1 if you want. And the valued infinity of them are, of course, as you quickly see, is 0, 0, 1. So let me now mention what is it known concerning existence of blow-up patterns. In fact, there are very few examples known. And in fact, the examples known concern single blow-up points where the profile is like this. But more than that, the examples concern radially symmetric, actually more precisely, radially symmetric settings of which this is the only steady state. Let me be more precise. One rotational solution of the harmonic map flow is a solution that has this form. This is, of course, the spherical coordinate is a way to represent a vector, say, in S2, a point in S2. And I assume that this angular variable is always e to the i theta. And polar coordinates, where R and theta are the polar coordinates. And these functions are radial, radially symmetric. So what is very nice is that with these ansatz, the harmonic map flow reduces to scalar equation. And this scalar parabolic equation is actually very simple. It's vt equal to d Laplacian modified by this guy, which is sort of, by the way, an encant type non-miniarity. But I will not really elaborate on all that. What something that can be observed is that this guy has the following steady state, the only monotone or finite energy or whatever steady state is given by decreasing, say. It's given by this function. This is explicit solution of stationary solution of the equation. And if we evaluate it with this v, say, we obtain exactly the u0 that I mentioned before, the standard power. Okay, so concerning the literature, in fact it had been, as I saw in the, in the paper, it had been actually previously conjectured that it may be the case that no blow up exists at all, but there's no blow up in the, in the way that it is to the harmonic map flow. But Chang, Ding, and Ye in 1991 found a very nice example within the one-corrotational class. In fact, what they are able to find is a solution of the scalar equation v, which looks like the steady state w, scale this way. Scale by some small number lambda of t or equivalently, say, for the actual original harmonic map flow, the solution you associated looks like the u0 at x over lambda of t. Yeah. Where this lambda of t goes to 0 as t approaches capital T. So what they do is they, is a very, is a very, very nice arguments. The, they capture the blow up using barriers. So they prove that there is blow up using some special barriers and those barriers are very sensitive to what the boundary conditions are. Yeah. So it is necessary to take the boundary conditions in certain range in order to, to observe the phenomenon. They take initial conditions 0 and boundary, and boundary values, say bigger than pi or less than pi, say, and they observe this phenomenon. Well, for the one-corrotational solution, this was proven many years later, in 2009, Angen and Hulshof and Matano found a much stronger estimate for this setting, for this special solutions, for this special harmonic map flow into S2 and one-corrotational. The rate turns out to be much smaller than t minus t to one-half. It turns out to be little order of capital T minus little t. Yeah, this is what they proved in 2009. Actually, before them, Van der Wer Hulshof and King in 2003 made a very, very, wrote a beautiful paper, formal paper, in which they describe a lot of phenomena around the one-corrotational blow up. They blow up in the one-corrotational case. And when they discovered, formally, is that the blow up rate, oh, here is a mistake, important. Here is squared. There is a two there, not the one. Yeah. It's very different life with the two than with the one in these rates, by the way. So what they discovered is that lambda of t is of this order. So it's not just little order of t minus t, but actually modified with the log squared in the denominator. So the result of Angen and Matano and Hulshof is weaker than this statement, but of course, it is a rigorous result. And a result that is much more recent, in fact, I think the year is 2013, is due to Raphael and Schweyer. They prove a very important fact. They succeeded to make a construction of an actual solution, one-corrotational solution, that blows up exactly with this rate. Yeah. So the construction is very explicit. It's not based on barriers. It is actually machinery closer to dispersive, dispersive equations to analysis of blow up in dispersive problems. And they found a very, very, very nice description of their solution with the right blow up. And they were able also to establish this very interesting fact. They proved that this rate, say, in the language, in the formal language of these authors, they say that they find the, quote, generic rate. So they say the generic rate should be this with the two. So what Raphael and Schweyer proved sort of that, but, well, not exactly. What they proved is that for the solution they constructed, if they take small perturbations of the initial condition, of the initial condition for that special solution, then they still see the same phenomenon. I mean, they see still blow up with the same rate, of course, at a slightly different time. Yeah. This is, I insist, in the one-corrotational radial asymmetric class. So these are the examples now, yeah, for the flow from R2 into S2, as far as we know. Now, there are, of course, a natural question now is to investigate what happens in the non-radial situation, or in a situation where there are boundary conditions. In the case of the result I just mentioned, the setting was entire space. So the question is, natural question is, well, are there solutions that are in non-radial asymmetric settings? For instance, is it possible to have more elements like in Struve's decomposition, the bubbling resolution, you know, like several bubbles in different places? Can you have towers of bubbles or bubble trees? I mean, et cetera, there are many questions around. Yeah. And, of course, the question of their stability, because it might perfectly be the case that the solution is stable within the radial asymmetric class, but when you make perturbations which get off the radial asymmetric class, perhaps the stability is destroyed. And our result, just as a summary, is the following. Say, we prove that if we take any finite number of points of omega, in fact, I should say the following. Yeah. It's roughly this. You fix k points arbitrarily. Then you can cook up initial conditions and boundary conditions, boundary conditions independent of time, so that a blow-up solution with exactly those k given points exists. This is essentially the result. And the profile is like a financial error scaling or shankding and yeah, is a scaling of the U0 with rates that can be computed explicitly and they coincide with the one formally derived and rigorously found by later. Yeah. And, you know, and this is also a kind of result that supports the quote's generosity statement. Okay. At least the bubbling at one point, at exactly one point is stable. Yeah. This, in fact, very connected with some work by Frank Merlin and Hatem Zag in the subcritical heat flow where solutions with finite number of points were found, arbitrary points in the domain. So, so it, the result shares the flavor. Okay. So, very quickly, let me, let me say how, yes? When you had the initial cartoon of the multiple points, it looked like there was a hole in the domain. Oh, yeah, yeah, yeah. No, no, but there's no, no, there's no, no, no need of the hole. No, no. In fact, if you are looking for this type of phenomenon in elliptic problems that where the bubbling is triggered by parameters, many times you need topology. So, I actually borrowed the picture from an elliptic setting. But many times you need, you need some topological assumption to get multiple blow-up. But here we don't need anything. A way, a way, a way to put it is that here we, we put all the blame in the initial condition. Say, all topology needed, say, is in the initial condition, which I don't say what it is. If you were to question is whether there's any numerics that shows this? Actually, the only numerical result that we are aware of, in fact, provided evidence opposite to one of our claims that was the, that was the stability. In fact, there is a paper, there is a paper by Vandenberg, I think, and Williams that I learned from Peter, actually, that conjecture that numerically that if you destroy, if you destroy the symmetry then the solution may blow up. Sorry, the blow-up may be destroyed, exactly the opposite. But say, okay, so, as I mentioned, the energies of all these guys is 4 pi. Let me consider not just arbitrary rotations, arbitrary orthogonal transformations. I will consider special ones that correspond simply to rotate this vector around the Y3 axis, yeah, which means if you write it in complex form, simply multiply the first two components written in complex as a complex number by e to the i alpha. Okay, so this is our more precise statement, but we are able to prove it is the following. Given k points, you know, the distinct points, there exists initial amount, there is, there exists initial amount of conditions such that the solution of harmonic flow for those data blows up at exactly those k points. And the profile at those k points is given by say, I mean, by a convergence to zero like this in H1, an uniform sense, H1 and uniform, where the bubbling takes the form of simply suitable rotation around the axis times the corresponding scalings around QI, where, now I put correctly the two, where the blow up parameters are like this. So we have in particular this energy formula. The uniform convergence, by the way, is a property in the bubbling composition. Let me mention something very quickly. When the time goes to infinity, so if you go actually along a palest male sequence, say, for the flow, yeah, for the, if you go along the solution, but for time going to infinity, in fact, all the properties are known. For instance, this convergence to zero uniformly is sometimes called non-neck property, yeah? So it is not known if this goes in general in the finite blow up thing. Well, of course, when time goes to infinity, this guy sees the harmonic map. There is a paper by Peter Topik where it is proven that if towers are formed at infinity, then all the elements of the tower except the first one must be either holomorphic or conjugates of holomorphic. At the same time, not holomorphic, I think that's, that's, is that what you proved? No. Okay. So anyway, so, Raphael and Schwerer, as I mentioned, proved the stability within the one correlational class. And here I mentioned this numerical work by van der Weyden Williams. So, and actually, let me mention the stability, the stability statement is a little bit more precise. It's a little bit more precise. Say, well, for the single blow up, the single blow up is stable in this sense. If you perturb the initial data, then you observe exactly the same phenomenon where, of course, this number, alpha i star, this cap i star on the time capital T and the points Q i are slightly modified. Yeah, as it is natural. And if you consider the blow up point at exactly k points, that phenomenon is not stable, which is, I mean, no wonder. I mean, the blow up points, the blow up times may not coincide. This is what it means. But if you insist that the blow up takes place in exactly those k points, you can choose the initial data in a co-dimension k minus one manifold around the initial condition. Yeah. Okay. So let me mention another issue that we found quite interesting, that is also addressed in the formal paper by Vanderberg and Hulshof and King. And in rigorous work by Peter, which is continuation after blow up by a mechanism called reverse bubbling. I will say very quickly what that is. Say, what happens is that the definition of Struve has disamplified some features of the energy jumps. Okay. But the energy jumps means, say, a change in the homotopy class. I mean, actually, it turns out that that means a change in the homotopy class of the image of you. Yeah. Change of degree, if you want. Yeah. As you, not just a change of degree, a change of homotopy class. So the question is, are there other type of continuations after blow up that are kind of nicer in the sense that they possibly preserve some of the topology of the manifold, of the original manifold. And a topping proposes a way to do this and prove that there is a continuation with this feature in the one rotational case. He proved that Chang, Ding and J solution can be continued with a profile very similar. However, with one difference, it's not the one rotation standard bubble what you see after the blow up time. But after the blow up time, you see the same thing, but in reverse orientation, the conjugate if you want. Yeah. And that is this map. So asymptotically it's the same. So this is zero one, like the other. But around the origin, it winds in the opposite direction. So this type of procedure is what is called reverse bubbling. So in the reverse bubbling has in the energy a feature that is different from Strube. In fact, it's nicer in a way because the energy in fact continues continuously except exactly at time T when it goes down, goes down and up. Remember that decreasing implies unique. But if you leave the decreasing but you allow that there is a fixable discontinuity of the energy, then this solution does it. What we proved is that this continuation is possible for the solutions we built. In the solutions we built can be continued with exactly the same profile except that after when you go from above the blow up time, say the solution is defined up to T plus some delta. And as you go down, you see this blow up. So concerning the construction, the something that may be reasonable to think is that this type of profiles that we built may actually be quite general. Say, may even be generic, really, say in the way blow up takes place. Why do we say this? It's not obvious at all. It's not obvious at all. But the question is, is it possible to have bubbles other than those induced by use zero or its conjugate? Because there are other harmonic maps. There are k-corotational harmonic maps with higher degree. So is it possible that there are profiles like that? And it turns out that in the k-corotational class, which also leads to some similar scalar equation, it turns out that for high, in the high-corotational class, these authors, one Gustavson and Sey, prove that this is not possible. In fact, they prove that this was not possible for high-corotational number, but they conjecture that it is true also for lower ones. Say, there is, I think it's Wanderberg or Wanderhout, the author. Sorry, I didn't include it. Actually, there are no bubble trees in finite time, so no towers. This is another thing. But just in the one-corotational class, in the infinite time, to find them, in the infinite time bubbling, there exists bubble trees, at least with some special right-hand sites, I mean target manifolds. And all the rates are also possible. They have been found by Raphael and Schwerer in the one-corotational class, but they are non-generic. They come with different powers of t minus t. Sorry. This shouldn't be here. Anyway, no, just let me, in one second, let me say how the solution we find is made. In fact, I don't have time to discuss it, unfortunately. Yeah, it was like for two talks. No, okay. The solution is made in the following way. Okay. The solution is made as the bubble, and actually this whole thing that I had written in the previous pages was to attempting to justify how the scaling parameter is chosen, and the way it is chosen is connected with the following. The initial condition is made in the following way, by the bubble, but plus some remainders, and these remainders are crucial. And I will not say, I mean, what this does is to correct the error, what this time does is to correct the error due to the lambda far away. I mean, when you scale this, you create an error, okay? And this just corrected far away. But this guy is less obvious. Yeah? This is a function projected in the tangent space of the sphere, say, orthogonal to U, and this C star is a solution to hit equation, and this is a lower order term. Yeah? That is explicit, but doesn't matter. So, and this guy has to have the following properties. He has to have, he's a vector field in R2 actually, with two components are zero. C star tilde is some C star, and with zero, third component. And we need divergence positive, currently equal to zero, and value equal to zero. So, you cannot put C star equal to zero. This is what I mean. Yeah? You really need this guy. You need it, you really need the perturbation of this type in order to find a reasonable equation for lambda. And that reasonable equation for lambda is an equation like this. It's an equation, it's a non-local equation. It's a non-local equation like this. And this non-local equation, when this guy is negative, is actually something that we cannot solve exactly, but we can solve approximately at a very high precision. Yeah? It looks like we cannot really solve it exactly. And this leads us to, I mean, this is not difficult to see. I mean, an equation like this leads, it's a non-local equation, but it leads us to a solution of this type where k is exactly proportional to the very divergence of C star. So, this is what, yeah, I'm sorry, I, it was the first time I tried to give these talks, so it was too long. Okay, thank you very much. Any questions, comments? Sure. The continuation with the results, Bob? Yes. Does it come as a limit or something? Some parameter goes for 0 or whatever, or you just throw it out? The construction is actually explicit. Say, let me put it this way. In the limit, you see the bubble, which exactly at time capital T you drop, then you get a new function there, right? It's kind of smooth, yeah? Yes. Let me precise your question. If you look at another question, please do not blow up and I'm going to see my back. What is the really? I think, I think that's a very good question. No, we have asked ourselves that question. Now I understood the question. Yes, we have asked is... One question is... Is this the right continuation? I know what you mean. Yeah. No, the guess one is this is very... I think that, I think in fact, I think it's a very good question. I mean, we have asked our question ourselves. No, the question is this. Say you regularize your flow. Say you put gradient U to power 2 minus epsilon or something like that. I don't know if that's the right thing. So, say, and then you analyze what is happening around the blow up time for same initial conditions when you let epsilon go to 0. So presumably the solution will blow up. But the question is, but the solution exceeds after... Also after blow up. So is the solution we are constructing the right limit? It's a good question. In fact, my opinion, my opinion is yes, but my co-authors opinion is the opposite. But I... We don't really have a clear evidence to that. No, absolutely. Absolutely. No, we don't know. This is somehow the choice that seems to make the energy more continuous. On the other hand, you know, if you... The convergence is not so nice in a way, okay, because you have the bubbling one with one rotation up to the blow up time, but then the rotation changes, changes direction. So if you go from degree 1 to minus 1, so there is a sort of discontinuity, even though the topology is preserved. Same degree. You just rotate more. Yeah, yeah, yeah. This is this. I'm sorry. This is what I mean. It's same degree. This is why the homotopy class is preserved. But if you see what is happening immediately after, immediately before, say there is no uniform convergence. The difference between time t plus epsilon and t and times t minus epsilon is not something that goes to zero. Yes. Yeah. And the first effect is you have this change. I think it's similar to what Merle has in his configuration in the MLS that it comes in and opens up. Yes. So you have this change of the class going to minus and everything else. Ah, no, I didn't think those terms could be. Do you think so? It's unstable. Yeah. But what is missing here is something should be lost in the single level, after the single level maybe. And in your continuation, I didn't see anything get lost. No, there's nothing lost actually. Say the energy, the energy is sort of continuous nicely. It's not, there's nothing else. This is, yeah, but this is a diffusion phenomenon. So maybe, maybe, maybe slightly different. I'm thinking, okay, but anyway, okay. But it is, it is a geometric equation. So you have some constraints coming from the geometry. So it could, that's the reason why probably there is only like, oh, a reasonable way of quantumization. It is possible. I don't know, I don't know if you have, do you have that opinion? Do you, for example, have different speed of deprofusing? Ah, let me, this is a mention. When you, when you construct the reverse bubbling, in fact, in fact, there are many reverse bubbling rates that you can get. In fact, the genetic one is not this one. If you go, if you go from, from positive time to zero, you can construct a blow up solution, but, but you can get other rates. And the typical rate is not with log square, but with log. Yes. So it's not. It depends of your approximation. You choose just one rate. Yes. So, so, so the, the rate you choose may depend on it. I don't know. Yeah. What do you say? Depends on the approximation of your equation. The, the, you don't have the freedom of the rate. This, this, this way to do it is just approximations. Yes. If you do an approximation. Yes. Of course, an alien gas around it. Yeah. It differs. Yeah. Many of them do not appear. Yeah. Yeah, yeah, yeah, yeah. So it may actually not, okay, so this has bearing on the stability that you're talking about, which I don't understand, but it may, you're not approximating the equation, but just approximating the initial data. So it will be a reasonable way to construct a continuation by taking approximations of the initial data flow. Maybe with smoothness and take a limit, but somehow the, the claim you're making about the stability has bearing on this. I must say I don't understand that. So in general, this finite time singularity can't be stable because you can smoothly perform. Is the only example is. Final time singularity to a harmonic map, which has no singularity. So unless the singular time is going back to infinity, which is. The singular time. No, it's not. You're saying you can perturb in the case of one, but what you seem to be claiming, you can perturb the initial data. And you, and you still get that. Yes. But in, not in full generality though, right? Because you can take an initial data which generates a final time singularity to a harmonic map. This is, this is absolutely, what, our construction is something extremely local. Say this function C star has an isolated zero, non degenerate, positive, there are a lot of constraints in it. If you perturb slightly from it, you, the, the stability is not, is not, is not lost. Okay. So you're saying you construct a solution and then in a neighborhood. In a neighborhood of that special solution, I get it. Yeah. In general, you know, maybe who knows what could happen if there are some degeneracy involved, maybe the singularities could escape. I mean, it's possible. But then do you include the changing in the example in your, in your examples? Does that, the changing in the, is actually absolutely. I mean, in fact, this, in fact, they are boundary condition is consistent with our divergence negative assumption. Say we, we, we, we analyze that. In fact, it is consistent with it. Last question maybe? Well, we got a lot of answers to some of the things I wanted to ask. But, but one is that every blow up you give is somehow isotropic. You have one target, one, one queue, there's no non-isotropic. So you don't get elongated bubbles. And it's kind of a question of uniqueness. Right. Here, say, yeah, I mean, we are really standing at the, at the standard, at the standard bubble. And the standard bubble is sort of isolated. In fact, the key element in the proof is the non-degeneracy of the, of the standard bubble. Say, if you analyze the linear ice equation for the harmonium flow around it, you can invert it with flat-home alternative where the kernel of the linear ice operator just is due to rigid motions. So there's nothing by furcating from it, if you wish. So, so, so I, so non-isotropic, I don't, I don't, I don't know other type of shapes I don't know if you can get. Not nearby at least. Not nearby at least. Yes. Yes. Very good. Okay. Thank you again. Thank you.
 Well, I would like to thank Ivan for the introduction, for the invitations, for Frank and the other  organizers. So it's great to be here. It's my first time here. So the topic of my talk  is actually parabolic equations, so not really connected with wave phenomenon. However, what  I will discuss is blow up phenomenon. So that, of course, as it has been deeply studied already,  it has a lot of parallels with singularity formation in parabolic problems, except that  here the mechanism that triggers it is diffusion. So I will mention next three problems that  have rather similar features. Of course, with a lot of technical differences, one with respect  to each other, but with a common flavor. I will only have time to speak about one of them.  So I prefer to say a little bit more about one of them specifically. So one of the problems,  one of the examples, actually, of the type of phenomenon I would like to see, where perhaps  it is easier to explain what happens, is say the three problems that I'm going to mention  are actually have been studied a lot. They are quite standard in the literature. So one  of them is that reaction diffusion equation with this special power, which is the critical  solar exponent. Say we can consider, for instance, the problem of finding positive solutions  in the domain, maybe with boundary conditions, and define up to some time capital T that  could even be infinity. So a second problem, which is the one I will really talk about,  is the harmonic map flow in 2s2 from R2 into 2s2. This is the special case, of course,  the flow of harmonic maps, the evolution problem for harmonic maps. But in the case  of values in 2, it takes a particularly simple form, which is this semi-linear equation,  where the non-linearity carries this gradient square term. And some special feature that  is actually preserved along the flow is the fact that the absolute value of u is 1. So  u really has values in 2s2. So we will consider this problem in a bounded domain, omega, although  most of what I will mention is valid also in the space, or certain classes of bounded  domains, but doesn't really matter, and up to finite time T. I will consider this problem  with a given boundary condition and corresponding initial data.  So a third problem of this type is the Kepler-Siegel equation. The Kepler-Siegel equation is  an unlocal equation, if you want, or a system. So this is simply the heat equation, the standard  heat equation, modified by a drift operator that depends on u and on this v. And what  is this v? This v is inverse Laplacian of u, which by definition, for me, will be simply  the convolution of u with the fundamental solution. This is a two-dimensional problem,  which you can also write this problem in divergence form like this.  Okay, what do these problems have in common? One of them, one very important that makes  the problems somehow nice, is the fact that they all have Lyapunov functionals. Actually,  this is valid not just for the critical power, but for any power you wish, formally. Say  this functional defined like this, is a Lyapunov functional. In fact, it is strictly decreasing  unless we are at steady states, at this where we are standing at steady states, and more  precisely, it's just a computation, formal computation tells us that the derivative in  time of this guy is going to be minus integral of u t square, so it's negative.  So similar feature we have for the harmonium flow, harmonium flow by definition actually,  is the two-gradient flow or the two-negative gradient flow for the Dirichlet integral.  For here, I insist u has values into s2. And again, we have the same characteristic for  e2. So e2 is strictly decreasing along solutions, along smooth solutions of it, non-trivial  solutions, non-static solutions. And well, perhaps a little bit more complicated, but  still similar, is the Kepler-Siegel system, the Kepler-Siegel equation, the Kepler-Siegel  non-local equation which can be written in this form as I mentioned, and it turns out  that the Lyapunov functional is given by this quantity, which is a little bit different.  Now the derivative of this guy along solutions, along smooth solutions, is negative, but now  the right-hand side is given by this. Okay, so this is one feature, yeah? This is one  feature, the fact that they have this structure, this variational structure. Yeah, but another  one that they share that is very special for these problems is the following. It's the  presence of a special family of steady states, as a matter of fact, a continuum of steady  states, but a continuum of steady states depending on scaling parameter. The three problems  have special scaling variances, and that asymptotically becomes singular. Say, what  do I mean? I mean, let us consider this, which is a special solution of this entire space,  but in fact all solutions of this can be written as a translation and a scaling of this function,  so they are all given by this formula, this is known, the positive solutions, I mean.  And a characteristic they have is that they all have the same energy, the same energy  that I wrote before, so it's actually constant along this scaling. So we have energy invariant  family, an energy invariant continuum of steady states, which as the parameter takes to the  limit to lambda going to zero, say this function becomes singular. Of course, away from the  origin this goes to zero, but exactly at the origin, this is one over lambda raised to  the power, it goes to infinity, so it has spike form, so like that. Okay, so what is  the corresponding object that I will consider, because it's not the only one, unlike the  critical exponent equation, say an object that seems to play a similar role is the following.  This is a harmonic map, this function is a harmonic map, a harmonic map is by definition  steady state of the harmonic map flow, and this function indeed solves this elliptic  equation with absolute value one. We call it the one-corrotation or harmonic map or  the standard one-corrotation or harmonic map. Again, we have a same type of translation  invariant, we also have a dilation invariant, if you simply scale the argument of it, the  energy is preserved, the energy meaning the Dirichlet integral this time, no lambdas in  front of course, because you have values in each one, but there is an extra, there is  an extra invariance that is this, say if you rotate, u is a vector, so it's an element  in S2, if you rotate it by an orthogonal transformation of R3, then you still get a solution of the  equation and they all have the same energy once more. As lambda goes to zero, this of  course does not blow up itself, it has absolute value one, but what blows up here is its gradient,  the gradient of u is actually going to infinity. Now what is the corresponding object in the  Keller-Siegel system? In the Keller-Siegel system, say in the Keller-Siegel equation in R2, well,  an actual object to refer here is this function and this function has again the characteristic  that if you scale it now this way, you still get a solution and this scaling preserves  again the energy, this funny energy, as I wrote at the beginning. They have an additional  feature that they all have mass 8 pi. So what is the topic of this talk? The topic of this  talk or actually the issue that I would like to analyze for the three problems, but in  reality we will refer only to the second one now, is the construction of solutions that  have blow up phenomenon, the exhibit blow up phenomenon as the approaches finite time  or even infinite time. And what is the special pattern of blow up that these equations exhibit  or that is already more or less known that they blow up a rise, that there is blow up  arising in this way, is precisely like this, say, is precisely in the form of blow up around  the finite number of points, maybe one or maybe more, where the profile of blow up near  the, near each of the blow up points is scaling asymptotically singular or resembles the scaling  and asymptotically singular scaling of a steady state of one of these energy preserving steady  states. This is not so crazy because since the energy is preserved, say, the energy of  an object like I am picturing here actually gets stabilized, because the scaling doesn't  change the energy. So this is what is special in this problem, in these problems. That's  feature sometimes called criticality as a matter of fact in this setting. Okay, so now  I would like to refer to the harmonic map flow from R2 into S2. Then I consider this  problem which from now on I will call HMF. So I consider this parabolic equation up to  some given finite time capital T in a bounded domain or in entire R2, but if omega is not  the whole space, I consider boundary conditions and an initial condition. So of course both  the initial and the boundary condition have to go into S2. If that's the case, that property  of having absolute value one is indeed preserving along the flow. Okay, so there is a lot of  literature around this problem. In fact, I would like to make a sort of review of known  resource. A lot of people have worked on this since long ago. Well, the harmonic map flow  was introduced not just in the setting, the two dimensional setting, but as say as the  L2 gradient flow or L2 negative gradient flow for the Dirichlet integral for maps between  two more general remaining manifolds. And it was introduced as a way to build harmonic  maps in reality. So these flows was introduced by Elson and Samson many years ago, 50 years  ago. And concerning for instance the problem of existence, local existence in time of classical  solutions, this was first seen by Elson and Samson themselves, then Struve, and then for  the case of compact manifolds and for the case of the boundary value problem by Casey  Chanck, more or less in 85. Okay, so I repeat what I said before. Say the Dirichlet integral  is actually strictly decreasing along non-trivial trajectory, non-steady trajectories, non-stationary  trajectories of the flow, and we have this identity. So the energy is decreasing. It's  decreasing along the flow, along the flow as long as the solution remains smooth. But  something that has become, that became known is that in general we don't expect global  existence, global existence of a smooth solution. That's typically not the case, not typically,  but it may not be the case. Say that the solutions remain smooth at all times, while this didn't  really become known up to much later. However, Struve in 84, say this is an important result  of the theory, actually, proved the following, proved that even though the solution may develop  singularities, those singularities can more or less be understood. And it can be understood  enough in the case, so that the solution can actually be continued in a natural way. In  fact, it can be continued in a way, continued as a weak solution. It can be continued as  an H1 weak solution, where the energy has discontinuities. The energy still decreases,  but it has jumps. This is what happens with the definition of Struve. Well, as a matter  of fact, it turns out that only at a finite number of points in space-time, so only up  to only for finite number of times, and for each of those times at a finite number of points,  we can have low up. And at those points, at those times, jumps down in the energy take  place. Say, there is this resolve by Freyr in 2002. He proved that for a certain class  of solutions where energy is decreasing even if it has jumps, this solution is unique.  So that means that the definition of Struve is, in a way, a good definition. Okay. More  recently, I don't remember exactly the reference, but very recently, a student of Struve proved  that if the jumps, say, the solution may not have decreasing jumps. Say, even if you have  a weak solution with jumps in the energy that could even go a little bit up, but not too  much, the solution is still unique. What I mean is that the jumps, as I will try to argue  a little bit later, the jumps in the energy are quantized. In fact, in all examples known,  this jump cannot be less than 4 pi. And I think this is a general fact indeed. So this  is Struve's solution, the energy in Struve's solution, when you reach a singularity. Okay.  So as I mentioned, the, say, U itself does not blow up, has absolute value 1, but the  gradient blows up when we have blow up. So in fact, if I call capital T the first time  at which regularity is lost, which is strictly positive, as we know, we must have that the  gradient goes to infinity as T goes from below to the blow up to the time capital T. Well,  I will mention, although maybe the statement is not completely precise, I will make, I  will state the property that is a summary of results by several people who work on this  in the 80s and 90s, starting with Struve and several other authors, as I mentioned here.  So the question is, how does the solution looks like when it blows up? And it turns  out that quite a bit of it can be set. The solution actually does blow up, if it does,  if it blows up, it blows up in the form that I mentioned before, namely as scaling around  a finite number of points of steady states, steady states, steady states Ui, which are  not necessarily the U that I wrote before. So more precisely, so the result, the statement  is a little bit weaker, but still quite a strong assumption. If we approach capital  T along a sequence, then I can extract a subsequence such that for a finite number of points Q1  up to Qk, the following holds. U along that sequence resembles an H1 function plus the  sum of these bubbles. This is the subtraction of the value of infinity of the bubble, which  makes sense. And say where this guy, say this is just a normalization, where this actually  goes to zero, this difference, say pointwise, pointwise away from the points Qi, Qin is approaching  each of the Qis, and this lambda In goes to zero. In fact, a little bit of important information  can be obtained about the relation between the scaling parameters and the mutual distances  of the points, which is this fact. This number is here, for I different from J, go to infinity.  What does this mean? What happens is that if the points, for instance, Qi are all well  separated, one from each other, say this is automatically satisfied, doesn't matter what  this is. This goes to infinity because this goes to zero. But it could be the case at  all points Qi or finding the number of them coincide. What this says is that if they coincide,  if they coincide, for instance, then this goes to infinity, which means that the rates  are not comparable. One of the rates is much smaller than the other. So if we scale around  the point, that means that you see just a single object. Say, if you use lambda i or lambda  gegeo, we'll just see one object. So this is what that property means. So the UIs, as  I mentioned, are harmonic maps. So harmonic maps with finite energy in entire ones. So  what I mean is the UIs add entire solutions of this equation with values in S2 and with  finite energy, which is natural. I mean, we are starting with finite energy configuration.  So in the limit, we should see something with finite energy. This is also a standard fact.  Is that U is a function that goes from R2 into S2. But I could regard it after stereographic  projection as a function from R2 into R2 or from the complex numbers into the complex  numbers or with the composition in the other way from S2 into, say, I can regard it as  a function from complex to complex or from S2 into S2. So this is known. This is a known  fact. These maps are, say, conformal and in S2. And that makes, well, in fact, when it  is lifted to S2, it becomes a smooth function. So that the, in particular, the value U at  infinity, say infinity is one of the poles in S2, is well defined. Some fact that is  known is that using correspondence in the complex numbers with a rational function.  And in fact, these also holds. So if we compute, again by conformality in S2, say the integral  of gradient U squared turns out to be a multiple of 4 pi. Say 4 pi is the area of S1, of S2.  So it's an integer times the area of S2. And this fact holds for some M in the national  numbers. So in particular, the least energy, non-trivial, harmonic maps, non-constant harmonic  maps have energy 4 pi. So this is the reason of the 4 pi I mentioned before. So in particular,  say what are we going to have? We are going to have that this U along Tn goes weekly to  U star in H1. And the energy, say the defect with respect to the gradient of U star squared,  are precisely asymptotic for the Dirichlet measure. It's given by this asymptotically.  Simply integer multiples of 4 pi times Dirichlet delta set, the corresponding lower points.  So this is a way to visualize quickly what the statement says.  OK. So of course, there are many questions around it. This is a very interesting phenomenon.  By the time of Struves results or many others that I mentioned, it was not really much known  about existence, for instance, of such patterns or further properties of it. Stopping in 2004  proved this very interesting fact. Proved that the rates at which the lambda is, remember,  are these blow-up rates around each of the bubbles in this decomposition. This is what  I mean. So what Topping proved is, say, along subsequences, but I mean, it doesn't matter,  say he proved that these blow-up rates have to be of this form. So they go to 0 faster  than T minus T to power 1 half. In fact, the estimate he finds, I think, is a little bit  better, say, with the log in the denominator. But what this says in the language of these  type of geometric problems, or actually of many singularity formation phenomena, what  this says is that the singularities developed must necessarily be of type 2. Type 2 means  a singularity that cannot be captured with a subsimilar form. This is more or less what  it means. And the right rate for that in the harmonium flow is this 1 half. So lambda i  actually goes to 0 faster than that. This is also valid in more general targets, not  just in S2. In fact, Topping finds examples that show that this estimate is, in fact,  optimal or approximately optimal. There are examples of manifolds for which you can get  blow-up rates as T minus T to 1 half plus delta for any arbitrarily small delta.  So let me consider now the function U that I am calling from now 1 U0. This standard  harmonic map, this standard harmonic map, actually when you is just lifting, say, this  is a function from R2 into S2. But if you take it from R2 into R2, this is the identity.  So this just comes from a stereographic projection. So these guys have exactly energy given by  4 pi, degree 1 if you want. And the valued infinity of them are, of course, as you quickly  see, is 0, 0, 1. So let me now mention what is it known concerning existence of blow-up  patterns. In fact, there are very few examples known. And in fact, the examples known concern  single blow-up points where the profile is like this. But more than that, the examples  concern radially symmetric, actually more precisely, radially symmetric settings of  which this is the only steady state. Let me be more precise. One rotational solution of  the harmonic map flow is a solution that has this form. This is, of course, the spherical  coordinate is a way to represent a vector, say, in S2, a point in S2. And I assume that  this angular variable is always e to the i theta. And polar coordinates, where R and  theta are the polar coordinates. And these functions are radial, radially symmetric.  So what is very nice is that with these ansatz, the harmonic map flow reduces to scalar equation.  And this scalar parabolic equation is actually very simple. It's vt equal to d Laplacian  modified by this guy, which is sort of, by the way, an encant type non-miniarity. But  I will not really elaborate on all that. What something that can be observed is that this  guy has the following steady state, the only monotone or finite energy or whatever steady  state is given by decreasing, say. It's given by this function. This is explicit solution  of stationary solution of the equation. And if we evaluate it with this v, say, we obtain  exactly the u0 that I mentioned before, the standard power. Okay, so concerning the literature,  in fact it had been, as I saw in the, in the paper, it had been actually previously conjectured  that it may be the case that no blow up exists at all, but there's no blow up in the, in  the way that it is to the harmonic map flow. But Chang, Ding, and Ye in 1991 found a very  nice example within the one-corrotational class. In fact, what they are able to find  is a solution of the scalar equation v, which looks like the steady state w, scale this  way. Scale by some small number lambda of t or equivalently, say, for the actual original  harmonic map flow, the solution you associated looks like the u0 at x over lambda of t. Yeah.  Where this lambda of t goes to 0 as t approaches capital T. So what they do is they, is a very,  is a very, very nice arguments. The, they capture the blow up using barriers. So they  prove that there is blow up using some special barriers and those barriers are very sensitive  to what the boundary conditions are. Yeah. So it is necessary to take the boundary conditions  in certain range in order to, to observe the phenomenon. They take initial conditions 0  and boundary, and boundary values, say bigger than pi or less than pi, say, and they observe  this phenomenon. Well, for the one-corrotational solution, this was proven many years later,  in 2009, Angen and Hulshof and Matano found a much stronger estimate for this setting,  for this special solutions, for this special harmonic map flow into S2 and one-corrotational.  The rate turns out to be much smaller than t minus t to one-half. It turns out to be  little order of capital T minus little t. Yeah, this is what they proved in 2009. Actually,  before them, Van der Wer Hulshof and King in 2003 made a very, very, wrote a beautiful  paper, formal paper, in which they describe a lot of phenomena around the one-corrotational  blow up. They blow up in the one-corrotational case. And when they discovered, formally,  is that the blow up rate, oh, here is a mistake, important. Here is squared. There is a two  there, not the one. Yeah. It's very different life with the two than with the one in these  rates, by the way. So what they discovered is that lambda of t is of this order. So it's  not just little order of t minus t, but actually modified with the log squared in the denominator.  So the result of Angen and Matano and Hulshof is weaker than this statement, but of course,  it is a rigorous result. And a result that is much more recent, in fact, I think the  year is 2013, is due to Raphael and Schweyer. They prove a very important fact. They succeeded  to make a construction of an actual solution, one-corrotational solution, that blows up  exactly with this rate. Yeah. So the construction is very explicit. It's not based on barriers.  It is actually machinery closer to dispersive, dispersive equations to analysis of blow  up in dispersive problems. And they found a very, very, very nice description of their  solution with the right blow up. And they were able also to establish this very interesting  fact. They proved that this rate, say, in the language, in the formal language of these  authors, they say that they find the, quote, generic rate. So they say the generic rate  should be this with the two. So what Raphael and Schweyer proved sort of that, but, well,  not exactly. What they proved is that for the solution they constructed, if they take  small perturbations of the initial condition, of the initial condition for that special  solution, then they still see the same phenomenon. I mean, they see still blow up with the same  rate, of course, at a slightly different time. Yeah. This is, I insist, in the one-corrotational  radial asymmetric class. So these are the examples now, yeah, for the flow from R2 into  S2, as far as we know. Now, there are, of course, a natural question now is to investigate  what happens in the non-radial situation, or in a situation where there are boundary  conditions. In the case of the result I just mentioned, the setting was entire space. So  the question is, natural question is, well, are there solutions that are in non-radial  asymmetric settings? For instance, is it possible to have more elements like in Struve's  decomposition, the bubbling resolution, you know, like several bubbles in different places?  Can you have towers of bubbles or bubble trees? I mean, et cetera, there are many questions  around. Yeah. And, of course, the question of their stability, because it might perfectly  be the case that the solution is stable within the radial asymmetric class, but when you  make perturbations which get off the radial asymmetric class, perhaps the stability is  destroyed. And our result, just as a summary, is the following. Say, we prove that if we  take any finite number of points of omega, in fact, I should say the following. Yeah.  It's roughly this. You fix k points arbitrarily. Then you can cook up initial conditions and  boundary conditions, boundary conditions independent of time, so that a blow-up solution with exactly  those k given points exists. This is essentially the result. And the profile is like a financial  error scaling or shankding and yeah, is a scaling of the U0 with rates that can be computed  explicitly and they coincide with the one formally derived and rigorously found by later.  Yeah. And, you know, and this is also a kind of result that supports the quote's generosity  statement. Okay. At least the bubbling at one point, at exactly one point is stable.  Yeah. This, in fact, very connected with some work by Frank Merlin and Hatem Zag in the  subcritical heat flow where solutions with finite number of points were found, arbitrary  points in the domain. So, so it, the result shares the flavor. Okay. So, very quickly,  let me, let me say how, yes? When you had the initial cartoon of the multiple points,  it looked like there was a hole in the domain. Oh, yeah, yeah, yeah. No, no, but there's  no, no, there's no, no, no need of the hole. No, no. In fact, if you are looking for this  type of phenomenon in elliptic problems that where the bubbling is triggered by parameters,  many times you need topology. So, I actually borrowed the picture from an elliptic setting.  But many times you need, you need some topological assumption to get multiple blow-up. But here  we don't need anything. A way, a way, a way to put it is that here we, we put all the  blame in the initial condition. Say, all topology needed, say, is in the initial condition,  which I don't say what it is.  If you were to question is whether there's any numerics that shows this?  Actually, the only numerical result that we are aware of, in fact, provided evidence opposite  to one of our claims that was the, that was the stability. In fact, there is a paper,  there is a paper by Vandenberg, I think, and Williams that I learned from Peter, actually,  that conjecture that numerically that if you destroy, if you destroy the symmetry then  the solution may blow up. Sorry, the blow-up may be destroyed, exactly the opposite. But  say, okay, so, as I mentioned, the energies of all these guys is 4 pi. Let me consider  not just arbitrary rotations, arbitrary orthogonal transformations. I will consider special ones  that correspond simply to rotate this vector around the Y3 axis, yeah, which means if you  write it in complex form, simply multiply the first two components written in complex  as a complex number by e to the i alpha. Okay, so this is our more precise statement, but  we are able to prove it is the following. Given k points, you know, the distinct points,  there exists initial amount, there is, there exists initial amount of conditions such that  the solution of harmonic flow for those data blows up at exactly those k points. And the  profile at those k points is given by say, I mean, by a convergence to zero like this  in H1, an uniform sense, H1 and uniform, where the bubbling takes the form of simply  suitable rotation around the axis times the corresponding scalings around QI, where, now  I put correctly the two, where the blow up parameters are like this. So we have in particular  this energy formula. The uniform convergence, by the way, is a property in the bubbling  composition. Let me mention something very quickly. When the time goes to infinity, so  if you go actually along a palest male sequence, say, for the flow, yeah, for the, if you go  along the solution, but for time going to infinity, in fact, all the properties are  known. For instance, this convergence to zero uniformly is sometimes called non-neck property,  yeah? So it is not known if this goes in general in the finite blow up thing. Well, of course,  when time goes to infinity, this guy sees the harmonic map. There is a paper by Peter  Topik where it is proven that if towers are formed at infinity, then all the elements  of the tower except the first one must be either holomorphic or conjugates of holomorphic.  At the same time, not holomorphic, I think that's, that's, is that what you proved? No.  Okay. So anyway, so, Raphael and Schwerer, as I mentioned, proved the stability within  the one correlational class. And here I mentioned this numerical work by van der Weyden Williams.  So, and actually, let me mention the stability, the stability statement is a little bit more  precise. It's a little bit more precise. Say, well, for the single blow up, the single blow  up is stable in this sense. If you perturb the initial data, then you observe exactly  the same phenomenon where, of course, this number, alpha i star, this cap i star on the  time capital T and the points Q i are slightly modified. Yeah, as it is natural. And if you  consider the blow up point at exactly k points, that phenomenon is not stable, which is, I  mean, no wonder. I mean, the blow up points, the blow up times may not coincide. This is  what it means. But if you insist that the blow up takes place in exactly those k points,  you can choose the initial data in a co-dimension k minus one manifold around the initial condition.  Yeah. Okay. So let me mention another issue that we found quite interesting, that is also  addressed in the formal paper by Vanderberg and Hulshof and King. And in rigorous work  by Peter, which is continuation after blow up by a mechanism called reverse bubbling.  I will say very quickly what that is. Say, what happens is that the definition of Struve  has disamplified some features of the energy jumps. Okay. But the energy jumps means, say,  a change in the homotopy class. I mean, actually, it turns out that that means a change in the  homotopy class of the image of you. Yeah. Change of degree, if you want. Yeah. As you, not  just a change of degree, a change of homotopy class. So the question is, are there other  type of continuations after blow up that are kind of nicer in the sense that they possibly  preserve some of the topology of the manifold, of the original manifold. And a topping proposes  a way to do this and prove that there is a continuation with this feature in the one  rotational case. He proved that Chang, Ding and J solution can be continued with a profile  very similar. However, with one difference, it's not the one rotation standard bubble  what you see after the blow up time. But after the blow up time, you see the same thing,  but in reverse orientation, the conjugate if you want. Yeah. And that is this map. So asymptotically  it's the same. So this is zero one, like the other. But around the origin, it winds in  the opposite direction. So this type of procedure is what is called reverse bubbling. So in  the reverse bubbling has in the energy a feature that is different from Strube. In fact, it's  nicer in a way because the energy in fact continues continuously except exactly at time  T when it goes down, goes down and up. Remember that decreasing implies unique. But if you  leave the decreasing but you allow that there is a fixable discontinuity of the energy,  then this solution does it. What we proved is that this continuation is possible for  the solutions we built. In the solutions we built can be continued with exactly the same  profile except that after when you go from above the blow up time, say the solution is  defined up to T plus some delta. And as you go down, you see this blow up. So concerning  the construction, the something that may be reasonable to think is that this type of profiles  that we built may actually be quite general. Say, may even be generic, really, say in the  way blow up takes place. Why do we say this? It's not obvious at all. It's not obvious  at all. But the question is, is it possible to have bubbles other than those induced by  use zero or its conjugate? Because there are other harmonic maps. There are k-corotational  harmonic maps with higher degree. So is it possible that there are profiles like that?  And it turns out that in the k-corotational class, which also leads to some similar scalar  equation, it turns out that for high, in the high-corotational class, these authors, one  Gustavson and Sey, prove that this is not possible. In fact, they prove that this was  not possible for high-corotational number, but they conjecture that it is true also for  lower ones. Say, there is, I think it's Wanderberg or Wanderhout, the author. Sorry, I didn't  include it. Actually, there are no bubble trees in finite time, so no towers. This is  another thing. But just in the one-corotational class, in the infinite time, to find them,  in the infinite time bubbling, there exists bubble trees, at least with some special right-hand  sites, I mean target manifolds. And all the rates are also possible. They have been found  by Raphael and Schwerer in the one-corotational class, but they are non-generic. They come  with different powers of t minus t. Sorry. This shouldn't be here. Anyway, no, just let  me, in one second, let me say how the solution we find is made. In fact, I don't have time  to discuss it, unfortunately. Yeah, it was like for two talks. No, okay. The solution  is made in the following way. Okay. The solution is made as the bubble, and actually this whole  thing that I had written in the previous pages was to attempting to justify how the scaling  parameter is chosen, and the way it is chosen is connected with the following. The initial  condition is made in the following way, by the bubble, but plus some remainders, and  these remainders are crucial. And I will not say, I mean, what this does is to correct  the error, what this time does is to correct the error due to the lambda far away. I mean,  when you scale this, you create an error, okay? And this just corrected far away. But this  guy is less obvious. Yeah? This is a function projected in the tangent space of the sphere,  say, orthogonal to U, and this C star is a solution to hit equation, and this is a lower  order term. Yeah? That is explicit, but doesn't matter. So, and this guy has to have the following  properties. He has to have, he's a vector field in R2 actually, with two components  are zero. C star tilde is some C star, and with zero, third component. And we need divergence  positive, currently equal to zero, and value equal to zero. So, you cannot put C star equal  to zero. This is what I mean. Yeah? You really need this guy. You need it, you really need  the perturbation of this type in order to find a reasonable equation for lambda. And  that reasonable equation for lambda is an equation like this. It's an equation, it's  a non-local equation. It's a non-local equation like this. And this non-local equation, when  this guy is negative, is actually something that we cannot solve exactly, but we can solve  approximately at a very high precision. Yeah? It looks like we cannot really solve it exactly.  And this leads us to, I mean, this is not difficult to see. I mean, an equation like  this leads, it's a non-local equation, but it leads us to a solution of this type where  k is exactly proportional to the very divergence of C star. So, this is what, yeah, I'm sorry,  I, it was the first time I tried to give these talks, so it was too long. Okay, thank you  very much.  Any questions, comments? Sure. The continuation with the results, Bob? Yes. Does it come as  a limit or something? Some parameter goes for 0 or whatever, or you just throw it out?  The construction is actually explicit. Say, let me put it this way. In the limit, you see  the bubble, which exactly at time capital T you drop, then you get a new function there,  right? It's kind of smooth, yeah? Yes. Let me precise your question. If you look at another  question, please do not blow up and I'm going to see my back. What is the really? I think,  I think that's a very good question. No, we have asked ourselves that question. Now I  understood the question. Yes, we have asked is... One question is... Is this the right  continuation? I know what you mean. Yeah. No, the guess one is this is very...  I think that, I think in fact, I think it's a very good question. I mean, we have asked  our question ourselves. No, the question is this. Say you regularize your flow. Say you  put gradient U to power 2 minus epsilon or something like that. I don't know if that's  the right thing. So, say, and then you analyze what is happening around the blow up time  for same initial conditions when you let epsilon go to 0. So presumably the solution will blow  up. But the question is, but the solution exceeds after... Also after blow up. So is  the solution we are constructing the right limit? It's a good question. In fact, my opinion,  my opinion is yes, but my co-authors opinion is the opposite. But I... We don't really  have a clear evidence to that. No, absolutely. Absolutely. No, we don't know. This is somehow  the choice that seems to make the energy more continuous. On the other hand, you know, if  you... The convergence is not so nice in a way, okay, because you have the bubbling one  with one rotation up to the blow up time, but then the rotation changes, changes direction.  So if you go from degree 1 to minus 1, so there is a sort of discontinuity, even though  the topology is preserved. Same degree. You just rotate more. Yeah, yeah, yeah. This is  this. I'm sorry. This is what I mean. It's same degree. This is why the homotopy class  is preserved. But if you see what is happening immediately after, immediately before, say  there is no uniform convergence. The difference between time t plus epsilon and t and times  t minus epsilon is not something that goes to zero. Yes. Yeah. And the first effect is  you have this change. I think it's similar to what Merle has in his configuration in  the MLS that it comes in and opens up. Yes. So you have this change of the class going  to minus and everything else. Ah, no, I didn't think those terms could be. Do you think so?  It's unstable. Yeah. But what is missing here is something should be lost in the single  level, after the single level maybe. And in your continuation, I didn't see anything  get lost. No, there's nothing lost actually. Say the energy, the energy is sort of continuous  nicely. It's not, there's nothing else. This is, yeah, but this is a diffusion phenomenon.  So maybe, maybe, maybe slightly different. I'm thinking, okay, but anyway, okay. But  it is, it is a geometric equation. So you have some constraints coming from the geometry.  So it could, that's the reason why probably there is only like, oh, a reasonable way of  quantumization. It is possible. I don't know, I don't know if you have, do you have that  opinion? Do you, for example, have different speed of deprofusing? Ah, let me, this is  a mention. When you, when you construct the reverse bubbling, in fact, in fact, there  are many reverse bubbling rates that you can get. In fact, the genetic one is not this  one. If you go, if you go from, from positive time to zero, you can construct a blow up  solution, but, but you can get other rates. And the typical rate is not with log square,  but with log. Yes. So it's not. It depends of your approximation. You choose just one  rate. Yes. So, so, so the, the rate you choose may depend on it. I don't know. Yeah. What  do you say? Depends on the approximation of your equation. The, the, you don't have the  freedom of the rate. This, this, this way to do it is just approximations. Yes. If you  do an approximation. Yes. Of course, an alien gas around it. Yeah. It differs. Yeah. Many  of them do not appear. Yeah. Yeah, yeah, yeah, yeah. So it may actually not, okay, so this  has bearing on the stability that you're talking about, which I don't understand, but it may,  you're not approximating the equation, but just approximating the initial data. So it  will be a reasonable way to construct a continuation by taking approximations of the initial data  flow. Maybe with smoothness and take a limit, but somehow the, the claim you're making about  the stability has bearing on this. I must say I don't understand that. So in general,  this finite time singularity can't be stable because you can smoothly perform. Is the only  example is. Final time singularity to a harmonic map, which has no singularity. So unless the  singular time is going back to infinity, which is. The singular time. No, it's not. You're  saying you can perturb in the case of one, but what you seem to be claiming, you can  perturb the initial data. And you, and you still get that. Yes. But in, not in full generality  though, right? Because you can take an initial data which generates a final time singularity  to a harmonic map. This is, this is absolutely, what, our construction is something extremely  local. Say this function C star has an isolated zero, non degenerate, positive, there are  a lot of constraints in it. If you perturb slightly from it, you, the, the stability is  not, is not, is not lost. Okay. So you're saying you construct a solution and then in  a neighborhood. In a neighborhood of that special solution, I get it. Yeah. In general,  you know, maybe who knows what could happen if there are some degeneracy involved, maybe  the singularities could escape. I mean, it's possible. But then do you include the changing  in the example in your, in your examples? Does that, the changing in the, is actually  absolutely. I mean, in fact, this, in fact, they are boundary condition is consistent  with our divergence negative assumption. Say we, we, we, we analyze that. In fact, it  is consistent with it.  Last question maybe?  Well, we got a lot of answers to some of the things I wanted to ask. But, but one is that  every blow up you give is somehow isotropic. You have one target, one, one queue, there's  no non-isotropic. So you don't get elongated bubbles. And it's kind of a question of uniqueness.  Right. Here, say, yeah, I mean, we are really standing at the, at the standard, at the standard  bubble. And the standard bubble is sort of isolated. In fact, the key element in the  proof is the non-degeneracy of the, of the standard bubble. Say, if you analyze the linear  ice equation for the harmonium flow around it, you can invert it with flat-home alternative  where the kernel of the linear ice operator just is due to rigid motions. So there's nothing  by furcating from it, if you wish.  So, so, so I, so non-isotropic, I don't, I don't, I don't know other type of shapes  I don't know if you can get.  Not nearby at least.  Not nearby at least. Yes. Yes.  Very good. Okay. Thank you again.  Thank you.
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Well, I would like to thank Ivan for the introduction, for the invitations, for Frank and the other organizers. So it's great to be here. It's my first time here. So the topic of my talk is actually parabolic equations, so not really connected with wave phenomenon. However, what I will discuss is blow up phenomenon. So that, of course, as it has been deeply studied already, it has a lot of parallels with singularity formation in parabolic problems, except that here the mechanism that triggers it is diffusion. So I will mention next three problems that have rather similar features. Of course, with a lot of technical differences, one with respect to each other, but with a common flavor. I will only have time to speak about one of them. So I prefer to say a little bit more about one of them specifically. So one of the problems, one of the examples, actually, of the type of phenomenon I would like to see, where perhaps it is easier to explain what happens, is say the three problems that I'm going to mention are actually have been studied a lot. They are quite standard in the literature. So one of them is that reaction diffusion equation with this special power, which is the critical solar exponent. Say we can consider, for instance, the problem of finding positive solutions in the domain, maybe with boundary conditions, and define up to some time capital T that could even be infinity. So a second problem, which is the one I will really talk about, is the harmonic map flow in 2s2 from R2 into 2s2. This is the special case, of course, the flow of harmonic maps, the evolution problem for harmonic maps. But in the case of values in 2, it takes a particularly simple form, which is this semi-linear equation, where the non-linearity carries this gradient square term. And some special feature that is actually preserved along the flow is the fact that the absolute value of u is 1. So u really has values in 2s2. So we will consider this problem in a bounded domain, omega, although most of what I will mention is valid also in the space, or certain classes of bounded domains, but doesn't really matter, and up to finite time T. I will consider this problem with a given boundary condition and corresponding initial data. So a third problem of this type is the Kepler-Siegel equation. The Kepler-Siegel equation is an unlocal equation, if you want, or a system. So this is simply the heat equation, the standard heat equation, modified by a drift operator that depends on u and on this v. And what is this v? This v is inverse Laplacian of u, which by definition, for me, will be simply the convolution of u with the fundamental solution. This is a two-dimensional problem, which you can also write this problem in divergence form like this. Okay, what do these problems have in common? One of them, one very important that makes the problems somehow nice, is the fact that they all have Lyapunov functionals. Actually, this is valid not just for the critical power, but for any power you wish, formally. Say this functional defined like this, is a Lyapunov functional. In fact, it is strictly decreasing unless we are at steady states, at this where we are standing at steady states, and more precisely, it's just a computation, formal computation tells us that the derivative in time of this guy is going to be minus integral of u t square, so it's negative. So similar feature we have for the harmonium flow, harmonium flow by definition actually, is the two-gradient flow or the two-negative gradient flow for the Dirichlet integral. For here, I insist u has values into s2. And again, we have the same characteristic for e2. So e2 is strictly decreasing along solutions, along smooth solutions of it, non-trivial solutions, non-static solutions. And well, perhaps a little bit more complicated, but still similar, is the Kepler-Siegel system, the Kepler-Siegel equation, the Kepler-Siegel non-local equation which can be written in this form as I mentioned, and it turns out that the Lyapunov functional is given by this quantity, which is a little bit different. Now the derivative of this guy along solutions, along smooth solutions, is negative, but now the right-hand side is given by this. Okay, so this is one feature, yeah? This is one feature, the fact that they have this structure, this variational structure. Yeah, but another one that they share that is very special for these problems is the following. It's the presence of a special family of steady states, as a matter of fact, a continuum of steady states, but a continuum of steady states depending on scaling parameter. The three problems have special scaling variances, and that asymptotically becomes singular. Say, what do I mean? I mean, let us consider this, which is a special solution of this entire space, but in fact all solutions of this can be written as a translation and a scaling of this function, so they are all given by this formula, this is known, the positive solutions, I mean. And a characteristic they have is that they all have the same energy, the same energy that I wrote before, so it's actually constant along this scaling. So we have energy invariant family, an energy invariant continuum of steady states, which as the parameter takes to the limit to lambda going to zero, say this function becomes singular. Of course, away from the origin this goes to zero, but exactly at the origin, this is one over lambda raised to the power, it goes to infinity, so it has spike form, so like that. Okay, so what is the corresponding object that I will consider, because it's not the only one, unlike the critical exponent equation, say an object that seems to play a similar role is the following. This is a harmonic map, this function is a harmonic map, a harmonic map is by definition steady state of the harmonic map flow, and this function indeed solves this elliptic equation with absolute value one. We call it the one-corrotation or harmonic map or the standard one-corrotation or harmonic map. Again, we have a same type of translation invariant, we also have a dilation invariant, if you simply scale the argument of it, the energy is preserved, the energy meaning the Dirichlet integral this time, no lambdas in front of course, because you have values in each one, but there is an extra, there is an extra invariance that is this, say if you rotate, u is a vector, so it's an element in S2, if you rotate it by an orthogonal transformation of R3, then you still get a solution of the equation and they all have the same energy once more. As lambda goes to zero, this of course does not blow up itself, it has absolute value one, but what blows up here is its gradient, the gradient of u is actually going to infinity. Now what is the corresponding object in the Keller-Siegel system? In the Keller-Siegel system, say in the Keller-Siegel equation in R2, well, an actual object to refer here is this function and this function has again the characteristic that if you scale it now this way, you still get a solution and this scaling preserves again the energy, this funny energy, as I wrote at the beginning. They have an additional feature that they all have mass 8 pi. So what is the topic of this talk? The topic of this talk or actually the issue that I would like to analyze for the three problems, but in reality we will refer only to the second one now, is the construction of solutions that have blow up phenomenon, the exhibit blow up phenomenon as the approaches finite time or even infinite time. And what is the special pattern of blow up that these equations exhibit or that is already more or less known that they blow up a rise, that there is blow up arising in this way, is precisely like this, say, is precisely in the form of blow up around the finite number of points, maybe one or maybe more, where the profile of blow up near the, near each of the blow up points is scaling asymptotically singular or resembles the scaling and asymptotically singular scaling of a steady state of one of these energy preserving steady states. This is not so crazy because since the energy is preserved, say, the energy of an object like I am picturing here actually gets stabilized, because the scaling doesn't change the energy. So this is what is special in this problem, in these problems. That's feature sometimes called criticality as a matter of fact in this setting. Okay, so now I would like to refer to the harmonic map flow from R2 into S2. Then I consider this problem which from now on I will call HMF. So I consider this parabolic equation up to some given finite time capital T in a bounded domain or in entire R2, but if omega is not the whole space, I consider boundary conditions and an initial condition. So of course both the initial and the boundary condition have to go into S2. If that's the case, that property of having absolute value one is indeed preserving along the flow. Okay, so there is a lot of literature around this problem. In fact, I would like to make a sort of review of known resource. A lot of people have worked on this since long ago. Well, the harmonic map flow was introduced not just in the setting, the two dimensional setting, but as say as the L2 gradient flow or L2 negative gradient flow for the Dirichlet integral for maps between two more general remaining manifolds. And it was introduced as a way to build harmonic maps in reality. So these flows was introduced by Elson and Samson many years ago, 50 years ago. And concerning for instance the problem of existence, local existence in time of classical solutions, this was first seen by Elson and Samson themselves, then Struve, and then for the case of compact manifolds and for the case of the boundary value problem by Casey Chanck, more or less in 85. Okay, so I repeat what I said before. Say the Dirichlet integral is actually strictly decreasing along non-trivial trajectory, non-steady trajectories, non-stationary trajectories of the flow, and we have this identity. So the energy is decreasing. It's decreasing along the flow, along the flow as long as the solution remains smooth. But something that has become, that became known is that in general we don't expect global existence, global existence of a smooth solution. That's typically not the case, not typically, but it may not be the case. Say that the solutions remain smooth at all times, while this didn't really become known up to much later. However, Struve in 84, say this is an important result of the theory, actually, proved the following, proved that even though the solution may develop singularities, those singularities can more or less be understood. And it can be understood enough in the case, so that the solution can actually be continued in a natural way. In fact, it can be continued in a way, continued as a weak solution. It can be continued as an H1 weak solution, where the energy has discontinuities. The energy still decreases, but it has jumps. This is what happens with the definition of Struve. Well, as a matter of fact, it turns out that only at a finite number of points in space-time, so only up to only for finite number of times, and for each of those times at a finite number of points, we can have low up. And at those points, at those times, jumps down in the energy take place. Say, there is this resolve by Freyr in 2002. He proved that for a certain class of solutions where energy is decreasing even if it has jumps, this solution is unique. So that means that the definition of Struve is, in a way, a good definition. Okay. More recently, I don't remember exactly the reference, but very recently, a student of Struve proved that if the jumps, say, the solution may not have decreasing jumps. Say, even if you have a weak solution with jumps in the energy that could even go a little bit up, but not too much, the solution is still unique. What I mean is that the jumps, as I will try to argue a little bit later, the jumps in the energy are quantized. In fact, in all examples known, this jump cannot be less than 4 pi. And I think this is a general fact indeed. So this is Struve's solution, the energy in Struve's solution, when you reach a singularity. Okay. So as I mentioned, the, say, U itself does not blow up, has absolute value 1, but the gradient blows up when we have blow up. So in fact, if I call capital T the first time at which regularity is lost, which is strictly positive, as we know, we must have that the gradient goes to infinity as T goes from below to the blow up to the time capital T. Well, I will mention, although maybe the statement is not completely precise, I will make, I will state the property that is a summary of results by several people who work on this in the 80s and 90s, starting with Struve and several other authors, as I mentioned here. So the question is, how does the solution looks like when it blows up? And it turns out that quite a bit of it can be set. The solution actually does blow up, if it does, if it blows up, it blows up in the form that I mentioned before, namely as scaling around a finite number of points of steady states, steady states, steady states Ui, which are not necessarily the U that I wrote before. So more precisely, so the result, the statement is a little bit weaker, but still quite a strong assumption. If we approach capital T along a sequence, then I can extract a subsequence such that for a finite number of points Q1 up to Qk, the following holds. U along that sequence resembles an H1 function plus the sum of these bubbles. This is the subtraction of the value of infinity of the bubble, which makes sense. And say where this guy, say this is just a normalization, where this actually goes to zero, this difference, say pointwise, pointwise away from the points Qi, Qin is approaching each of the Qis, and this lambda In goes to zero. In fact, a little bit of important information can be obtained about the relation between the scaling parameters and the mutual distances of the points, which is this fact. This number is here, for I different from J, go to infinity. What does this mean? What happens is that if the points, for instance, Qi are all well separated, one from each other, say this is automatically satisfied, doesn't matter what this is. This goes to infinity because this goes to zero. But it could be the case at all points Qi or finding the number of them coincide. What this says is that if they coincide, if they coincide, for instance, then this goes to infinity, which means that the rates are not comparable. One of the rates is much smaller than the other. So if we scale around the point, that means that you see just a single object. Say, if you use lambda i or lambda gegeo, we'll just see one object. So this is what that property means. So the UIs, as I mentioned, are harmonic maps. So harmonic maps with finite energy in entire ones. So what I mean is the UIs add entire solutions of this equation with values in S2 and with finite energy, which is natural. I mean, we are starting with finite energy configuration. So in the limit, we should see something with finite energy. This is also a standard fact. Is that U is a function that goes from R2 into S2. But I could regard it after stereographic projection as a function from R2 into R2 or from the complex numbers into the complex numbers or with the composition in the other way from S2 into, say, I can regard it as a function from complex to complex or from S2 into S2. So this is known. This is a known fact. These maps are, say, conformal and in S2. And that makes, well, in fact, when it is lifted to S2, it becomes a smooth function. So that the, in particular, the value U at infinity, say infinity is one of the poles in S2, is well defined. Some fact that is known is that using correspondence in the complex numbers with a rational function. And in fact, these also holds. So if we compute, again by conformality in S2, say the integral of gradient U squared turns out to be a multiple of 4 pi. Say 4 pi is the area of S1, of S2. So it's an integer times the area of S2. And this fact holds for some M in the national numbers. So in particular, the least energy, non-trivial, harmonic maps, non-constant harmonic maps have energy 4 pi. So this is the reason of the 4 pi I mentioned before. So in particular, say what are we going to have? We are going to have that this U along Tn goes weekly to U star in H1. And the energy, say the defect with respect to the gradient of U star squared, are precisely asymptotic for the Dirichlet measure. It's given by this asymptotically. Simply integer multiples of 4 pi times Dirichlet delta set, the corresponding lower points. So this is a way to visualize quickly what the statement says. OK. So of course, there are many questions around it. This is a very interesting phenomenon. By the time of Struves results or many others that I mentioned, it was not really much known about existence, for instance, of such patterns or further properties of it. Stopping in 2004 proved this very interesting fact. Proved that the rates at which the lambda is, remember, are these blow-up rates around each of the bubbles in this decomposition. This is what I mean. So what Topping proved is, say, along subsequences, but I mean, it doesn't matter, say he proved that these blow-up rates have to be of this form. So they go to 0 faster than T minus T to power 1 half. In fact, the estimate he finds, I think, is a little bit better, say, with the log in the denominator. But what this says in the language of these type of geometric problems, or actually of many singularity formation phenomena, what this says is that the singularities developed must necessarily be of type 2. Type 2 means a singularity that cannot be captured with a subsimilar form. This is more or less what it means. And the right rate for that in the harmonium flow is this 1 half. So lambda i actually goes to 0 faster than that. This is also valid in more general targets, not just in S2. In fact, Topping finds examples that show that this estimate is, in fact, optimal or approximately optimal. There are examples of manifolds for which you can get blow-up rates as T minus T to 1 half plus delta for any arbitrarily small delta. So let me consider now the function U that I am calling from now 1 U0. This standard harmonic map, this standard harmonic map, actually when you is just lifting, say, this is a function from R2 into S2. But if you take it from R2 into R2, this is the identity. So this just comes from a stereographic projection. So these guys have exactly energy given by 4 pi, degree 1 if you want. And the valued infinity of them are, of course, as you quickly see, is 0, 0, 1. So let me now mention what is it known concerning existence of blow-up patterns. In fact, there are very few examples known. And in fact, the examples known concern single blow-up points where the profile is like this. But more than that, the examples concern radially symmetric, actually more precisely, radially symmetric settings of which this is the only steady state. Let me be more precise. One rotational solution of the harmonic map flow is a solution that has this form. This is, of course, the spherical coordinate is a way to represent a vector, say, in S2, a point in S2. And I assume that this angular variable is always e to the i theta. And polar coordinates, where R and theta are the polar coordinates. And these functions are radial, radially symmetric. So what is very nice is that with these ansatz, the harmonic map flow reduces to scalar equation. And this scalar parabolic equation is actually very simple. It's vt equal to d Laplacian modified by this guy, which is sort of, by the way, an encant type non-miniarity. But I will not really elaborate on all that. What something that can be observed is that this guy has the following steady state, the only monotone or finite energy or whatever steady state is given by decreasing, say. It's given by this function. This is explicit solution of stationary solution of the equation. And if we evaluate it with this v, say, we obtain exactly the u0 that I mentioned before, the standard power. Okay, so concerning the literature, in fact it had been, as I saw in the, in the paper, it had been actually previously conjectured that it may be the case that no blow up exists at all, but there's no blow up in the, in the way that it is to the harmonic map flow. But Chang, Ding, and Ye in 1991 found a very nice example within the one-corrotational class. In fact, what they are able to find is a solution of the scalar equation v, which looks like the steady state w, scale this way. Scale by some small number lambda of t or equivalently, say, for the actual original harmonic map flow, the solution you associated looks like the u0 at x over lambda of t. Yeah. Where this lambda of t goes to 0 as t approaches capital T. So what they do is they, is a very, is a very, very nice arguments. The, they capture the blow up using barriers. So they prove that there is blow up using some special barriers and those barriers are very sensitive to what the boundary conditions are. Yeah. So it is necessary to take the boundary conditions in certain range in order to, to observe the phenomenon. They take initial conditions 0 and boundary, and boundary values, say bigger than pi or less than pi, say, and they observe this phenomenon. Well, for the one-corrotational solution, this was proven many years later, in 2009, Angen and Hulshof and Matano found a much stronger estimate for this setting, for this special solutions, for this special harmonic map flow into S2 and one-corrotational. The rate turns out to be much smaller than t minus t to one-half. It turns out to be little order of capital T minus little t. Yeah, this is what they proved in 2009. Actually, before them, Van der Wer Hulshof and King in 2003 made a very, very, wrote a beautiful paper, formal paper, in which they describe a lot of phenomena around the one-corrotational blow up. They blow up in the one-corrotational case. And when they discovered, formally, is that the blow up rate, oh, here is a mistake, important. Here is squared. There is a two there, not the one. Yeah. It's very different life with the two than with the one in these rates, by the way. So what they discovered is that lambda of t is of this order. So it's not just little order of t minus t, but actually modified with the log squared in the denominator. So the result of Angen and Matano and Hulshof is weaker than this statement, but of course, it is a rigorous result. And a result that is much more recent, in fact, I think the year is 2013, is due to Raphael and Schweyer. They prove a very important fact. They succeeded to make a construction of an actual solution, one-corrotational solution, that blows up exactly with this rate. Yeah. So the construction is very explicit. It's not based on barriers. It is actually machinery closer to dispersive, dispersive equations to analysis of blow up in dispersive problems. And they found a very, very, very nice description of their solution with the right blow up. And they were able also to establish this very interesting fact. They proved that this rate, say, in the language, in the formal language of these authors, they say that they find the, quote, generic rate. So they say the generic rate should be this with the two. So what Raphael and Schweyer proved sort of that, but, well, not exactly. What they proved is that for the solution they constructed, if they take small perturbations of the initial condition, of the initial condition for that special solution, then they still see the same phenomenon. I mean, they see still blow up with the same rate, of course, at a slightly different time. Yeah. This is, I insist, in the one-corrotational radial asymmetric class. So these are the examples now, yeah, for the flow from R2 into S2, as far as we know. Now, there are, of course, a natural question now is to investigate what happens in the non-radial situation, or in a situation where there are boundary conditions. In the case of the result I just mentioned, the setting was entire space. So the question is, natural question is, well, are there solutions that are in non-radial asymmetric settings? For instance, is it possible to have more elements like in Struve's decomposition, the bubbling resolution, you know, like several bubbles in different places? Can you have towers of bubbles or bubble trees? I mean, et cetera, there are many questions around. Yeah. And, of course, the question of their stability, because it might perfectly be the case that the solution is stable within the radial asymmetric class, but when you make perturbations which get off the radial asymmetric class, perhaps the stability is destroyed. And our result, just as a summary, is the following. Say, we prove that if we take any finite number of points of omega, in fact, I should say the following. Yeah. It's roughly this. You fix k points arbitrarily. Then you can cook up initial conditions and boundary conditions, boundary conditions independent of time, so that a blow-up solution with exactly those k given points exists. This is essentially the result. And the profile is like a financial error scaling or shankding and yeah, is a scaling of the U0 with rates that can be computed explicitly and they coincide with the one formally derived and rigorously found by later. Yeah. And, you know, and this is also a kind of result that supports the quote's generosity statement. Okay. At least the bubbling at one point, at exactly one point is stable. Yeah. This, in fact, very connected with some work by Frank Merlin and Hatem Zag in the subcritical heat flow where solutions with finite number of points were found, arbitrary points in the domain. So, so it, the result shares the flavor. Okay. So, very quickly, let me, let me say how, yes? When you had the initial cartoon of the multiple points, it looked like there was a hole in the domain. Oh, yeah, yeah, yeah. No, no, but there's no, no, there's no, no, no need of the hole. No, no. In fact, if you are looking for this type of phenomenon in elliptic problems that where the bubbling is triggered by parameters, many times you need topology. So, I actually borrowed the picture from an elliptic setting. But many times you need, you need some topological assumption to get multiple blow-up. But here we don't need anything. A way, a way, a way to put it is that here we, we put all the blame in the initial condition. Say, all topology needed, say, is in the initial condition, which I don't say what it is. If you were to question is whether there's any numerics that shows this? Actually, the only numerical result that we are aware of, in fact, provided evidence opposite to one of our claims that was the, that was the stability. In fact, there is a paper, there is a paper by Vandenberg, I think, and Williams that I learned from Peter, actually, that conjecture that numerically that if you destroy, if you destroy the symmetry then the solution may blow up. Sorry, the blow-up may be destroyed, exactly the opposite. But say, okay, so, as I mentioned, the energies of all these guys is 4 pi. Let me consider not just arbitrary rotations, arbitrary orthogonal transformations. I will consider special ones that correspond simply to rotate this vector around the Y3 axis, yeah, which means if you write it in complex form, simply multiply the first two components written in complex as a complex number by e to the i alpha. Okay, so this is our more precise statement, but we are able to prove it is the following. Given k points, you know, the distinct points, there exists initial amount, there is, there exists initial amount of conditions such that the solution of harmonic flow for those data blows up at exactly those k points. And the profile at those k points is given by say, I mean, by a convergence to zero like this in H1, an uniform sense, H1 and uniform, where the bubbling takes the form of simply suitable rotation around the axis times the corresponding scalings around QI, where, now I put correctly the two, where the blow up parameters are like this. So we have in particular this energy formula. The uniform convergence, by the way, is a property in the bubbling composition. Let me mention something very quickly. When the time goes to infinity, so if you go actually along a palest male sequence, say, for the flow, yeah, for the, if you go along the solution, but for time going to infinity, in fact, all the properties are known. For instance, this convergence to zero uniformly is sometimes called non-neck property, yeah? So it is not known if this goes in general in the finite blow up thing. Well, of course, when time goes to infinity, this guy sees the harmonic map. There is a paper by Peter Topik where it is proven that if towers are formed at infinity, then all the elements of the tower except the first one must be either holomorphic or conjugates of holomorphic. At the same time, not holomorphic, I think that's, that's, is that what you proved? No. Okay. So anyway, so, Raphael and Schwerer, as I mentioned, proved the stability within the one correlational class. And here I mentioned this numerical work by van der Weyden Williams. So, and actually, let me mention the stability, the stability statement is a little bit more precise. It's a little bit more precise. Say, well, for the single blow up, the single blow up is stable in this sense. If you perturb the initial data, then you observe exactly the same phenomenon where, of course, this number, alpha i star, this cap i star on the time capital T and the points Q i are slightly modified. Yeah, as it is natural. And if you consider the blow up point at exactly k points, that phenomenon is not stable, which is, I mean, no wonder. I mean, the blow up points, the blow up times may not coincide. This is what it means. But if you insist that the blow up takes place in exactly those k points, you can choose the initial data in a co-dimension k minus one manifold around the initial condition. Yeah. Okay. So let me mention another issue that we found quite interesting, that is also addressed in the formal paper by Vanderberg and Hulshof and King. And in rigorous work by Peter, which is continuation after blow up by a mechanism called reverse bubbling. I will say very quickly what that is. Say, what happens is that the definition of Struve has disamplified some features of the energy jumps. Okay. But the energy jumps means, say, a change in the homotopy class. I mean, actually, it turns out that that means a change in the homotopy class of the image of you. Yeah. Change of degree, if you want. Yeah. As you, not just a change of degree, a change of homotopy class. So the question is, are there other type of continuations after blow up that are kind of nicer in the sense that they possibly preserve some of the topology of the manifold, of the original manifold. And a topping proposes a way to do this and prove that there is a continuation with this feature in the one rotational case. He proved that Chang, Ding and J solution can be continued with a profile very similar. However, with one difference, it's not the one rotation standard bubble what you see after the blow up time. But after the blow up time, you see the same thing, but in reverse orientation, the conjugate if you want. Yeah. And that is this map. So asymptotically it's the same. So this is zero one, like the other. But around the origin, it winds in the opposite direction. So this type of procedure is what is called reverse bubbling. So in the reverse bubbling has in the energy a feature that is different from Strube. In fact, it's nicer in a way because the energy in fact continues continuously except exactly at time T when it goes down, goes down and up. Remember that decreasing implies unique. But if you leave the decreasing but you allow that there is a fixable discontinuity of the energy, then this solution does it. What we proved is that this continuation is possible for the solutions we built. In the solutions we built can be continued with exactly the same profile except that after when you go from above the blow up time, say the solution is defined up to T plus some delta. And as you go down, you see this blow up. So concerning the construction, the something that may be reasonable to think is that this type of profiles that we built may actually be quite general. Say, may even be generic, really, say in the way blow up takes place. Why do we say this? It's not obvious at all. It's not obvious at all. But the question is, is it possible to have bubbles other than those induced by use zero or its conjugate? Because there are other harmonic maps. There are k-corotational harmonic maps with higher degree. So is it possible that there are profiles like that? And it turns out that in the k-corotational class, which also leads to some similar scalar equation, it turns out that for high, in the high-corotational class, these authors, one Gustavson and Sey, prove that this is not possible. In fact, they prove that this was not possible for high-corotational number, but they conjecture that it is true also for lower ones. Say, there is, I think it's Wanderberg or Wanderhout, the author. Sorry, I didn't include it. Actually, there are no bubble trees in finite time, so no towers. This is another thing. But just in the one-corotational class, in the infinite time, to find them, in the infinite time bubbling, there exists bubble trees, at least with some special right-hand sites, I mean target manifolds. And all the rates are also possible. They have been found by Raphael and Schwerer in the one-corotational class, but they are non-generic. They come with different powers of t minus t. Sorry. This shouldn't be here. Anyway, no, just let me, in one second, let me say how the solution we find is made. In fact, I don't have time to discuss it, unfortunately. Yeah, it was like for two talks. No, okay. The solution is made in the following way. Okay. The solution is made as the bubble, and actually this whole thing that I had written in the previous pages was to attempting to justify how the scaling parameter is chosen, and the way it is chosen is connected with the following. The initial condition is made in the following way, by the bubble, but plus some remainders, and these remainders are crucial. And I will not say, I mean, what this does is to correct the error, what this time does is to correct the error due to the lambda far away. I mean, when you scale this, you create an error, okay? And this just corrected far away. But this guy is less obvious. Yeah? This is a function projected in the tangent space of the sphere, say, orthogonal to U, and this C star is a solution to hit equation, and this is a lower order term. Yeah? That is explicit, but doesn't matter. So, and this guy has to have the following properties. He has to have, he's a vector field in R2 actually, with two components are zero. C star tilde is some C star, and with zero, third component. And we need divergence positive, currently equal to zero, and value equal to zero. So, you cannot put C star equal to zero. This is what I mean. Yeah? You really need this guy. You need it, you really need the perturbation of this type in order to find a reasonable equation for lambda. And that reasonable equation for lambda is an equation like this. It's an equation, it's a non-local equation. It's a non-local equation like this. And this non-local equation, when this guy is negative, is actually something that we cannot solve exactly, but we can solve approximately at a very high precision. Yeah? It looks like we cannot really solve it exactly. And this leads us to, I mean, this is not difficult to see. I mean, an equation like this leads, it's a non-local equation, but it leads us to a solution of this type where k is exactly proportional to the very divergence of C star. So, this is what, yeah, I'm sorry, I, it was the first time I tried to give these talks, so it was too long. Okay, thank you very much. Any questions, comments? Sure. The continuation with the results, Bob? Yes. Does it come as a limit or something? Some parameter goes for 0 or whatever, or you just throw it out? The construction is actually explicit. Say, let me put it this way. In the limit, you see the bubble, which exactly at time capital T you drop, then you get a new function there, right? It's kind of smooth, yeah? Yes. Let me precise your question. If you look at another question, please do not blow up and I'm going to see my back. What is the really? I think, I think that's a very good question. No, we have asked ourselves that question. Now I understood the question. Yes, we have asked is... One question is... Is this the right continuation? I know what you mean. Yeah. No, the guess one is this is very... I think that, I think in fact, I think it's a very good question. I mean, we have asked our question ourselves. No, the question is this. Say you regularize your flow. Say you put gradient U to power 2 minus epsilon or something like that. I don't know if that's the right thing. So, say, and then you analyze what is happening around the blow up time for same initial conditions when you let epsilon go to 0. So presumably the solution will blow up. But the question is, but the solution exceeds after... Also after blow up. So is the solution we are constructing the right limit? It's a good question. In fact, my opinion, my opinion is yes, but my co-authors opinion is the opposite. But I... We don't really have a clear evidence to that. No, absolutely. Absolutely. No, we don't know. This is somehow the choice that seems to make the energy more continuous. On the other hand, you know, if you... The convergence is not so nice in a way, okay, because you have the bubbling one with one rotation up to the blow up time, but then the rotation changes, changes direction. So if you go from degree 1 to minus 1, so there is a sort of discontinuity, even though the topology is preserved. Same degree. You just rotate more. Yeah, yeah, yeah. This is this. I'm sorry. This is what I mean. It's same degree. This is why the homotopy class is preserved. But if you see what is happening immediately after, immediately before, say there is no uniform convergence. The difference between time t plus epsilon and t and times t minus epsilon is not something that goes to zero. Yes. Yeah. And the first effect is you have this change. I think it's similar to what Merle has in his configuration in the MLS that it comes in and opens up. Yes. So you have this change of the class going to minus and everything else. Ah, no, I didn't think those terms could be. Do you think so? It's unstable. Yeah. But what is missing here is something should be lost in the single level, after the single level maybe. And in your continuation, I didn't see anything get lost. No, there's nothing lost actually. Say the energy, the energy is sort of continuous nicely. It's not, there's nothing else. This is, yeah, but this is a diffusion phenomenon. So maybe, maybe, maybe slightly different. I'm thinking, okay, but anyway, okay. But it is, it is a geometric equation. So you have some constraints coming from the geometry. So it could, that's the reason why probably there is only like, oh, a reasonable way of quantumization. It is possible. I don't know, I don't know if you have, do you have that opinion? Do you, for example, have different speed of deprofusing? Ah, let me, this is a mention. When you, when you construct the reverse bubbling, in fact, in fact, there are many reverse bubbling rates that you can get. In fact, the genetic one is not this one. If you go, if you go from, from positive time to zero, you can construct a blow up solution, but, but you can get other rates. And the typical rate is not with log square, but with log. Yes. So it's not. It depends of your approximation. You choose just one rate. Yes. So, so, so the, the rate you choose may depend on it. I don't know. Yeah. What do you say? Depends on the approximation of your equation. The, the, you don't have the freedom of the rate. This, this, this way to do it is just approximations. Yes. If you do an approximation. Yes. Of course, an alien gas around it. Yeah. It differs. Yeah. Many of them do not appear. Yeah. Yeah, yeah, yeah, yeah. So it may actually not, okay, so this has bearing on the stability that you're talking about, which I don't understand, but it may, you're not approximating the equation, but just approximating the initial data. So it will be a reasonable way to construct a continuation by taking approximations of the initial data flow. Maybe with smoothness and take a limit, but somehow the, the claim you're making about the stability has bearing on this. I must say I don't understand that. So in general, this finite time singularity can't be stable because you can smoothly perform. Is the only example is. Final time singularity to a harmonic map, which has no singularity. So unless the singular time is going back to infinity, which is. The singular time. No, it's not. You're saying you can perturb in the case of one, but what you seem to be claiming, you can perturb the initial data. And you, and you still get that. Yes. But in, not in full generality though, right? Because you can take an initial data which generates a final time singularity to a harmonic map. This is, this is absolutely, what, our construction is something extremely local. Say this function C star has an isolated zero, non degenerate, positive, there are a lot of constraints in it. If you perturb slightly from it, you, the, the stability is not, is not, is not lost. Okay. So you're saying you construct a solution and then in a neighborhood. In a neighborhood of that special solution, I get it. Yeah. In general, you know, maybe who knows what could happen if there are some degeneracy involved, maybe the singularities could escape. I mean, it's possible. But then do you include the changing in the example in your, in your examples? Does that, the changing in the, is actually absolutely. I mean, in fact, this, in fact, they are boundary condition is consistent with our divergence negative assumption. Say we, we, we, we analyze that. In fact, it is consistent with it. Last question maybe? Well, we got a lot of answers to some of the things I wanted to ask. But, but one is that every blow up you give is somehow isotropic. You have one target, one, one queue, there's no non-isotropic. So you don't get elongated bubbles. And it's kind of a question of uniqueness. Right. Here, say, yeah, I mean, we are really standing at the, at the standard, at the standard bubble. And the standard bubble is sort of isolated. In fact, the key element in the proof is the non-degeneracy of the, of the standard bubble. Say, if you analyze the linear ice equation for the harmonium flow around it, you can invert it with flat-home alternative where the kernel of the linear ice operator just is due to rigid motions. So there's nothing by furcating from it, if you wish. So, so, so I, so non-isotropic, I don't, I don't, I don't know other type of shapes I don't know if you can get. Not nearby at least. Not nearby at least. Yes. Yes. Very good. Okay. Thank you again. Thank you.
construction of solutions that have blow up steady states, steady states, steady states Ui, holds for some M in the national numbers. So in uniform, where the bubbling takes the form of