 I would first like to thank the organizers for this wonderful conference. It's the first time I'm here. So before I say anything about math, everything I will talk about is a joint work with Juhi Jiang, who is based at University of Southern California and this year at the IAS. So the good thing about this long title is that it basically tells you the statement of the main theorem. There are certain special solutions to the system of PDEs I will look at, Euler Poisson system. It comes from astrophysics. It's a model of a star. And we will be studying the nonlinear stability of that class of solutions. So let me give you an outline. Well, first, I intend to spend quite a bit of time introducing the system and talking about general properties, perhaps the expense of giving you some technical details in the second half of the talk. But I can promise that there will be two different stability results pertaining to two different types of special solutions that we will look at. And one of them, the first one is easier. This is the one where I will probably give you a pretty complete idea how the proof works. And hopefully there will be time to talk about a second type of expansion. So before any of that, let me just tell you that the gravitational Euler Poisson system is the model you would find in a typical astrophysics book describing a star. So it is a Newtonian model. The idea is that the star is a gas or a fluid kept together by gravitational forces. So let me tell you what this talk is not about. It's not about relativistic models of stars. But nevertheless, part of the motivation for studying this model indeed comes from relativity. And I will mention that hopefully at the end of the talk. So as I said, the star is idealized as a compact fluid body. And the word compact is crucial because it suggests that there is a boundary between the fluid and the vacuum. And if you want to model such a problem and if you want to track the behavior of the boundary, you have to allow for that boundary to move. So this causes this gives us a moving star vacuum interface. So what is the model? Well, first I'll tell you the unknowns. The first three are unsurprising, and you're used to them, the fluid density rho and the pressure p, the velocity field u, and the gravitational field phi that you need to couple to the Euler equations. The fourth unknown or the further unknown is the support of the fluid because we are trying to treat a free boundary problem, OK? And the equations, again, you have seen some fluid models this time around. Don't be scared. We will soon forget about this slide. The first equation is the continuity equation. The second one expresses the conservation of momentum. And on the right-hand side, you will see a term caused by the gravitational field phi. And the gravitational field phi is self-consistent. It's generated by the mass density here. So Laplace phi is 4 pi rho. This is the Poisson equation. And these are the equations of compressible fluid dynamics. On top of that, we have to supply this problem with some boundary conditions. And remember, this boundary is moving. So the kinematic boundary condition is the natural statement that the boundary is moved by the fluid particles. So the normal velocity of the boundary is, in fact, the normal component of the velocity field at the free boundary. And the pressure vanishes at the boundary, OK? Now, if you count the unknowns, you will notice that there is one superfluous unknown here, namely the pressure. There is no evolution equation for pressure. And you need to prescribe a further equation or a relation to actually close this system. There is one equation for rho, three for u, one for phi, but there is nothing for pressure. And this is typically done by prescribing the so-called equation of state. Now, equations of state is something that, in the physics literature, you would look at a particular model of a star. And you would hope to have a good understanding of what the equation of state is for that particular star. What is commonly done, people study the so-called polytropic equation of state where p of rho is rho to a certain power gamma. And this gamma is the so-called adiabatic index, and it's allowed to vary between 1 and 2. So the way I would like you to think about this model, I would like to think of this nonlinear problem as parameterized by gamma. OK? And as we vary gamma, the qualitative properties of the solutions of this system will change. OK. So let me comment a little bit on the literature. The vacuum interface itself is known to cause severe analytical difficulties in improving even local well-posedness. So the basic result in the theory of compressible fluids with vacuum interfaces are basically these two results. They came independently in 2010 by Couton and Scholar and Jang and Masmoudi. And so what they have to contend with is a certain degeneracy. I will explain it as we go through the talk that is caused by the presence of the vacuum. But one of the key insights is the use of the Lagrangian coordinates. So if you use Lagrangian coordinates and if you rephrase the compressible Euler equations surrounded by vacuum, you will discover that there is a wave equation, a quasi-linear wave equation lurking in the background. OK. So that's somehow one of the big revelations. It is a degenerate wave equation, degenerate in the sense that I will specify later. But the word degenerate has to do with the fact that some of the coefficients of the wave equation will vanish at the vacuum boundary. And another key assumption in these works is that of a physical vacuum boundary condition. So here I'm stating what it is. It's a statement that the fluid enthalpy, which is this quantity, rho to the power gamma minus 1, has to have its normal derivative at the vacuum boundary, has to have a sign. This is perhaps not a well-motivated condition. So physically, it expresses the fact that particles have to accelerate in the normal direction at the boundary. But I will soon actually motivate this. Sorry. I will soon motivate this condition by finding particular steady-state solutions of the Euler-Poisson system that satisfy this. And indeed, if you open an astrophysics textbook, the star models, the steady-state star models that they look at, all satisfy this type of condition, the compactly supported ones. OK, so there is a little bit of a discrepancy here. Here I'm citing works that pertain to a compressible Euler equation. You could ask me, what about Euler-Poisson? Because I'm adding a gravitational field. From the point of view of well-poseness theory, this is a lower-order perturbation. So if you understand how to prove well-poseness here, you will understand how to prove well-poseness for Euler-Poisson system. But nevertheless, there is a reference that I'm missing. Where is the chalk? Oh, fantastic. Thank you. So prior to this work, there's one result which deals only with a priori estimates, where with the Lienblad, Couton, Lienblad, and Scholar. So this is one missing. So on a priori, bounds, Couton, Lienblad, and Scholar. And for the Euler-Poisson system, this came much later entirely based on these ideas. There's a work of Luol, Schinn, and Zang. OK, this is for it. Not for the vacuum interface. So for the liquid case, when the density is not vanishing at the boundary. So this is a significantly less degenerate model. OK, so instead of I will now try to give you to do three things for you. I will give you an example of this famous class of steady states known as Lane-Emden-Stars. Then I'm expecting a psi of relief. I will produce an invariant rescaling for this problem and explain a certain notion of criticality that enters. And that will motivate the main results. So most famous class of special solutions are the steady states. And the way you produce them, you set rho to be only a function of x. And u to be 0. So you plug it in. And the whole equation reduces to this statement here, gradient of rho star gamma plus rho star gradient phi star equals 0. So if you use the fact that this satisfies the Poisson equation, and if you call this object here, this is the fluid enthalpy. If you call it w, then in spherical symmetry. And this is a very classical stuff. You will see this in astrophysics textbooks. You reduce the problem to solving this ordinary differential equation. And the key question is, for what powers of gamma can I find a compactly supported solution with finite mass? This is your star. So the answer is, if gamma lies in this magical range, then you can find profiles that are indeed compactly supported, say, if it supports 0, 1, and they look exactly like that. Once gamma hits 6-fifths and below, no such solutions exist. It's just the ODE analysis. And moreover, these special solutions in fact satisfy the physical vacuum conditions. So w prime at 1 is strictly less than 0. So the density profile is not smooth across the vacuum interface. And this is one of the critical points that works of Couton, Scholar, Jang, and Nassmoudi had to address, the fact that the non-smoothness is a generic part of the theory. So you have to find a way, what does it even mean to differentiate the equation in a proper sense? You can ask, however, and this is what physicists did, are these solutions stable? So now the numerology gets even more interesting. If gamma lies in this range here, the lane end and stars are linearly unstable. And if gamma, sorry, this is wrong. And if gamma is bigger, equal than 4-thirds, it's in fact linearly stable. So it's just some basic linear analysis. Of course, it begs a question about two questions. Are these results, can they be upgraded to a nonlinear statement? So the first result can, it is not terribly surprising. Once you have a growing mode, which is indeed the main source of instability in this regime, and once you have a good well-posedness theory, then you can prove nonlinear instability. And this was shown by Jang in 2014. Nonlinear stability is a much, much more contentious point. So there is a conditional nonlinear stability result in a very, sort of a fairly weak topology generated by basically just the energy by Ryan from 2003, which says, if the solution exists, it will be nonlinearly stable. However, there is absolutely no reason why the solution should exist. Why small perturbations of these steady states should. And it's in the same regime, 4-thirds to 2-thirds. Pardon? The result of Ryan, is it in that regime? Right. So the result of Ryan, the topology is so weak that it doesn't really see the free boundary. In particular, this type of stability result, if you think about it, will actually allow you to have support splitting up and things like that, the topology will not see it. So you really need something else. And in fact, I will talk about it at the very end. I will highlight this as one of the important open problems in this development. OK, so let me try to perhaps explain these exponents that show up in a different way. And this hopefully will help you appreciate them. There is an invariant rescaling of this problem. To each gamma, you can associate this invariant rescaling, where you just work it out, and you realize that if rho and u are solutions, then rho tilde and u tilde are also solutions. And the way you rescale the time depends on gamma. So really, the value of gamma has a lot to say about the structure of the solutions of the equation. In particular, the pressure and the potential, you can work it out from these how to rescale. OK, the second ingredient that I'm sure you will appreciate, what are the conserved quantities? Well, the mass, this is easy, the integral of the density, and the total energy. Let's just briefly talk about the total energy. This bit here is the kinetic energy. This object here, remember, rho is Laplace phi. So if you integrate by parts, this becomes negative gradient phi squared. So this is just the gravitational energy, which gives a negative contribution. And this thing here, rho to the power of gamma, is some sort of thermal energy of the star. OK, and the question that is quite natural in this context is, what is the behavior under the rescaling? And then you discover that the mass remains in variant exactly when gamma equals 4 thirds, and energy remains in variant exactly when gamma equals 6 fifths. And these are the two numbers that are popped up here. So this is to suggest that this numerology has a special meaning. All of the results for the remainder of the article of this presentation will pertain to the mass critical case gamma equals 4 thirds. I said article because the actual article is supposed to appear tomorrow on the preprint server on the archive, and there are several spots where we said and erased all the results for the remainder of this article pertain to this case. Anyway, but OK, this is the key point to remember. So gamma equal 4 thirds is the mass critical case. I'll tell you why physicists like this particular value of gamma. You see, it's a rescaling that keeps the mass preserved. So it's a very appealing to think of gravitational collapse or expansion, and some process that sort of cascades through these scales, lambda, but it preserves the mass. It's very natural. So the beautiful thing about this is that, in fact, there is a result of Goldreich and Weber, who were physicists in 1980, where they found explicit examples of collapsing and expanding solutions of examples of stars in this mass critical case. And now let me do some computation. So if gamma equals 4 thirds, and if my self-similar rescaling is this, so you can work out what this coefficient is here. So 2 minus 4 thirds is 2 thirds. 1 over that is 3 halves. So this becomes p over lambda to the power 3 halves. So to have a self-similar collapse or a self-similar rescaling would correspond to having a collapsing solution whose radius scales like t minus t to the power 2 thirds, or an expanding solutions that expands at the same rate. This would be the corresponding coefficient if you wanted to call this a self-similar collapse. However, if you follow through the paper of Goldreich and Weber, already implicitly there and then explicitly in the works of Makino and Lin, you will find a second family of expanding and collapsing solutions that expand and collapse at the linear rate. And this was part of the mystery for us to understand, first of all, the reasons why this is so. Yes? Which one corresponds to the explicit solution also? This one. This is not explicit. This is just, you see, I'm putting this sign here. So this is informal. In a few slides, I will give you the exact formula. There is a correction term. For this guy, there is no correction term. This is perfect. This is corrected. By a log? No, no, no. It's just something linear. You will see. You will see. No logs. Well, we'll come to that at the end of the talk as well. So there are two families of such. There are two families. No, no, no, no. When I say self-similar, I mean this. But there are two families of collapsing and expanding solutions. For whatever reasons, in these works, you will never read this off. They will never talk about the rates. They sort of analyze this and they say there exist these solutions. But if you really look at the rates, this is what will happen. I will give a more precise statement later on. So this is an informal statement. Informal statement. So don't ask me for any norms and so on at this moment. This will also come later. The basic result is that these self-similar expanding profiles, so I'm not talking about collapse, the ones that behave like 2 to the power 2 thirds, are co-dimension 1 stable. And the linearly expanding guys will be just stable. In this case, we can actually talk about asymptotic stability in this co-dimension 1 set of perturbations, whereas here, we do not characterize the asymptotic attractors. There are just some objects that do not necessarily belong to some sort of nearby linearly expanding profile. This is the gist of the results. I will get into the technical statement slightly later. In particular, these are, to my knowledge at least, first non-trivial examples of global solutions for this free boundary problem. And what is very important for us, we can actually say that the support of the star, of the nearby perturbed star, actually grows at approximately the same rate as the underlying background solution. So now let me take you back to what ideas go into the well-posed in this theory, and this will bring us then to the proofs of these results. As I mentioned, the fundamental idea in this work is to use the Lagrangian coordinates. So you've seen them, eta is simply the flow of particle trajectories. And because I will now make an assumption of spherical symmetry, it is actually not fundamental to these results. But I want to make it because it will display, in a very concise way, the fundamental structure that we need to worry about. OK, so assume everything is spherical symmetric. Now maybe it's slightly pompous to say structural miracle, but it is a beautiful structure of these equations. If you use Lagrangian coordinates in spherical symmetry, you discover that chi, which is now my Lagrangian map, satisfies a second order equation, chi tt plus fw of chi equals 0, where this fw is some nonlinear operator, which is of second order. And this is an effective quasi-linear wave equation on a compact domain. Now there's this thing here. I don't want to display fw because it's not beautiful. But I will tell you later how it linearizes. And you will be convinced that it is a second order. It's a second order self-adjoint operator in a certain space. Is this an artifact of the fact that this would be irrotational? Correct, absolutely. So now that you raised this question, so if it were not spherical symmetric or irrotational, if it were pure, if it were the full system, the flow map eta would satisfy an effective wave map type equation. The divergence of the flow map, which roughly corresponds, which captures the irrotational part, would satisfy any equation like this. So divergence of eta t of the velocity vector field. Whereas the curl would satisfy because, OK, so let me explain this, because remember in the Lagrangian coordinates rho dt u, where u is now pullback with respect to the t is my Lagrangian time and u is my Lagrangian velocity, plus the pullback of the pressure would be something. And your curl of this guy vanishes. So this would give you, if you wish, a transport equation for curl and the sort of estimates of curl of u for free. So the idea is that when you deal with the full problem, you estimate the divergence in the curl in two separate ways, OK? So you have a, but you can't decouple it. Absolutely, yeah. So this is in some sense at the heart of the approach of Couton, Scholar, Jeng, and Nussbaum. The framework, the functional framing is slightly different, but this is the main idea. OK, so there is a wave equation. That's my point. And it's posed on a compact domain because you pulled it back onto the, to fix the free boundary, you had to pull it back onto something fixed. So it's a wave equation on a compact domain. So let me now re-derive, really, those solutions of Goldreich and Weber and other people. I will call them homogeneous solutions because I will make the ansatz that chi is just lambda of t. It doesn't depend on r. Now this will tell you this ansatz will reduce the previous equation to this thing here, lambda squared, lambda double dot, plus something which doesn't depend on t equals 0. So you can separate variables, and if you do that, you discover that there is an ODE, satisfied by lambda. And there is another ODE in r, this time satisfied by w. You are interested in finding compactly supported solutions because that's what qualifies as a star with a vacuum interface. And you can do that. There is a magical value of delta star such that for any delta bigger than delta star, you can find compactly supported solutions to this equation. And this is just an ODE. You can really, for a range of values of delta and initial conditions, you can really classify and see what are the rates of collapse or expansion and so on. So I will do that for you in a second. As I said, you can solve this ODE, prescribe the initial conditions. There are three parameters in the problem, the delta that I mentioned, lambda 1, initial velocity, and lambda 0, the initial radius of the star. There is a conserved energy associated with this ODE. It's trivial. It's this quantity here, lambda dot squared plus 2 delta over lambda. In the Eulerian description, so you can always go back and recover the original solution in the Eulerian description, the density and the velocity will look like this. So the key point, the density is like lambda to the minus 3. So if lambda is shrinking, density is blowing up. If lambda is expanding, the density, the fluid wants to tear itself apart. It's decaying. And remember, as I mentioned, self-similarity would correspond to these numbers here, 2 thirds. So it turns out that self-similar solutions exist, but only when the energy is 0. If energy is not 0, they're not going to obey the 2 third law. And instead of solving the ODE, I also could not resist not putting this on the slides. This is a kind of a bifurcation diagram that explains what's going on. So this is a delta axis. This is the lambda 1 axis. And I'm fixing initial radius to be exactly 1. So what you see here, the light blue region, this is the range of parameters that will give you always a linearly expanding profile. So a profile that behaves asymptotically like some constant times t. These guys here, the thick blue line, this is exactly the self-similar expansion. And there is an explicit formula that I will give you for them on the next slide. But you have to insist that the energy is 0. So they're obviously unstable, because if you can go either way from the surface, the question is if you look at perturbations of energy 0, are they going to remain stable? And then you have the collapsing. So as you can see, most of the collapsing solutions in this range here exhibit self-similar collapse. And then there is this lower dimensional line where the collapse happens at a linear rate. So this is the structure of this space in this two-dimensional parameter space. And the basic results are this light blue region is stable, and this thick blue region is a co-dimension 1 stable. So for this line, you have an explicit formula, which is just a statement that E equals 0. Do you have a question? And below also. You mean here? Yes. Well, I can tell you a very weak statement. I can prove that these are non-linearly stable in the self-similar frame. But this is not a good statement. You really need to. I don't know whether. So for instance, I don't know if you perturb an element here, whether it will collapse. OK, so this is an open problem. I will state it at the end as well. Is a linear rate of expansion the physical one, the one with astrophysics, not interesting? So Goldreich and Weber completely discarded this for whatever reason. And they were interested in collapse and expansion at self-similar rate. If you look at other physics literature, they actually mostly don't want to deal with the free boundary. They are aware of the difficulties that come with that. So in their literature, they think of Goldreich and Weber as the only example where you have a sharp boundary. In most cases, they are worried about constructing some type of solution that exhibits either collapse or supernova or the expansion, where you have infinite support but some sort of decay at infinity. This would be, for them, good enough. And this also leads to good problems because most of it is also not rigorous. So part two, how to study the stability of the expanding guys. This is the formula that Frank, I believe, asked me about. This is the explicit formula for self-similar expansion, whereas you don't have explicit formulas for the linear expansion. They are sort of approximate. But they are to the leading order just this. So I will, from now on, focus only on the expanding profiles and perhaps pompously call it supernova, but let's just say expansion. And I want you to recall that they satisfy a wave equation of this type. And FW is some second-order operator. Now, how to study this object? Well, you adapt your unknown to what you believe to be stable by dividing by it. Basically, you define psi to be e divided by lambda bar. And then you expect psi to be a small perturbation of 1, which happens to be a steady state of this equation, in this case. So if you make this ansatz and plug it back in here, you get this equation, second-order PDE. Now, how to deal with this? Well, self-similar expansion really behaves well with respect to the scaling and variances of the problem. So the natural thing to do is to rescale time with respect to the corresponding self-similar rescaling. As you will see, the linear expansion doesn't sort of, the equations don't digest it very well. So it will cause certain difficulties that will not be present in the self-similar case. OK, so let's move on to the disk. Are there any questions about this? So is it absolutely clear what's going on? So let me tell you the basic structure that will allow you to understand the stability. So let's now focus purely on the self-similar profiles. So remember, let me write it down. Self-similar profile is something that expands like t to the 2 thirds. So this is the radius of the star. OK, xc is a steady state of the equation on this slide. So I will write xc as 1 plus a perturbation. xc equal 1 is a steady state. You do that. You demand that the initial energy is 0, that the energy of perturbation is 0, energy is conserved. So this is a conserved quantity. And of course, the key idea, as I mentioned, is to pass to the self-similar time. If you do that, so this roughly means that the new time behaves like log t, because lambda expands like 2 thirds. If you do that, you discover that this coefficient here is constant. I call it b to honor some of the notation that's popular in the literature. This quantity is a constant which is strictly less than 0. What about the nonlinearity? So you expand the nonlinearity around 1. And you discover, and this is now the point that will perhaps clarify one of the questions, you discover that in the linearization of this operator is precisely this second order elliptic operator here. But now I want you to stare at it for a second. Notice that when r is 0, this term vanishes. And when r is 1, remember w delta is the density profile, is a density profile, which roughly looks like this. And it also vanishes at r equal 1. OK, so in this sense, this operator is degenerate, because it has these vanishing weights, if you wish, at 0 and at 1. This is not surprising. This is the type of difficulty you will see in the study of just pure Euler. But it is suggestive. It tells you what sort of functional framework to use to, in fact, deal with this problem. It tells you that you need to use some type of weighted spaces. Now the cool thing about this, if you pass to the self-similar time and you express, you now try to find the equation for phi, this is what you discover. So phi ss, the second derivative of phi, plus a strictly positive term times phi s, minus something negative times phi, plus this linearized operator equals the right-hand side, which is nonlinear. OK, and you immediately see, realize what should be the mechanism for stability. It is precisely this blue term here, which acts as an effective damping. This is a consequence of the fact that we are expanding our profile around an expanding solution. If I did it around a collapsing solution, the sign here would change, and this would be sort of the opposite of damping. So it's a wave equation, quasi-linear wave equation, on a compact domain, but with a disdamping term. So the correct statement is that there is a high-order energy. If it is initially sufficiently small, and if the physical energy initially is exactly 0, then there is a global unique solution to the above equation, and in fact it decays exponentially fast in the s variable. In the t variable, this translates into some sort of algebraic decay. How? The high-order energy is bound to the cation theory, I guess, right? Precisely. This is what you need. This thing, it's you, the derivative, right? Thank you for the question, because I now, in next slide, I will write down the energy. The L delta is a self-adjoint operator with respect to this weight. Remember, this vanishes at 1. This vanishes at 0. Spectral gap, OK, 1 is in the null space of this operator. So if you're orthogonal to the null space, you're good. You have a coercivity of this operator. This per se is obvious. What I do want to pay attention to, this first term is just the L2 norm in the corresponding weighted space. If you want to control the first derivative, you have to raise the level of degeneracy in the weight W delta. So the derivative gets first? Correct. Correct. So this is a critical observation? Not in one, each other. No, it's only in this direction. Absolutely. Excuse me, the B you mentioned is a constant? It's a constant. It's a constant. B is a constant due to the because of the self-similar structure. Because I'm expanding around self-similar solutions. So it really captures that. This is the correct information about the self-similarity that you need to address this. Term is obviously the good guy. This term here, this is the bad guy. This will give you exactly one unstable mode. But this is not surprising. I told you that it has to be unstable. Remember, the self-similar expansion was a co-dimension one phenomenon. So clearly this has to be the case. But this unstable mode, which is caused by this negative term, is remember, we are starting with zero energy data. And so the unstable direction you can check this is transversal to this zero energy surface. So you can really control the instability, which translates in this case in controlling the inner product of phi and the generator of the null space. So this allows you to carry on the estimates. So think of this as the relationship that you get by linearizing energy equals zero. This is the quantity that you will control just by doing some simple energy identity. And this is the quantity that you want to control to get a spectral gap. Now answer to Pierre's question. Indeed, the energy is motivated in this case quite concretely in the second case more philosophically in the case of linear expansion by the works of these people. And now we come to your observation. Indeed, I don't want you to gain any intuition from this. All I want you to see is that each time I take a further normal derivative, I have to raise some index in my weight. And this is how this works. These delta k spaces are basically mean you add a k-fold power of w delta to your weight. And this is the only way you can do it. So more spatial derivatives implies more degenerate weights. And now you will trust me if I tell you how the energy method works. You try to prove something like this. This term here kind of comes from that damping term that I mentioned. Of course, you need to control the energy by this dissipative term here. You can do that because of the spectral gap property. It's a technical thing, but you can do it. And this, in principle, gives you the exponential decay and completes the proof. Perhaps I want to say, going back one slide, this is a mixed space time norm. In reality, you do the estimates only with the time vector field because it commutes with the equation entirely. And then you use elliptic estimate to control the higher space derivatives with the time derivatives. So this is what I just said. I used the time derivatives to build an energy, elliptic estimates. Now what are the technical tools that go into closing the estimates? Remember, there is a non-linear right-hand side, which actually looks quite dirty when you write it down. So the tools are hardy inequalities, which exactly allow you to deal with those weights that degenerate at 0 and 1 and embeddings between weighted solvable spaces. If you take enough derivatives, you can control lower norm by the L infinity norms of lower order terms by the energy. So I will not go into any of this. This is technical. I have eight minutes left. So let me then say just a few words about this second problem here, and maybe try to hint at least at why is it more difficult in the first one. The difficulty appears not to be conceptual, but more technical. But it is there, and it took us some time to understand. So the reason why this problem is harder conceptually is because you cannot, it doesn't honor the self-similar structure of the problem. It doesn't honor the scaling invariances. The solutions are there, but how do you prove that they are stable? So it will necessitate the use of a, so first of all, the fact that it is not honoring the invariance will disallow us from using time vector fields. As you will see, they will not commute well with the problem. Because now when I write down the equation for my perturbation phi, it will have time-dependent coefficients. So we have to come up with a correct spatial derivative that will capture the degeneracies at the vacuum. And this is precisely this vector field here, which just happens to be a five-dimensional Laplacian. But this is a coincidence. OK. So what we do, we rescale time so that tau, the new time, becomes log t again. So this means dividing by this linearly expanding profile. Let me write it down here. So lambda tilde of t, think of it as something that behaves like a constant times t. And if you write down the equation for the perturbation, now you discover that you have these tau-dependent factors in front. And lambda tilde is growing exponentially in this new variable. OK, so you have, let's say, e to the tau plus e to the tau plus this stuff equals right inside. This is still causing a certain damping effect. It is well known that the expansion produces a sort of a stabilizing effect. It's reminiscent of works, for instance, in the general relativity, when you have cosmological constant, for instance. And this is the statement. There is a high-order energy, e tilde, that will allow you that will give you global existence for small data. However, you cannot classify the asymptotic attractor. So all you can say is that the tau derivative of the solution decays, but you cannot say that in the limit, you belong to a nearby member of the linearly expanding family. This energy is constructed by using powers of this elliptic operator that I just wrote a second ago. And so to be honest, I can talk about it. I have a few more slides at the end, but let me not go into the technical aspects of this proof. Let me just say that this operator is designed to capture very precisely. So in other words, you cannot just blindly apply dr derivative or dx. If you do that, you will create singularities at zero and they are hard to handle. So you have to do it with a particular combination. And this operator S is a reflection of the structure of that weighted-generate operator L delta. And let's leave it at that. So I will not go into the technical details. Yeah, that's a good question. So it's just something that sits nearby, but it is not one of the proof. Yeah, you can prove that the time derivative of that is decaying. So that gives you convergence to something. Yeah, and it's nearby. It's actually reminiscent of a result I heard recently in a talk by Falker. Falker Schluhe. And sort of in these expanding space times. Who's here, I believe. Yeah, very good. Let me spend the remaining three minutes discussing, perhaps, questions that I find very interesting that are open. So here's this kind of a silly lemma. I call it lemma, and perhaps it's too ambitious to call it lemma. Do there exist collapsing or expanding compactly supported stars? So this is the key for other values of gamma. They should exist in the supercritical range, which is between 6 fifths and 4 thirds. This is an open question. What you can show, however, is that self-similarity and a spherical symmetry cannot coexist for this type of collapse. Gamma being 4 thirds is absolutely critical for the two to be able to combine the two. And this is a very silly thing to show. It's just some conservation law type calculation. So the question is, can you construct axisymmetric self-similar collapse in this mass supercritical range? I think this is a very interesting problem. And now we come to the question that I believe Sergio asked. Are the Lane-Emden stars that I mentioned before in the subcritical range, where they are shown to be conditionally stable, are they, in fact, stable? So a possible mechanism, and you will see this discussed sometimes in the physics literature, is some type of shock formation. This is not known, and this is really, really important. So let me remind you of this picture. The big, the pink elephant in the room is, of course, this region here, the collapsing region. So as I said, it is not difficult to show that it's nonlinearly unstable. The moment you have a correct well-posed in this theory and you have a growing mode, it's very easy. So again, this should be a lemma. But it's really unsatisfactory, because it doesn't tell you anything about the real instability. In particular, it could it be that the nearby guys decay, collapse at perhaps a corrected rate? Do they collapse at all? So this is just in spherical symmetry. You don't have to go outside spherical symmetry to ask these questions. And finally, I would like to point out this viewpoint that I believe studying this family of problems is really the right thing, because you really want to have a robust understanding of how equations of state affect the collapse. This is partly motivated by general relativity. If you think of the most famous example of a collapsing star in general relativity, it's the Oppenheimer-Sneider solution, which corresponds to the case where you don't have any pressure. So by a result of Christodulo from 83, I believe, you can show that these are generically unstable solutions. If you go outside the homogeneity class, you generically form naked singularities. So you want to, well, you can devise the various conclusions out of that. One conclusion that I like to draw out of that is that studying non-trivial equations of state is an important problem. And this is just the Newtonian version of the problem. But realistic equations of state are, this is just a final comment, is in principle quite a hard problem, even for physicists. So this is my favorite relativistic astrophysics book by Zeldovich and Noviko, where I sort of learned this stuff about. And they discuss one of the sort of high-density regimes for the equations of state. And here's what Zeldovich says. So it reminds him of Averchenko's parody. And it is precisely like this. So if you look at the astrophysics literature, there's a huge disparity between what people think is the right equation of state. Anyway, yeah, so think of it as a teaser trailer for this wonderful textbook because it's a very serious book. But as I browse through it every now and then, I find a gem of this type. Thank you. Questions or comments? In case we need the answer. So it seems the non-year-expanding rate is unstable. Does it mean that there's a type of solutions which will converge to the stable one, which is a linear one? OK, so if you generically perturb a self-similar expanding rate, if you generically perturb it with perturbations that have positive energy, then they will converge to the linear guy. So it's generically unstable. And these guys are stable really only in the co-dimension one sense. The linear is the stable. The linear takes over, not the self-similar. Yeah, yeah, exactly. So do the faces know this part? It should be of great interest to them, I guess. Well, if you're really. The supermodel should be linear. Yeah, exactly. So if you read Goldreich and Weber, which was one of the starting points for this project, for some reason they really say we will not be interested in these linear rates. They drop a constant of integration in a second order ODE, which allows them to go down one level and get an explicit solution, which is this. And they literally say, ah, we kind of don't want to look at that. So but this is the finding. I mean, part of the mess was also to understand the exact structure of this parameter space. Yes? The non-linear unstable result covers the whole regime of gamma written there? Actually, it actually goes up to here. I did not tell you what happens at 6-fifths. At 6-fifths you still get a steady state, but it is supported all the way up to it's infinite. It has an infinite support. And it turns out to be, so in this case, the steady state is exactly the steady state of the critical Quintic wave equation. So it's one of these guys. I don't know, there's some constant here. And then I believe one over two, something like that. So this is the rate of decay of this guy. So it is the most complicated case before the last week? In this work, well, it's quite different, because here you are working on a compactly supported domain. So you really have to rely on the well-posed theory for vacuum interface problems. Here it's different. Here the weights are at infinity. So it has a flavor of it, but it's quite different. I'm not sure how to. So this result is also due to Jang. So this equality case, all of this is due to Jang. Sorry, in this range, I have to be careful. This is the instability range. What this result shows in a way is that the fourth-third case, which is linearly stable, is, in fact, widely unstable. So the result is radial from the stability? This is entirely radial. So the paper that will show up is entirely radial, but it holds also for non-radial perturbations. So I sort of alluded earlier to that. There's a reason why the radial case is simpler, not just for obvious reasons, but also because it reduces the whole problem to just a quasi-linear wave equation. When you work with a non-radial case, so the fluid is not irrotational anymore in general, so you have to account for the vorticity, which does not have this wave-like structure, but it satisfies a certain transport equation that allows you to control it. Well, what makes the vorticity better than in the standard order equation, where you know that you cannot control the vorticity? Is it because of the? Well, it's a local in-time result. I mean, but. No, this is drawback. Yeah, yeah, so all I'm saying, I'm not sure if I understand your question, but all I'm saying is that. Right, I'm talking about the stability, the long time. Ah, for the long time. For the long time result, you don't have a long time. Oh, if you haven't produced it yet, no, no. So for the irrotational case, it will be straightforward. And for the vorticity in the long-time result, I don't want to make the claim yet. I don't think. Because I don't see why it should be any better than. Than in the regular case, yeah, maybe. No, no, you have this damping effect, so it will be better. It will be better. But will affect the vorticity? No, no, you have, for instance, for Friedman, LeMette, Robertson, Walker space times, you have a stability without irrotational assumption. So when you have a cosmological constant, for instance, and you perturb away from that, you see this, you can control the full flow. Of course, this is not a free boundary problem, because this free boundary problem in GR is more. Okay, but the mechanism is not the mechanism. Yeah, yeah, you will kill it with damping. All right, so thank you. So we start again at 11.30. Thank you. Thank you.