 Yes, I very much appreciate the invitation from the organizers to come here and visit ITS and be able to tell you something about this work. I don't think the organizers knew about this topic because I hadn't started it, but at the time I got the invitation. So it'll be a bit of a surprise. It's a bit intimidating for me in this audience because I think about every expert in the world having to do with the initial value problem for the water waves is here and I don't know anything about that, so there's no dispersive estimates, but I have a kind of different point of view on the subject. So it'll be quite different, I think, from a lot of talks in this summer school, but nevertheless it's, I think, a perspective that's interesting. So yeah, it's really related. So the title should be about Least Action Principles, and those things were brought into the subject by Vladimir Arnold in the 60s. And really what I'm doing here has to do with discussing how least action principles for incompressible flow are related to least action principles that determine optimal transport distances. And here I'm focusing on the problem involving comparing shapes. And so let's see. I wanted to start with a bit of motivation, a bit of a long prologue, just almost just a minute really to say I didn't come at this problem by starting to think about water waves. We were thinking really about modeling animal groups, things like swarms and things like that. So the idea there is animals kind of wanting to maintain a distance between themselves and a kind of continuum limit level, there's a kind of constant density. And we were interested in various models to do with that, flocking models, gradient flow models. And we're going to try to understand the geometry of configurations like that. So another place where this kind of geometry shows up is in understanding how images deform. So this goes back way back to work of Darcy Thompson to understand evolution somehow and lately, since after Arnold's work on incompressible flow, a subject called computational anatomy arose, and that has to do with putting Riemannian structures on spaces of images, spaces of diffeomorphisms, and trying to understand the geometry that goes on there. So there's kind of a history of work there. My colleague, Dayan Slipchev, actually has a pattern to do with implementing these kind of methods to kind of analyze cancer images. So it's a kind of a big subject. Okay, so go back, assuming not everybody remembers differential geometry, just some elementary things really. So the length of a path into Riemannian manifold, the so-called metric on the manifold is an inner product on the tangent space at every point. So the length of the velocity vector is given in terms of that metric, and the total length of a path is the integral. So Cauchy-Schwarz tells us, okay, the square of the thing inside is an upper bound for length squared, and this action, if you minimize it, you have to have a constant speed geodesic, a constant speed distance minimizing geodesic path. So later I'll be talking about geodesic paths. These are critical paths for the action, so the variation of action is zero, but not necessarily distance minimizing, so that's a distinction I'll maintain. Okay, so least action principles, they go back a long way. Of course, one of the first ones one thinks of is Fermat. So here I'm in just an Rd, and there's not really water and air where the light bends going from trying to get from point A to point B in the fastest time, but really the trivial case in which if you're trying to minimize kinetic energy squared, the action, then particles want to go in a straight line from source to target. So here we're going to have a continuum, Z is a label for the particles in a setting Rd. So I think of the, let's see, capital X as a flow map at time t, and let's see. So the push forward image, I'd like to think of that as the image. So let's see. So Arnold said, well, if we require incompressibility for this flow map, then actually we compute the geodesic equations and they actually correspond incompressible Euler fluid flow. So he's working in a fixed domain for that, typically, but okay, it hardly matters. We'll see, well, I want to do a calculation about this a little bit later. So minimize the action then subject to incompressibility is a thing you could try to do to construct solutions. And Jan Brania in particular did a lot of work to try to understand how well that can work. And we'll see some variation of one of his approaches a bit later. But something that turned out to be extremely successful was to drop the incompressibility condition and specify not the positions of all the particles at time 0 and time 1, but the Eulerian densities, basically the images of the diffeomorphisms, the shapes. So if you specify only that, you end up with something called Wasserstein distance or Manj Kantorovich distance related to optimal transportation. And of course, as many people know, this subject has just burgeoned enormously in the last 20 years and there are many, many results in this direction. So in a sense what we wanted to do here was to try to impose the endpoint conditions for minimizing action in terms of Eulerian densities, but try to maintain the volume-preserving condition of our null. So we should have incompressible flow and yet we won't get the full Euler equation. There will be some kind of extra freedom to do with rearranging endpoints of particles. It won't matter to me how the particles are arranged as long as the density is the same. The reason for the green dot over here is because somehow all these other cases, okay, they're either trivial or just not much that you can say about them. What in a sense we show is that the completion of looking at this last possibility is the Wasserstein distance. So in a kind of deep sense the Wasserstein distance is kind of the good theory in this setting. Okay, so what does Wasserstein distance say? So it's somewhat, not quite the most general form, but you specify two measures, I'm assuming here they have densities, rho zero and rho one. They should have equal total mass because we're going to be transporting with a continuity equation preserving mass. And let's say compact support for convenience. So what that's computed by according to the result of Betemou and Brenier is you minimize the action, you minimize kinetic energy, according to the Eulerian form it looks like this, rho v squared, where the densities as a path of function of t are transported by a continuity equation. You fix the endpoint conditions and v just has the regularity needed to make the action finite. So they showed that Wasserstein's distance is characterized this way by minimizing kinetic energy and of course well as we'll see in a little bit the density is not constant. It's not incompressible. Nevertheless it has a lot of wonderful and amazing properties. One of these is that if you consider measures of constant total mass supporting the fixed compact set K, the distance you get this way actually metrises the topology of weak star convergence. So it's a complete space and you have a lot of nice metric properties. So we're going to try to restrict to characteristic function densities, shapes. So we're only interested in what we can see in a sense. And so we'll define, I'll kind of overload the notation, the induced Wasserstein distance on sets is the corresponding distance between characteristic function densities. And the first qualm you should have about that is that the set of characteristic function densities is not weak star closed. So that's we're going to see that come up again. Okay, so one more slide about optimal transport distance. It's very well understood in RD and in flat space what the optimal transport paths look like. Now the particle paths follow just straight lines. There's no incompressibility condition, it's just Fermat, I just think things follow straight lines. The extra information though that Bernie added to the theory is that, okay, you start at position z and you end up at position capital T of z, this image map actually turns out to be the gradient of a convex function. So that's a consequence of his polar decomposition theorem from 25 years ago. That convex function in this case that we're talking about here is unique and characterizes this optimal transport path. So those are the length minimizing GEDs for Wasserstein distance and I'll drop the length minimizing when I talk about Wasserstein. So those are the important things. So let's see, so the density then is the inverse Jacobian of the flow map and the velocity straight line transport is constant along particle paths. So that means that the system rho v satisfies pressureless Euler equation. So density is transported by continuity, velocity is advected by the Eulerian flow, so with no pressure. So here's a little cartoon of what happens when you go from a short fat rectangle to a tall narrow one, it doesn't just rotate, it follows straight line transport from source to target and in the middle there's this kind of big rectangle. So you can guess that a density one here and density one here, the total mass is conserved and it'll have to be a small density in the middle. And that's an example of a more general phenomenon which is that along the particle paths the mass density is log convex. So it's in particular convex and so for my endpoint densities that are characteristic functions it'll be stuck between zero and one. Actually the proof is you don't actually have to compute, it's just you take the log and minus the log of a linear function is convex, that's all that's really going on. Okay but those things are properties I'm going to be using. So now the distance that we're trying to study is a little more restricted. Okay so we're looking at actions, so Eulerian form like this and we're required that along the path we have, call them shape densities, characteristic functions of sets that are deforming according to L2 velocity fields. So there's a continuity equation, it holds in weak form and we want to define the shape distance to be the infimum of the action. So we'll fix the initial and final states, source and target to be characteristic functions. And we were motivated in this by some work of some people in actually in the image processing community who tried a similar idea. Well so we've imposed this constraint that here it is, that along the path, not just at the beginning and the end, that the density is a characteristic function and so the admissible set is smaller than it is for Vosserstein distance. So the infimum is going to be no smaller, it could be bigger. So that's the first thing we observe, in Vosserstein distance the path is free. Okay well so there's a bunch of natural questions about this then so what I want to do is write down the Geodesic equations for critical paths of this action with this constraint. I'd like to understand, okay where can we start and end up? What states can be connected by Geodesics? Can we say something about this gap between shape distance and Vosserstein distance? That's an infimum, the distance is defined by an infimum, is it really a minimum? At least locally. And let's see, and somehow what's the completion of the minimization problem? What's the completion of the topology that the distance provides? Okay so there's a sketch of the answers, well there'll be water wave equations, so there's some nice ones. Kind of surprising that essentially any shapes can be connected by Geodesics, at least approximately, and I'll say what that means exactly. The shape distance turns out exactly equal to Vosserstein distance. There's no gap. Length minimizing Geodesics almost never exists for our shape distance. I think there's actually a fun problem to try to classify the ones that can possibly exist. So that kind of is different than in the case of Euler's equation in a fixed domain. And the completion and relaxation actually is the Benoum-Webrenier form of Vosserstein distance. So somehow in the completion you lose the incompressibility constraint. And that happens essentially due to weak star convergence. And we're going to see that quite explicitly in the construction. Okay that's sort of right now, under the prologue there's a kind of sketch of the things I'm going to really talk about. Okay, so I want to go back then and talk about Arnold's characterization and really derive the Geodesic equations in the context here. Okay, so smooth Geodesics in the lead group of diffeomorphisms of a fixed domain correspond to incompressible Euler fluid flows. So that's the incompressible Euler equations. This is what Arnold showed and what we'll see in a minute. Abin and Marsden used this to do some analysis and some geometry. In particular, they showed a local subjection theorem that if you have a diffeomorphism close to the identity in a high Sobolev norm, then there's a Geodesic that connects the identity diffeomorphism to that particular one. So you can rearrange the points basically in a fixed domain omega using Euler flow as long as the diffeomorphism close to the identity. So it's this local, it's kind of a controllability theorem I guess almost for Geodesics and Brandeis was trying to study how you can use least action to construct solutions or generalize solutions in some sense of incompressible Euler. He's got a long series of works on these things and I'm gonna try to discuss one version which involves a kind of mixture of different fluids which arises in the context where we've got here. So we have, in a sense, we have some explicit examples of how the relaxation works and Euler's and Brandeis work. Okay, so get to business. So they want to derive the Geodesic equations for our shape distance. So let's see, so omega t is gonna be the image domain under the flow map, capital X. And the velocity v is the just the advected Eulerian flow velocity. We start with label z and end up with, well, there's gonna be some target domain omega one that I want at the end. Let's see, so we compute variations of the density, so variations. Let's see, find a little chalk. So just sometimes the notion of variation is confusing. It's some derivative with respect to a supplemental parameter taken at the initial part of some curve of solutions parameterized by epsilon say, the variation is just a derivative. So we wanna compute the variation of density. So we've gotta differentiate. Well, here I'm differentiating log determinant. So we need the differentiation formula for determinant. Abel's formula involves the trace, the trace of the variation of the matrix times the inverse of the matrix. If I switch back to Eulerian coordinates, that's the Eulerian velocity gradient trace. And that's okay, the divergence of the velocity perturbation. Okay, we're interested in motions constrained by being characteristic functions. So, inside our domain that's evolving, velocity will have zero divergence. And so will the perturbations. But that's not true in the weak sense in our D. We have the continuity equation. Okay, so we go ahead and compute the variation of the action. And I basically do it on one slide here. We'll have Euler's equation at the next line. So I just replace Arnold's Lagrange multiplier by Helmholtz's decomposition argument. This makes it nice and clean. Let's see, we take the variation of the action, get this quantity, integrate by parts in time, get an endpoint condition here at time zero, it's delta x is zero. So here's the acceleration, Lagrangian acceleration, pull back to Eulerian coordinates, and that's the Eulerian acceleration. It's dotted with the Eulerian variation of the position. So Eulerian variation of position has got divergence zero. And so let's see, we want to use the Helmholtz decomposition. What, L2, just the classic result, is that what? L2 of a domain is a gradient of H1 functions on a domain. Direct sum with, okay, some space W, so this is things with zero divergence because they're L2 orthogonal to gradients and they should have zero normal component on the boundary because of integration by parts. The zero normal component on the boundary is in, I guess, H minus a half according to classical results in fluid dynamics. Okay, so these are the conditions for the part of a vector field U orthogonal to gradients are v tilde, we'd like to consider v tilde is that have that property. And that means that the acceleration should be a gradient. And that's the, basically, the result of our null. So incompressible Euler should hold with zero, with velocity field zero, zero divergence inside the moving domain. For us, we actually have a free boundary. So we can consider variations of position that vary the boundary. And that's going to force the pressure to be zero. That's a slight white lie, actually. There's a, the integral of v tilde dot n is constant, or it is zero. And so pressure is actually a function of t in the boundary. But I can subtract that off without changing it. So pressure is zero on the boundary. And now, as far as the end, as far as the endpoint condition at time one is concerned, that's what's going to give me that velocity is actually a gradient. So I've got, by, by specifying only the, only the final Eulerian density, only the final shape of the image domain, all I can do is consider variations of velocity that involve the divergence free part with no normal component in the boundary. So that won't change, variations like that won't change the shapes of the image domains. And, but the total action because of our, our, L2 orthogonality breaks into these two pieces. When I differentiate with respect to the variational parameter epsilon, stationarity tells me that the W, W is zero. So, so in this context here, the velocity, Eulerian velocity should be potential. We should have potential flow. Okay, so in sum, actually these are water wave equations. So, velocity is potential flow, divergence of velocity is zero, so Laplacian holds. And this is the Bernoulli equation you get by integrating up once the Euler equation for potential flow, and on the boundary the pressure is zero. So the difference with a lot of work on Euler is on water wave equations is I don't have any gravity, that's just, I'm omitting that kind of potential force in the problem. Let's see, there's no vorticity, I've got great potential flow. Zero pressure and zero surface tension on the free boundary. Okay, so all the experts, almost all the experts in the world are on the analysis of the initial value problem right here. So my understanding which is very limited says that basically the work of the stitch way on the water wave problem with gravity, basically the same results extend to handle this problem. Explicitly the geometry that's considered here was handled by Lin-Baut and later Cooke-Tiner Scholar, they included vorticity. So the initial value problem for this is in some sense on the way to being well understood. But I won't need to solve it. I'm interested in this endpoint problem. So I have a family of solutions though, it's kind of cute. It's straining flow, so I'm meeting along the coordinate axes. We have equations like, let's see, I think in my coordinates here, it's xj dot is, I think it's aj dot over aj, xj. So there's just straining along each of the coordinate axes. And the motion of the, we're starting with a ball, and the motion is going to be an ellipsoid, the ellipsoid written in this way. aj as a function of t are just the major axis, the principal axis radii. So we get these kind of solutions, it turns out, by taking, look at the vector of these radii, these principal axes. And look at their product has to be constant because of constant volume. But take any geodesic with respect to the Euclidean metric in RAD on that constant volume surface. And that determines a solution of the Euler equations. So what's nice about that is you can connect any ball to any ellipsoid with the same volume. So no matter how stretched out or whatever. And we're going to make some use of that. So there's the formulas, the potential is essentially parabolic. So is the pressure, it vanishes on the boundary. And there's this coefficient beta that comes from, that comes from the straining rates. I can't, I refuse to believe that this is, the solution is less than 200 years old. But I haven't found it in literature yet. I think it must be related to Jacoby's self-gravitating fluid ellipsoids in some limit, but I haven't looked at it, looked for it yet. So these Euler droplets, they have a nice property of being nested inside Vosserstein ellipsoids. So starting from, so in orange here are the sort of particle paths of Vosserstein geodesics that connect a ball to an ellipsoid. And in blue are the, is the Euler droplet. So what's responsible for this is that the principal axis lengths turn out to be convex. And for the Euler droplets, and for Vosserstein ellipsoids they're straight lines. Everything moves in straight lines. So the Euler droplet is inside. It's not very much inside, but this is about as far as I can get it inside. But it's inside, I'm going to use this. So now I can say what an Euler spray is. It's some kind of countable disjoint superposition of droplets like this. So I'm going to have an infinite number of them. And we're going to build some interesting solutions out of that. Okay, so here's the result. Take two open sets, two bounded open sets in RD with equal volume. Then there's an Euler spray, which connects the, a source, which is, let's see, up to set of, up to set of measure zero. It's agrees with omega zero. And there's a target which in transport distance is very close to the desired target omega one. So it's L infinity Vosserstein distance. It means there's a map from omega one epsilon to omega one, which moves every point by distance no more than epsilon. So it's close in just in a sense of displacement. And the action of the spray, the total action of the spray, of course the, let's see, this is the shape distance I defined as the infimum over paths connecting omega zero epsilon to omega one epsilon. Of course the actual action is no smaller than that. But it's as close as you like to the Vosserstein distance between omega zero and omega one. So this tells us somehow that, okay, these Euler sprays are in some sense geodesics, they're critical paths of the action. And you can find, you can find a lot of them. You can, there's, there's somehow everywhere in this, in this space of shapes. Everywhere dense in this space of shapes. What this estimate says, actually, if you think about the completion is the following, so by, so by approximating omega one, better and better by a sequence of sets basically. That better and better approximate omega one. I can concatenate those paths and get, and get an exact connection. Not a geodesic connection, but an exact connection between omega zero and omega one. And I find out that the, that the infimum of paths connecting omega zero and omega one is Vosserstein distance. So there's no gap. So let's see. So what else can we say? Yeah, so now how, how do we construct these Euler sprays? So here we actually use some pretty powerful theorem about partial regularity of the Vosserstein optimal transport, optimal transport map that originates in work of Caffirelli and later Fygalli and Kim. So what it says in our present context where we have the density that's one inside, inside this, the open set omega zero. The optimal transport map is smoothed on a dense open set of full measure, omega zero hat. So there's infinite regularity there. And so we'll see an example in a second of how it can fail to be smooth everywhere. In some sense it's, it's, it's, it's, it's not hard. I'm not, there's no, there's no hypothesis of connectedness here. There's transport can, can rip the sudden in pieces. So there's no reason that the transport map should be continuous. But you have this partial regularity. So in the, in the, in the smooth set, we're going to use that. We're going to construct a Vitaly covering by disjoint balls with radii that are, that are chosen in a nice way that controls error accumulation in the, in the total, in the total action. So that I won't write down. It's, I can't do that in public. So it's just, it looks like a mess. And what we do is we're going to look to, to only have to deal with ellipsoidal solutions. So these are somehow the transport maps are linear. And so they'll only approximate the Vostostan transport map locally. We're going to, we need to spread out slightly by an epsilon the centers of the, of the target balls. The centers of the targets to make a little bit of room so that the, to make things not overlap. So, let's see. For the actual brandy map T, that's injective. There's no overlapping of images anywhere along the, along the Vostostan optimal transport path. But we're making this ellipsoidal approximation. We want to eliminate overlap. We just need an epsilon of room. And so we get to list this little bit of room for these transported ellipsoids. They're linear approximations of the transport map. And there's just a little bit of room. There's a countable number of these balls though, remember. So the boundary is pretty, pretty, quite a mess. So the Euler, Euler droplets though are nested inside the ellipsoids that you get this way. These are just, let's see, you do linear interpolation here. So each little ball is transported according to the Vostostan ellipsoid. Okay, for the whole collection, I don't know what the optimal Vostostan map is. But this is, this is close enough for, for approximation. So our Euler droplets at the end, we have a countable number of them. But we can control them and not to collide. And so they superimpose and provide a solution of the, well, sort of an action critical path, a geodesic. Okay, so this is a bit of an illustration of a computation. It's not quite with the Euler-Sprey construction I described here, but it's the only illustration I have at the moment. But it illustrates how you divide up the source domain into little balls. So the, in the proof, the balls are going to be a lot smaller and many more of them. Along the way, there's going to be a little room between them. These should really should be ellipsoids, but in this computation they weren't. What this computation indicates here over here on the right is the image, images of the balls under the actual Wasserstein map. So those get, those get approximated by ellipsoids and spread out a little bit in the proof. So over here you can see that there's two balls touching over here, orange and yellow. And over here the computed Wasserstein map has this cut along the inside. So you can see it's not necessarily smooth even in the case of connectedness. Okay, so it's a kind of illustration of what's, what's supposed to be happening here. Let's see, how am I doing with time? Let's see, 10 minutes. Let's see, so the, let's see. So the, all the sprays we get, so there's weak solutions of, of Euler equation. And written in terms of the density and momentum conservation laws, we can think of them as weak solutions of an equation like this. They, they have this pressure that comes from the, the formula, the formula as we, we, we wrote down before. Let's see, the, the version of, the version of velocity is, remember, is not zero in the weak sense. But the constraint really is that the density is this characteristic function. So that's the constraint that, that determines the pressure in a sense. In this, in this problem here. So what's happening to the, as Epsilon goes to zero in the previous theorem, is the weak solutions you get from, from incompressible Euler here are converging in a weak sense to solutions of the pressureless Euler system here. The pressure is actually going to zero uniformly. And the other ones they converge in, in the weak star sense. It looks, it looks fancier than it is. It's, it's just really has to do with the effective gaps between the, between the, between the ellipsoids for time between zero and one. The average density goes to the limiting, limiting row, which typically is going to be less than one. And the other weak convergences are kind of a very similar, similar fashion. So the only kind of non-trivial estimate here is the pressure estimate. I don't know if I quite have time to talk about it. I think I probably won't go through it in detail. But I will say that it has to do with that vector of the estimate. It doesn't, does use the choice of, the choice of those major axes, the principal axes as length minimizing geodesics on that constant volume surface. So what we do is we take the Wasserstein linear interpolant between the original, the ball of original and radius r and the final ball with, with principal axis lengths a, j hat, that linear interpolant you project onto the constant volume manifold and use that as an upper bound for the length minimizing geodesic length. Okay, and a couple of geometric factors end up with a bound like this. So we choose our Vitaly covering to make sure that everything comes out less than accumulates to something less than epsilon. Okay, so I'm actually, I'm kind of, okay, fairly close to the end then. So, so we have this minimization problem which we were trying to solve. I tried to minimize action to construct solutions of incompressible Euler. And we see that actually there's this instability of micro droplet formation. So in fact, typically no length minimizing path exists. And that has to do with the uniqueness of Wasserstein geodesics. So actually necessary and sufficient for a length minimizing path with shape densities everywhere along the, characteristic functions everywhere along the path is that the Wasserstein geodesic has to have characteristic function densities. So way back where, way back when, yeah, over here. That said, rho had to be identically one inside the domains. So all the eigenvalues of the, of the Hessian of the convex function have to be one. The convex function has to be the identity, essentially. Locally, well, some translate of the identity. So what possible, what possibilities are there? So kind of locally you have to have rigid body motion on components of the smooth set. Okay, how, what kind of ways can you break up a domain into pieces of rigid bodies flying apart? I don't want to make the experiment, but it's, it seems an interesting problem actually, there could be a lot of interesting structure there. So we've seen that Euler sprays exist approximately transporting a bounded open set to another one of equal volume. If you consider weak limits of the corresponding other solutions, you get the optimal transport flows, the Wasserstein geodesics, which have density going between zero and one. I have this little picture here which I was trying to think of an artist, right, that used this pixelation. I think there was an earlier French one, surat, right, pixelation. And the idea is in a lot of geometric analyses of shape deformations, everything's assumed to be smooth. And of course I have the opposite here. I have the things fragmented in a very bad way. Okay, but I'd like to argue that, okay, modern digital technology with ink drop printing and 3D printing really one should be able to think about micro droplet, building up images out of micro droplets. And so maybe this has some relevance, you know. Yeah, so the last issue has to do with how this connects with Bernie's work on relaxations of least action principles for incompressible flows. I don't have a lot to say about that. There's a lot of interesting work on this subject. But my goal here is basically to describe a well-posed relaxation or completion of the least action principle in a natural way. And something that Bernie didn't really consider was what happens when you have variable density. So that's what we're going to have here. We're going to have two fluids with different densities. Row one hat and row zero hat. And the interesting case is actually when row zero is zero. So Bernie's key argument, the key element in Bernie's work is the observation that kinetic energy is convex in mass and momentum considered jointly. And so let's see, where's mass and momentum? There's masses, basically, row hat c, that's density. And momentum is m, m is down here somewhere. m is row hat c times v, so that's the momentum. So that's a convex function of both variables. And okay, so using convex analysis, you can write this function as an indicator function of a certain set, parabolic set. And okay, it's a supremum of affine functions, so it's convex. And this is the one that one ends up using to construct a relaxation. So what's a relaxation? It should be something, you should be thinking about the closure in the weak start topology. And so that's why we're going to have densities that can be between zero and one. So C i is a concentration, you should think of it as between zero and one. In the least action problem here, you don't necessarily constrain it, but I will in a second. So the kinetic energy of the mixture is the sum of the kinetic energy of the components. We have two fluids, they basically occupy more or less, we're at the macroscopic level, they occupy the same space. At the microscopic level, who knows. And the object is to minimize that kinetic energy, infamize it over subject incompressibility constraints, so the sum of the concentration should be one. Transport of each component separately with conserved total mass and end point conditions, which here I'm taking as characteristic functions. So the kind of solutions I was looking for all along, actually the concentrations are characteristic functions for all time. But whether you can actually minimize and preserve that property in the limit, you don't know, but the point is that this kind of formulation does have an infimum, does have a minimum rather, so you can minimize in this framework. So you have a relaxation, a kind of completion of the least action principle. And in fact, when you have fluid vacuum mixture, this connects our incompressible, incompressible constrained problem explicitly to Wasserstein, so we can say that the relaxed least action problem actually has a unique solution given by the Wasserstein geodesic flow, the pressureless Euler flow coming from the Wasserstein geodesic. A minimizing family of unmixed paths, so where the CJs are given by characteristic functions, is provided by, well, concatenated Euler sprays. And so that's a kind of explicit connection of the geodesics from, with the incompressibility constraint. Looking at it from the point of view of really trying to relax the least action problem to something that has a minimizer, the minimization is actually given by the Wasserstein geodesic. Okay, so that's the end. Let's see, so those are some of the same things I said. We've added here the relaxed least action problem naturally related to work of Bernier. There's a preprint online, which is being heavily revised at the moment. We've got substantially simplified proofs, so we expect within a month or so to have a better version. For you to read. Thanks for your attention. So do you get something better, like if you start with two densities rather than the... Oh, interesting. So I didn't know a lot about this problem of where to go, of mixed your problem with two densities, but I've actually heard recently a talk by a student of Barbara Keefitz. It seems that this problem with two densities is actually connected to, it's a problem with singular shocks and a non-hybrid bollock system that has singular shocks. So it's really the same thing. In the case of characteristic function, you have a lot of possibilities, but if you start with two densities, you have less room. In terms of moving things around and constructing some analog of unmixed paths, I actually have no idea. How you would move things around to make room. There's a little bit connected with Bresson's conjectures about it. They're really tough problems. But for the mixture theory itself, it's connected with this non-hybrid bollock system with singular shocks.