The L2 exponential map in 2D and 3D hydrodynamics
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Misiolek, Gerard
In the 1960's V. Arnold showed how solutions of the incompressible Euler equations can be viewed as geodesics on the group of diffeomorphisms of the fluid domain equipped with a metric given by fluid's kinetic energy. The study of the exponential map of this metric is of particular interest and I will describe recent results concerning its properties as well as some necessary background.
 Some of you have seen some version of this talk. I mean, a little bit, but we'll see. I'm not sure how much people can cover it, but because in the audience there are also people who have not seen it. And from different fields, I feel that I should give some background and I'll try to riff off of Boris' talk. So my main goal is to discuss some recent results concerning the Romanian exponential map on the group of volume-preserving diffeomorphisms in three dimensions. This point of view has to do with Arnold's approach to incompressible hydrodynamics. I reveal that it's in visit incompressible fluids. That's an infinite-dimensional Romanian geometry in this case. This is in spirit of carton. It's pretty pure differential geometry in infinite dimensions. Okay, so I should perhaps begin with that setup. So as I said, I'll use Boris' introduction here, but nevertheless I would like to present here a summary of this approach in the picture. So MN is the fluid domain. I think compact Romanian manifold possibly with boundary dimensions, of course, two, three in general, the only relevant for hydrodynamics. But also R3 or Rm, maybe even more generally asymptotically Euclidean manifold, so complete Romanian manifold with ends such that it infinity the Euclidean spaces. Mu is the Romanian volume form. So this is fixed fluid domain. Now, because we, so fluid fills M, but because we do not allow particles, fluid particles to fuse or split, the configuration space of the fluid will be a group of diffeomorphisms of preserving the volume in the compressible carries of M. So now this is the picture I'm trying to summarize things here. So I will call it dif mu as in the previous talk. So these are volume-preserving diffeomorphisms, diffios of M. So we think of this as positions, fluid positions of fluid particles. Then because this is a group, so this is a group of volume-preserving diffeomorphisms with a group log given by the composition. If this is the identity, we also have an upstairs, so to speak, and that's the Lie Algebra infinite dimensional. In this case, consisting of divergence-free vector fields. So here we want to think of this as the spatial velocities, fluid velocities. Okay, now as in classical mechanics, Newtonian mechanics, fluid motions will obey Newton's law. So we can think of really fluid motions as tracing out geodesics in the configuration space, which is the group of diffeomorphisms, and the geodesics are of the metric given by the kinetic energy. Now that kinetic energy, so this is just like classical mechanics, now that kinetic energy will then induce essentially an L2 inner product at the Lie Algebra level, and also a right invariant L2 metric. So this is the right invariant L2 metric at the level of the group. And if we, say, have some initial condition, initial divergence-free vector field, then the idea is that we get fluid motions described by the corresponding geodesic, let's call it gamma of t, in the configuration space diffe mu. Now this is the Lagrangian star. So in fact, there is a very nice way, I mean this picture will provide a nice way of thinking about somewhat confusing for a novice in particular, somebody who enters the field and reads about Eulerian description of Eulerian coordinates and Lagrangian coordinates. This is really the geometric, nice geometric picture that combines the two. So the Lagrangian story happens in the group, the Eulerian story will happen in the Lie Algebra. So how do we pass, where is the, what are the Euler equations here? Well, one way to see them is to look at the tangent vectors to the geodesic, say gamma adult, at the point gamma of t, yeah. So this is by the way L2 geodesic flow, which happens here. And the right translate, use the group structure to get a curve, let me call this u of t, in the tangent space of divergence-free vector fields, and it turns out that this curve, so we get a dynamical system here, and this u, this vector field, which is a time-dependent vector field on m, satisfies Euler equations of hydrodynamics. Now here the nabla is the covalent derivative on m, p is the pressure, then we take the divergence equal to zero, and here is the initial condition, so we get a Cauchy problem, I'll call it star. Okay, so these are Euler equations. Okay, so this is really Arnold in his paper from 1966. However, in order to do analysis here, one needs to bring in topology. So I need to be able to work with manifolds that allow me to solve differential equations so that I can talk about these geodesics. So this was done by Eben and Marston in a subsequent paper. Eben, Marston, 1970. So what they did is they introduced topology, so one can topologize diff s, diff mu, with for example sub-left topology, so I'll write this diff s mu, or, so possible, held up topology, one alpha. So in sub-left hs, s here is greater than n over 2 plus 1, or some other reasonable topology, more exotic, so held as c1 alpha. And if that's the case, then the geodesic equations become a lonely E. So now completed, for example, I'll stick to sub-left in the completions of the configuration spaces and the algebra which now becomes, first of all, this becomes in hs case. So if s is greater than n over 2 plus 1, then this is an topological group, as well as smooth Hilbert manifold. And in that case, the geodesic flow here becomes really an ordinary differential equation, albeit in infinite dimensions. So this is the Lagrangian picture. And here we have the ODE's describing the fluids, and here we have the Eulerian picture, and that's the PDE, the final PDE that corresponds to this. So this is really the kind of one succinct way of looking at the Lagrangian and OEulerian story. I want to concentrate on the Lagrangian story. I mean, I want to describe the work on the Lagrangian side, but I still would like to say a few words about the OEulerian. So the Cauchy problem for OEulerian equations has a long history, and it goes back to the 1920s and 30s. So the work of Liechtenstein, Günther, Volibner, who proved, well, they were in C1-alpha category, and they only looked at local, well, Liechtenstein and Günther, existence and uniqueness results. Volibner proved global, first global result, global existence result in two dimensions, but they did not really study the dependence of solutions on the initial condition. So this was not the full Ademar well-positiveness. This was studied later, and the pivotal, I guess, name here is Kato. So Kato in the 60s, well, also in the Udowic, so Udowic and Kato, by the way, in the 60s, this is 1960s, studied the Cauchy problem, so this is well-positiveness, well, maybe I should, you can write, because of Kato, Ademar, well-positiveness of star in HS, actually even in WSP, where S here is greater than N over P cos 1. And with what I mean by Ademar, well-positiveness, I mean existence, uniqueness, and at least continuous dependence of solutions on the data. So there's been a lot of work that sprang from that project. We wrote several papers with Ponsa and Lai and some other collaborators. But the reason I'm bringing this up is because these two pictures, if one uses some finite smoothness topology, like, for example, the Soveless topology, are not equivalent. So here is a remark, and I'll come back to the Cauchy problem story in a second. But let me just point this out right here. The two pictures on the left side are not equivalent in HS or C1 alpha, or any other reasonable topology because of the properties of the solution map. So if we look at data to solution map in the Eulerian case, then we're looking at U0 being mapped to the solution U. Now in the Lagrangian case, we're looking at U0 mapped into gamma and gamma dot. Now in this case, the map is at best, in any reasonable topology like the ones I mentioned here, at best continuous. In this case, it's smooth. In fact, C infinity is smooth. There are lots of surprises here on this side. So for example, if I had more time, I would be able to maybe save a little bit more. But perhaps if I had time at the end, I'll return this. But let me just point out one thing. Especially in the view of the classical results, well-posed results in C1 alpha, it is a surprising result due to Burgen and Lee. It's a recent paper from 2015, I think, that the Euler equations, I mean this problem is not well-posed in CK spaces. So EG, here. I forgot I should mention two results. But okay, let me start with Burgen and Lee. Star is not well-posed in CK spaces. Mk greater than or equal to 1. The reason is essentially this. Those analytically inclined. Take a derivative of this equation. Here, so I will write this as du sub t plus more or less. This is more or less, I'm just speaking. U dot nabla du plus essentially something like du squared equals minus d nabla. And there is the pressure. So I'll come back to that. So now assume that du is in fact C1. And let's see what happens. Obviously this is C1. This is a transport equation. These are transport terms. So if the rest is C1, I can make this live in C1 for all time. Or at least short terms. No problem. What happens here? That's the trouble term. Let's look more closely at this. If I go back here, and this is really the gist of Eben and Maast. And this is what they found. They noticed that one can take the divergence here of both sides and observe that the divergence, when u has divergence zero, divergence of the covariant derivative of u in the direction of u doesn't lose derivatives. This is by the way why we will have smooth, why we have smooth map here. Because that's the what you call, picar, vanard, picar contractions. Okay, that's the ODE. That's the reason for the ODE. But let's go back here. If I take the divergence here, all I get is this. Plus Ricci if I'm on the manifold. So that's lower or lower. And let's forget the Ricci. Let's forget the curvature. That's an essential. So if I go back here, I'm saving that. Then what I get is a double-reze transform. What is this? Okay, if I want to solve for p, I need to take the divergence. And I have divergence gradient. I have a Laplacian. I have to invert the Laplacian. I have the inverse of the Laplacian. And now I have the trace of the u squared. Okay, this is C1. Trace doesn't do anything. And this is a singular integral operator of Kaldron signet type. And Cz type. And so it's not bounded in L infinity norms. That's the troublemake. Okay, so there are surprises here. Even in local TV. Another thing I wanted to point out, why at best continuous? Well, somehow this was not seen, although it could have been seen by Liechtenstein. Liechtenstein could have seen a lot of other things, I guess. Probably didn't bother rank them all up. He has the famous book from one of the first in the Springer series on hydrodynamics, with a lot of great ideas. But in 2010, together with Himalas. So this is 2010. What we showed is that this map in sub-Alippes space setting is not even uniformly continuous. This has been generalized to other spaces as well. So really, continuity is best, one can expect. One more, perhaps, result that is surprising. Let me go back to C1 alpha case. Even in this case, there are some surprises. But of different nature. So here I should mention again. And Lie, this is 2015. So for C1 alpha case, let me just say that full up in full space, in the helter space, C1 alpha, Adamar, well, positeness does not hold. One needs to pass to the little helter space to get existence, local existence, uniqueness, and continuous dependence of solutions from U0. So full Adamar for star is OK in little helter space, but not in big helter space. And I'm pretty sure this holds also for any other more exotic settings, functional settings like BASOF spaces, or 3D resarching, which are now popular in connection with Cauchy problem. So this is a result with Yonetta. I think it just appeared. All right. Now, how much time do I have left? Seven. Seven? I think I started five minutes late, right? Yeah, OK, that's fine. 12. All right, thank you. So let me now switch to the Lagrangian side. On the Lagrangian side, as I said, we have the smooth dependence there. And this immediately tells us that we have a nice infinite-dimensional model of Re-maniac geometry here, because in particular this means that we have a well-defined L2 exponential map. And that map, the L2 exponential map, will be defined initially in dimension 6, 2, and 3, or 3, on some open subset of the zero vector in the tangent space with values in difference mu. And it's defined, say it's T0, by definition, this is the unique L2 geodesic with the initial conditions that it starts at the identity in the direction of mu0. And we know, because we are dealing with an ODE, that this is perfectly fine, by contractions, as I said. OK, so then, locally, by the way, it's just like in finite dimensions. This will be a local diffeomorphism by the inverse vector there for Banach-Manifold, the Banach spaces. So locally, everything is fine. We have a perfect nice Re-maniac geometry. What happens globally? And that's, of course, of interest, because this is a solution map in Lagrangian coordinates, or fluids, two or three dimensions. So I'm interested in the singularities. So what can I say about the singularities of the L2x? Well, these are the conjugate points from classical Re-maniac geometry. The conjugate points in diffeur. But unlike in finite classical Re-maniac geometry, here, we have to be careful. So in infinite dimensions, in general, we have two times. Depending on whether dx fails to be, it's not one to one, and in which case we say that the conjugate point is a monoconjugate point, or it's derivative. The singularity is of the type that dx is not onto, in which case we talk about epiconjugate points. Then another problem is that they can have conjugate points, in this case, can have infinite order. Imagine the unit sphere in the space of square-solvable sequences. Easy exam. They can cluster. Conjugate points can cluster on finite units, due to the segments. Now, here the example again, an abstract example, is easy to see, is a football, an American football. So in infinite dimensions, so an infinite dimensional ellipsoid, in little L2 again. So take x1, x2, take a circle there, unit circle, and take x1, x2, x3, take a football, take an ellipsoid, and then x1, x2 still fixed, x4, make it squeeze it a little bit more, make it look more like a ball. x1, x2, x5, even more so, and so on. What you get then on the circle joining the northern and south pole, you'll get a cluster of sequence of conjugate points, which will converge at the south pole. So that's an example. Now, the point is that we know that all these, we have examples of all of these pathologies in diff mu. All these can be found in diff mu in general. Well, let me put diff mu and n, but now is the theorem that this n has to be 3, cannot be 2. Yes, what do you mean by n in order? If you take a sphere in R3, the north pole, take north pole and south pole, they're conjugates of order 1. There's one linearly independent, just one, really, Jacomi field, which is 0 at the top and 0 at the bottom, and it's non-zero. Now, if you take same setting, but in R4, you'll have the order of conjugacy increased by 1. So you have to put this here too. The infinity you have, infinite over. Sorry. So we have all these, but not in 2D hydrodynamics because of the result. Well, this result goes back to 2006, and it says that if n is 2, and well, boundary, m is compact, and compact. Originally, Riemannia, originally we assumed that it has to be without boundary, but this can be now deleted because of, I think I've written something here. Okay, deleted because of the recent work with Steve and James Brown. Again, I think this is 2018. So we can drop that. It took us a while, but it was a technical thing, quite a technical thing. So in this case, if n is 2, then the L2x is nonlinear fat-home map of index 0. In particular, what this means is that the L2 exponential map in 2D hydrodynamics behaves like an exponential map between finite-dimensional models, a classical. So 2D fluids want to become, want to be like classical finite-dimensional geometry, Riemannian geometry. And none of these pathologies can occur. We can find them, however, in three dimensions. But on the other hand, and this is the last thing that I will state, on the other hand, in 3D, we now know that one also can find some protagonist, provided that there is some, there's enough symmetry present in the manifold. So let's see. So I'll state this one, maybe, this is 2. And this is due to Newton-Pellets, myself and Steve. Well, in progress, I guess. And I won't be able to state it. Maybe we'll have a dig for it. Okay. Well, the statement is as of 2018. So, okay. If M3 is compact, or by the way, asymptotically euclidean, asymptotically euclidean, for example, R3, with a killing field. So this is HS class killing field. And U0, the initial condition I mentioned earlier in the Cauchy problem, which is the divergence free HS, of course, S here is greater than 3 halves, what is it, 5 halves. And U0 is a divergence free axisymmetric vector field with no square, or with small square. Then, axisymmetric, you mean the one commuting this Q? I will, yes, that's right. Then for each T, the X is 0. So I did not mention that there is a Fredholm operating, maybe, since I'm right here, in terms of DX. So by Fredholm, nonlinear Fredholm, what I mean is, so this is a notion introduced by Smale in the 60s when he generalized Sartz-Sterren to infinite dimensions, to Fredholm mappings. So the idea of Fredholm between a map is not only a Fredholm between two Helbert or Banach manifolds, if its derivative is a Fredholm operator. And its index, if the source is connected, the source manifold is connected, its index is the same as the index of the derivative. It's all classical. So here we have Fredholm, so we have a nice piece of, in fact, a subgroup of the group of all volume preserving diffeomorphisms of a three-dimensional manifold, those consisting of axisymmetric, no-swear-all flows, where the expansion map can be, in some cases, there are some topological obstructions in certain situations. That's why I'm actually stating this in terms of linear, meaning a linear statement here. Where? The expansion map will be Fredholm. Yes, so now, yeah, so let me, so first of all, what does it mean for V to be axisymmetric? Axisymmetric means that it commutes indeed on a manifold. So these are nice and very straightforward and very natural generalizations of really the only, the classically studied case of R3 of axially symmetric flows, and R3 goes back to Yudovic and Obukhov, 70s, I guess. Then there was nothing, nothing, and recently it's much more active field, and there are some nice results, and I think that's what Gindy and John very recent, kind of similar to what I was saying earlier, but I don't have time to discuss this, related to the norm inflation, but okay. So this is axi-symmetric, now no swirl, what is swirl? A swirl of a vector field, say U, will be defined simply as the inner product on the manifold of that vector field in K. And if this is zero, that's swirl three, we say that U is swirl three. Small swirl, well, it turns out that swirl is conserved. So even in the axi-symmetric case, which by the way, an example of that would be, take R3 with cylindrical coordinates R, theta, and Z, and consider, take a vector field in these cylindrical coordinates whose components do not depend on theta. That's an example of an axi-symmetric vector field. Swirl three, take the theta component of that vector field to be zero. An example, a draining jet. Open a water tap, at least, up to approximate, right, I mean with the swirl. But imagine a simple draining jet, that's a swirl three axi-symmetric flow, or smoke rings, ideal smoke rings, if you can do them, if you smoke. Okay, so these are the examples, and what I wanted to say here is that swirl turns out to be conserved along the swirl. So swirl, this, so that's a theorem from the, one of the results that we have, is that swirl is conserved along particle trajectories, in the three-dimensional case. What this kind of means is that, yeah, there is some conservation law that one can use, and maybe attempt to say something about global behavior of fluids. Thank you very much. Questions for Kjara? Because essentially when you had this commutation with Kjara, it was down to the fact that it's like 2D man, like the way it's 2D. This is not a problem really, because we know that we still, there are still people who suspect that there may be blow-up in the axi-symmetric case. So axi-symmetric world, let's put it this way, seems to display all the difficulties. We see all the same problems as we see in the full incompressible Euler case. In 3D. In 3D, yeah, in 3D. We don't know how to prove global existence. Now global existence can be proved in this, in the Eudovic, and if swirl is, if we have axi-symmetric and swirl zero, that's really what Eudovic and the book of Kahl axi-symmetric case, swirl three. And it's the swirl three, the really conservation of some additional conservation law that enable them to prove global existence. So what I'm really concerned about is, or what I'm interested, what we're interested in, is really to understand the axi-symmetric world. So we just concentrate on the subgroup of axi-symmetric diffeomorphisms, in the 3D dimensional case, and look at the Fretholm properties, vis-a-vis global properties, global persistence, and try to see if there is a relation. So for example, I mentioned here, if we have Fretholm, this type of Fretholm result, for the no swirl case, I mentioned here small swirl, that's just a cattle perturbation, perturbation of Fretholm operated steam, that gives us a free. I know that they've been attempts to prove global existence for small swirl, and there are some results, but they're very partial, and there is a lot of work to be done, and there are things to be understood. I would like to see if it's possible to connect these two. If for example Fretholm can inform us a little bit about what is really going on, well, with the global flow, in light of the conservation that we have. Because by the way, there is no reason, of course, in the even axi-symmetric world to expect that vorticity would be answered. So the standard to the methods right there. So you think of axi-symmetric as kind of two and a half dimensional world, right? Is that fair to say? Well, maybe two and a half, yeah, something like that. That's right. I don't want to specify the meaning of the word, to quantify it. So swirl three, axi-symmetric, these are like 2D. Exactly, this is all the work of Nikola and Kolmaharov. Yeah, that's fast rotation. Fast rotation implies, so really fast rotation, implies global resistance. So you can get it with methods like this, because later it was just classical PDs, just metal and all that. Yeah, I don't know, I should look at that. Yes, yes, you could. I remember, of course, with the result. It's quite old, demonstrate in the other one. Right. I think there'd be some developments, but you let Nikola and I have some work on that, but I must say I'm not probably even though this is my club. Any other questions? Alright, thanks again. We have a coffee break until 10.50, which is almost an hour, and then we'll have our last speaker for the morning.
 Some of you have seen some version of this talk.  I mean, a little bit, but we'll see.  I'm not sure how much people can cover it,  but because in the audience there are also people who have not seen it.  And from different fields, I feel that I should give some background  and I'll try to riff off of Boris' talk.  So my main goal is to discuss some recent results  concerning the Romanian exponential map on the group of  volume-preserving diffeomorphisms in three dimensions.  This point of view has to do with Arnold's approach to incompressible hydrodynamics.  I reveal that it's in visit incompressible fluids.  That's an infinite-dimensional Romanian geometry in this case.  This is in spirit of carton.  It's pretty pure differential geometry in infinite dimensions.  Okay, so I should perhaps begin with that setup.  So as I said, I'll use Boris' introduction here,  but nevertheless I would like to present here a summary of this approach in the picture.  So MN is the fluid domain.  I think compact Romanian manifold possibly with boundary dimensions,  of course, two, three in general, the only relevant for hydrodynamics.  But also R3 or Rm, maybe even more generally asymptotically Euclidean manifold,  so complete Romanian manifold with ends such that it infinity the Euclidean spaces.  Mu is the Romanian volume form.  So this is fixed fluid domain.  Now, because we, so fluid fills M,  but because we do not allow particles, fluid particles to fuse or split,  the configuration space of the fluid will be a group of diffeomorphisms  of preserving the volume in the compressible carries of M.  So now this is the picture I'm trying to summarize things here.  So I will call it dif mu as in the previous talk.  So these are volume-preserving diffeomorphisms, diffios of M.  So we think of this as positions, fluid positions of fluid particles.  Then because this is a group, so this is a group of volume-preserving diffeomorphisms  with a group log given by the composition.  If this is the identity, we also have an upstairs, so to speak,  and that's the Lie Algebra infinite dimensional.  In this case, consisting of divergence-free vector fields.  So here we want to think of this as the spatial velocities, fluid velocities.  Okay, now as in classical mechanics, Newtonian mechanics,  fluid motions will obey Newton's law.  So we can think of really fluid motions as tracing out geodesics  in the configuration space, which is the group of diffeomorphisms,  and the geodesics are of the metric given by the kinetic energy.  Now that kinetic energy, so this is just like classical mechanics,  now that kinetic energy will then induce essentially an L2 inner product  at the Lie Algebra level, and also a right invariant L2 metric.  So this is the right invariant L2 metric at the level of the group.  And if we, say, have some initial condition,  initial divergence-free vector field, then the idea is that we get fluid motions  described by the corresponding geodesic, let's call it gamma of t,  in the configuration space diffe mu.  Now this is the Lagrangian star.  So in fact, there is a very nice way, I mean this picture will provide a nice way  of thinking about somewhat confusing for a novice in particular,  somebody who enters the field and reads about Eulerian description  of Eulerian coordinates and Lagrangian coordinates.  This is really the geometric, nice geometric picture that combines the two.  So the Lagrangian story happens in the group,  the Eulerian story will happen in the Lie Algebra.  So how do we pass, where is the, what are the Euler equations here?  Well, one way to see them is to look at the tangent vectors to the geodesic,  say gamma adult, at the point gamma of t, yeah.  So this is by the way L2 geodesic flow, which happens here.  And the right translate, use the group structure to get a curve,  let me call this u of t, in the tangent space of divergence-free vector fields,  and it turns out that this curve, so we get a dynamical system here,  and this u, this vector field, which is a time-dependent vector field on m,  satisfies Euler equations of hydrodynamics.  Now here the nabla is the covalent derivative on m, p is the pressure,  then we take the divergence equal to zero, and here is the initial condition,  so we get a Cauchy problem, I'll call it star.  Okay, so these are Euler equations.  Okay, so this is really Arnold in his paper from 1966.  However, in order to do analysis here, one needs to bring in topology.  So I need to be able to work with manifolds that allow me to solve differential equations  so that I can talk about these geodesics.  So this was done by Eben and Marston in a subsequent paper.  Eben, Marston, 1970.  So what they did is they introduced topology, so one can topologize  diff s, diff mu, with for example sub-left topology,  so I'll write this diff s mu, or, so possible, held up topology, one alpha.  So in sub-left hs, s here is greater than n over 2 plus 1,  or some other reasonable topology, more exotic, so held as c1 alpha.  And if that's the case, then the geodesic equations become a lonely E.  So now completed, for example, I'll stick to sub-left in the completions of the configuration spaces  and the algebra which now becomes, first of all, this becomes in hs case.  So if s is greater than n over 2 plus 1, then this is an topological group,  as well as smooth Hilbert manifold.  And in that case, the geodesic flow here becomes really an ordinary differential equation,  albeit in infinite dimensions.  So this is the Lagrangian picture.  And here we have the ODE's describing the fluids,  and here we have the Eulerian picture, and that's the PDE,  the final PDE that corresponds to this.  So this is really the kind of one succinct way of looking at the Lagrangian and OEulerian story.  I want to concentrate on the Lagrangian story.  I mean, I want to describe the work on the Lagrangian side,  but I still would like to say a few words about the OEulerian.  So the Cauchy problem for OEulerian equations has a long history,  and it goes back to the 1920s and 30s.  So the work of Liechtenstein, Günther, Volibner, who proved,  well, they were in C1-alpha category,  and they only looked at local, well, Liechtenstein and Günther,  existence and uniqueness results.  Volibner proved global, first global result, global existence result in two dimensions,  but they did not really study the dependence of solutions on the initial condition.  So this was not the full Ademar well-positiveness.  This was studied later, and the pivotal, I guess, name here is Kato.  So Kato in the 60s, well, also in the Udowic,  so Udowic and Kato, by the way, in the 60s, this is 1960s,  studied the Cauchy problem, so this is well-positiveness,  well, maybe I should, you can write, because of Kato,  Ademar, well-positiveness of star in HS,  actually even in WSP, where S here is greater than N over P cos 1.  And with what I mean by Ademar, well-positiveness, I mean existence, uniqueness,  and at least continuous dependence of solutions on the data.  So there's been a lot of work that sprang from that project.  We wrote several papers with Ponsa and Lai and some other collaborators.  But the reason I'm bringing this up is because these two pictures,  if one uses some finite smoothness topology,  like, for example, the Soveless topology, are not equivalent.  So here is a remark, and I'll come back to the Cauchy problem story in a second.  But let me just point this out right here.  The two pictures on the left side are not equivalent in HS or C1 alpha,  or any other reasonable topology because of the properties of the solution map.  So if we look at data to solution map in the Eulerian case,  then we're looking at U0 being mapped to the solution U.  Now in the Lagrangian case, we're looking at U0 mapped into gamma and gamma dot.  Now in this case, the map is at best, in any reasonable topology like the ones I mentioned here,  at best continuous.  In this case, it's smooth.  In fact, C infinity is smooth.  There are lots of surprises here on this side.  So for example, if I had more time, I would be able to maybe save a little bit more.  But perhaps if I had time at the end, I'll return this.  But let me just point out one thing.  Especially in the view of the classical results, well-posed results in C1 alpha,  it is a surprising result due to Burgen and Lee.  It's a recent paper from 2015, I think, that the Euler equations,  I mean this problem is not well-posed in CK spaces.  So EG, here.  I forgot I should mention two results.  But okay, let me start with Burgen and Lee.  Star is not well-posed in CK spaces.  Mk greater than or equal to 1.  The reason is essentially this.  Those analytically inclined.  Take a derivative of this equation.  Here, so I will write this as du sub t plus more or less.  This is more or less, I'm just speaking.  U dot nabla du plus essentially something like du squared equals minus d nabla.  And there is the pressure.  So I'll come back to that.  So now assume that du is in fact C1.  And let's see what happens.  Obviously this is C1.  This is a transport equation.  These are transport terms.  So if the rest is C1, I can make this live in C1 for all time.  Or at least short terms.  No problem.  What happens here?  That's the trouble term.  Let's look more closely at this.  If I go back here, and this is really the gist of Eben and Maast.  And this is what they found.  They noticed that one can take the divergence here of both sides and observe that the divergence,  when u has divergence zero, divergence of the covariant derivative of u in the direction of u doesn't lose derivatives.  This is by the way why we will have smooth, why we have smooth map here.  Because that's the what you call, picar, vanard, picar contractions.  Okay, that's the ODE.  That's the reason for the ODE.  But let's go back here.  If I take the divergence here, all I get is this.  Plus Ricci if I'm on the manifold.  So that's lower or lower.  And let's forget the Ricci.  Let's forget the curvature.  That's an essential.  So if I go back here, I'm saving that.  Then what I get is a double-reze transform.  What is this?  Okay, if I want to solve for p, I need to take the divergence.  And I have divergence gradient.  I have a Laplacian.  I have to invert the Laplacian.  I have the inverse of the Laplacian.  And now I have the trace of the u squared.  Okay, this is C1.  Trace doesn't do anything.  And this is a singular integral operator of Kaldron signet type.  And Cz type.  And so it's not bounded in L infinity norms.  That's the troublemake.  Okay, so there are surprises here.  Even in local TV.  Another thing I wanted to point out, why at best continuous?  Well, somehow this was not seen, although it could have been seen by Liechtenstein.  Liechtenstein could have seen a lot of other things, I guess.  Probably didn't bother rank them all up.  He has the famous book from one of the first in the Springer series on hydrodynamics,  with a lot of great ideas.  But in 2010, together with Himalas.  So this is 2010.  What we showed is that this map in sub-Alippes space setting is not even uniformly continuous.  This has been generalized to other spaces as well.  So really, continuity is best, one can expect.  One more, perhaps, result that is surprising.  Let me go back to C1 alpha case.  Even in this case, there are some surprises.  But of different nature.  So here I should mention again.  And Lie, this is 2015.  So for C1 alpha case, let me just say that full up in full space,  in the helter space, C1 alpha,  Adamar, well, positeness does not hold.  One needs to pass to the little helter space to get existence,  local existence, uniqueness, and continuous dependence of solutions from U0.  So full Adamar for star is OK in little helter space,  but not in big helter space.  And I'm pretty sure this holds also for any other more exotic settings,  functional settings like BASOF spaces,  or 3D resarching, which are now popular in connection with Cauchy problem.  So this is a result with Yonetta.  I think it just appeared.  All right.  Now, how much time do I have left?  Seven.  Seven?  I think I started five minutes late, right?  Yeah, OK, that's fine.  12.  All right, thank you.  So let me now switch to the Lagrangian side.  On the Lagrangian side, as I said, we have the smooth dependence there.  And this immediately tells us that we have a nice infinite-dimensional model  of Re-maniac geometry here, because in particular this means that we have a well-defined L2 exponential map.  And that map, the L2 exponential map, will be defined initially in dimension 6, 2, and 3, or 3,  on some open subset of the zero vector in the tangent space with values in difference mu.  And it's defined, say it's T0, by definition, this is the unique L2 geodesic with the initial conditions  that it starts at the identity in the direction of mu0.  And we know, because we are dealing with an ODE, that this is perfectly fine, by contractions, as I said.  OK, so then, locally, by the way, it's just like in finite dimensions.  This will be a local diffeomorphism by the inverse vector there for Banach-Manifold, the Banach spaces.  So locally, everything is fine.  We have a perfect nice Re-maniac geometry.  What happens globally?  And that's, of course, of interest, because this is a solution map in Lagrangian coordinates,  or fluids, two or three dimensions.  So I'm interested in the singularities.  So what can I say about the singularities of the L2x?  Well, these are the conjugate points from classical Re-maniac geometry.  The conjugate points in diffeur. But unlike in finite classical Re-maniac geometry, here, we have to be careful.  So in infinite dimensions, in general, we have two times.  Depending on whether dx fails to be, it's not one to one,  and in which case we say that the conjugate point is a monoconjugate point, or it's derivative.  The singularity is of the type that dx is not onto, in which case we talk about epiconjugate points.  Then another problem is that they can have conjugate points, in this case, can have infinite order.  Imagine the unit sphere in the space of square-solvable sequences.  Easy exam.  They can cluster.  Conjugate points can cluster on finite units, due to the segments.  Now, here the example again, an abstract example, is easy to see, is a football, an American football.  So in infinite dimensions, so an infinite dimensional ellipsoid, in little L2 again.  So take x1, x2, take a circle there, unit circle, and take x1, x2, x3,  take a football, take an ellipsoid, and then x1, x2 still fixed, x4,  make it squeeze it a little bit more, make it look more like a ball.  x1, x2, x5, even more so, and so on.  What you get then on the circle joining the northern and south pole,  you'll get a cluster of sequence of conjugate points, which will converge at the south pole.  So that's an example.  Now, the point is that we know that all these, we have examples of all of these pathologies in diff mu.  All these can be found in diff mu in general.  Well, let me put diff mu and n, but now is the theorem that this n has to be 3, cannot be 2.  Yes, what do you mean by n in order?  If you take a sphere in R3, the north pole, take north pole and south pole,  they're conjugates of order 1.  There's one linearly independent, just one, really,  Jacomi field, which is 0 at the top and 0 at the bottom, and it's non-zero.  Now, if you take same setting, but in R4, you'll have the order of conjugacy increased by 1.  So you have to put this here too.  The infinity you have, infinite over.  Sorry.  So we have all these, but not in 2D hydrodynamics because of the result.  Well, this result goes back to 2006, and it says that if n is 2, and well, boundary, m is compact, and compact.  Originally, Riemannia, originally we assumed that it has to be without boundary,  but this can be now deleted because of, I think I've written something here.  Okay, deleted because of the recent work with Steve and James Brown.  Again, I think this is 2018.  So we can drop that.  It took us a while, but it was a technical thing, quite a technical thing.  So in this case, if n is 2, then the L2x is nonlinear fat-home map of index 0.  In particular, what this means is that the L2 exponential map in 2D hydrodynamics behaves like an exponential map between finite-dimensional models, a classical.  So 2D fluids want to become, want to be like classical finite-dimensional geometry, Riemannian geometry.  And none of these pathologies can occur.  We can find them, however, in three dimensions.  But on the other hand, and this is the last thing that I will state,  on the other hand, in 3D, we now know that one also can find some protagonist,  provided that there is some, there's enough symmetry present in the manifold.  So let's see.  So I'll state this one, maybe, this is 2.  And this is due to Newton-Pellets, myself and Steve.  Well, in progress, I guess.  And I won't be able to state it.  Maybe we'll have a dig for it.  Okay.  Well, the statement is as of 2018.  So, okay.  If M3 is compact, or by the way, asymptotically euclidean, asymptotically euclidean,  for example, R3, with a killing field.  So this is HS class killing field.  And U0, the initial condition I mentioned earlier in the Cauchy problem,  which is the divergence free HS, of course, S here is greater than 3 halves,  what is it, 5 halves.  And U0 is a divergence free axisymmetric vector field with no square,  or with small square.  Then, axisymmetric, you mean the one commuting this Q?  I will, yes, that's right.  Then for each T, the X is 0.  So I did not mention that there is a Fredholm operating, maybe, since I'm right here,  in terms of DX.  So by Fredholm, nonlinear Fredholm, what I mean is,  so this is a notion introduced by Smale in the 60s when he generalized  Sartz-Sterren to infinite dimensions, to Fredholm mappings.  So the idea of Fredholm between a map is not only a Fredholm between two Helbert or Banach manifolds,  if its derivative is a Fredholm operator.  And its index, if the source is connected, the source manifold is connected,  its index is the same as the index of the derivative.  It's all classical.  So here we have Fredholm, so we have a nice piece of, in fact, a subgroup  of the group of all volume preserving diffeomorphisms of a three-dimensional manifold,  those consisting of axisymmetric, no-swear-all flows,  where the expansion map can be, in some cases, there are some topological obstructions in certain situations.  That's why I'm actually stating this in terms of linear, meaning a linear statement here.  Where?  The expansion map will be Fredholm.  Yes, so now, yeah, so let me, so first of all, what does it mean for V to be axisymmetric?  Axisymmetric means that it commutes indeed on a manifold.  So these are nice and very straightforward and very natural generalizations of really the only,  the classically studied case of R3 of axially symmetric flows,  and R3 goes back to Yudovic and Obukhov, 70s, I guess.  Then there was nothing, nothing, and recently it's much more active field,  and there are some nice results, and I think that's what Gindy and John very recent,  kind of similar to what I was saying earlier, but I don't have time to discuss this,  related to the norm inflation, but okay.  So this is axi-symmetric, now no swirl, what is swirl?  A swirl of a vector field, say U, will be defined simply as the inner product on the manifold  of that vector field in K.  And if this is zero, that's swirl three, we say that U is swirl three.  Small swirl, well, it turns out that swirl is conserved.  So even in the axi-symmetric case, which by the way, an example of that would be,  take R3 with cylindrical coordinates R, theta, and Z,  and consider, take a vector field in these cylindrical coordinates whose components do not depend on theta.  That's an example of an axi-symmetric vector field.  Swirl three, take the theta component of that vector field to be zero.  An example, a draining jet.  Open a water tap, at least, up to approximate, right, I mean with the swirl.  But imagine a simple draining jet, that's a swirl three axi-symmetric flow,  or smoke rings, ideal smoke rings, if you can do them, if you smoke.  Okay, so these are the examples, and what I wanted to say here is that swirl turns out to be conserved along the swirl.  So swirl, this, so that's a theorem from the, one of the results that we have,  is that swirl is conserved along particle trajectories, in the three-dimensional case.  What this kind of means is that, yeah, there is some conservation law that one can use,  and maybe attempt to say something about global behavior of fluids.  Thank you very much.  Questions for Kjara?  Because essentially when you had this commutation with Kjara,  it was down to the fact that it's like 2D man, like the way it's 2D.  This is not a problem really, because we know that we still,  there are still people who suspect that there may be blow-up in the axi-symmetric case.  So axi-symmetric world, let's put it this way, seems to display all the difficulties.  We see all the same problems as we see in the full incompressible Euler case.  In 3D.  In 3D, yeah, in 3D.  We don't know how to prove global existence.  Now global existence can be proved in this, in the Eudovic,  and if swirl is, if we have axi-symmetric and swirl zero,  that's really what Eudovic and the book of Kahl axi-symmetric case, swirl three.  And it's the swirl three, the really conservation of some additional conservation law  that enable them to prove global existence.  So what I'm really concerned about is, or what I'm interested, what we're interested in,  is really to understand the axi-symmetric world.  So we just concentrate on the subgroup of axi-symmetric diffeomorphisms,  in the 3D dimensional case, and look at the Fretholm properties,  vis-a-vis global properties, global persistence, and try to see if there is a relation.  So for example, I mentioned here, if we have Fretholm, this type of Fretholm result,  for the no swirl case, I mentioned here small swirl,  that's just a cattle perturbation, perturbation of Fretholm operated steam,  that gives us a free.  I know that they've been attempts to prove global existence for small swirl,  and there are some results, but they're very partial, and there is a lot of work to be done,  and there are things to be understood.  I would like to see if it's possible to connect these two.  If for example Fretholm can inform us a little bit about what is really going on,  well, with the global flow, in light of the conservation that we have.  Because by the way, there is no reason, of course, in the even axi-symmetric world  to expect that vorticity would be answered.  So the standard to the methods right there.  So you think of axi-symmetric as kind of two and a half dimensional world, right?  Is that fair to say?  Well, maybe two and a half, yeah, something like that.  That's right.  I don't want to specify the meaning of the word, to quantify it.  So swirl three, axi-symmetric, these are like 2D.  Exactly, this is all the work of Nikola and Kolmaharov.  Yeah, that's fast rotation.  Fast rotation implies, so really fast rotation, implies global resistance.  So you can get it with methods like this, because later it was just classical PDs,  just metal and all that.  Yeah, I don't know, I should look at that.  Yes, yes, you could. I remember, of course, with the result.  It's quite old, demonstrate in the other one.  Right.  I think there'd be some developments, but you let Nikola and I have some work on that,  but I must say I'm not probably even though this is my club.  Any other questions?  Alright, thanks again.  We have a coffee break until 10.50, which is almost an hour,  and then we'll have our last speaker for the morning.
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