Semi-invariant metrics on groups of diffeomorphisms
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Modin, Klas
We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy-a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity-under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the Γ-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing, and on the infinite-dimensional shape manifold of viscous rods.
 And I would like to thank the organizers for inviting me here. It's very nice so the first two talks were About the few more face groups of the few more face I'm going to continue But but now I will change the metric structure, so I'm not going to use the L2 metric Well, I see you missing. And what I present here is joint work with Start with a little outline So first some background and motivation for what I want to do I will present some some results And essentially this is related to the analysis that Gerard was talking about We want to study what happens And the metric is no longer fully writing variables So there are some pitfalls when you try to emulate this a bit more student analysis So we'll talk about that and then how you can move from global to from local to global results We also heard a little bit about this And also what happens when you go to Banach from Banach to a for shade quality so from from so many spaces to Okay, so the background was presented very well by Boris Many of us consider this paper by Arnault from 1966 as a foundation of this field of geometric hydrodynamics So you know what it's about so the Euler equations describing an incompressible fluid evolves on the space of vector field, let's say So there's an interpretation of this as instead a geodesic flow on the group of Sorry, I should say divergence field and there's an interpretation of this as a flow on volume preserving the homomorphism And As Gerard presented the the relation is just take either the Lagrangian or the Ilarian point of view So so after an old paper the Evident Marson did analysis on this based on this and they came up with these local post in this results That's the paper contains a lot of things actually And since then many authors many were here today have been working in in this fashion, let's say and also trying to do the same thing not just on the volume preserving the homomorphism but on the whole and Normally then you work with higher order right invariant sub-left-right and the key point I guess is that Gerard also said is that the Geodesic equation that you end up with It's a smooth you obtain a smooth second order order differential equation of on a sub-left completion of your different morphism So here is the basic question that I will discuss today So what happens if you have a metric on the space of all the few morphism of some very compact manifold? Where the metric is not fully invariant it's just invariant with the subgroup of all in preserving the So an example of that metric is exactly the fluid metric that Boris was talking about this morning But we want to do higher order versions of this and and What happens then? So so we assume that we have some some metric structure or some remanus structure on the group of the few morphism And we assume also that it's invariant with respect to From the right with respect to elements living in the volume preserve What can we do? And before I tell you what we can do why why are we doing this at all? What is the the sort of underlying questions or where does this come from and of course? I think it's sort of interesting in itself to see just see how much of this seven Marston Analysis goes through if you drop if you drop the full writing variant assumption So that's one motivation the other one That's how we sort of started with this comes from this auto calculus framework, which Boris also told you about So let me summarize that maybe my point of view is the same picture we had before So this is the manifold of the few morphism, right? It has a group structure. So at some point there is the identity And then if you take the projection So you fix some reference density And here I actually think of density is not as volume forms, but as functions Relative to some fixed volume form your manifold So you take the projection and which I called the x here So you take the projection given by the the action the left action of the diffio on that volume form corresponding volume form you get the projection down to these densities probability densities, so we assume they integrate up to one and And this gives a result by most sir, but this is in fact the principal bundle So this this gives us a vibration of the space of the morphism and this works This works in the for Seattle Polydus in the case when you have smooth diffium or face It also works in the in the solar Okay Well, yeah some restrictions, maybe it's not a smooth principal bundle Okay But anyway So so what is the autocadabulous? Well you take this? L2 metric on this M Given by the expression here. That's the same one that was talked about earlier And because the metric is right invariant it induces metric on this quotient here, so on the probability densities and This is called sometimes the automatric and this the distance of this metric generates the L2 washerstein distance So that's the point and then The open problem here, which already has been addressed by some authors here Is how can you generalize this to to higher order? So that's that's one problem stated in the 800 pages book by a villainy to sort of try to find generalizations of this So that's one motivation the other one comes from our shallow water equations so in fact, there is a very closed relationship between shallow water equations and compressible oiler equations, so in fact and this equation up here which sort of the simplest Shallow water equation H describes the depth or the height of the wave if you like and V is the horizontal velocity on the surface this and Then the simplest model looks like this So you have the evolution of a vector field to say one or two dimensional typically and Then you also have the evolution of the height function Which is transported like a density? Okay, and the energy functional here contains Two terms and the first one is just H times V squared So so this is like a kinetic energy term and then you have a potential as well, which is just a square So this is the simplest possible one and if you like you can also think of this as a simple Compressible fluid where so this is the internal energy that Boris was talking about before and then H would be the density So H would be row And the corresponding so you can view this of course as a Newton system so the under Lagrangian on On the tangent bundle would look something like this where fee and fee dot are elements in the tangent bundle of the diffumes Okay So here again you have this standard metric plus some potential that only depends on how the diffused acting on the reference But this is only the simplest model what people tend to do in the shallow water in the shallow water field is that they find more More refined models So so one of them is the green 90 equations Which is sort of if you remember this first part here is just a standard one and then you add these terms here To sort of correct this to make a better model and did the transport of the height function or H is still the same if you if you look at this you see that From the point of view of energy the energy functional is you have still this term, but then you have a correction terms that depends on the derivatives of your vector field P So now the first two terms this is still quadratic in P right the first two terms that is positive can be interpreted as the kinetic energy And this is still the same potential you had before So this somehow if you think of this as a Lagrangian on the tangent bundle of the diffused this would then correspond to an H1 metric with just a potential and In fact if you look at this you see that it comes from just a Tailor expansion so you if you want even more refined models can continue this and get even more derivatives on your Yes Oh, so it's just the adjoint of the knowledge with respect to F2. So yeah, so not yes divergence generalized I guess because Well, let's see. Not exactly right depends on the curvature. It's a flat if it's flat Okay So so you can generate even more refined models and that corresponds to actually having higher order metric So that's one motivation Yes, so that's what I said this just corresponds to an H1 Okay, so let's get into business And see what we want to do now which kind of metrics structures do we want to study? So the first thing which is sort of implicitly why I already stated here, but let's state it more formally is that So you're on T The tangent bundle of the your different morphism you have a symmetry given by the volume preserving the morphism and then if you sort of look at The reduced phase space it is it consists of just a vector field and a density. That's what you have seen so so all of this would be equations evolving a vector in involving a vector field and a density and the Lagrangians that I would consider on the tangent bundle will always come from let's say a small Lagrangian depending just on on the Velocity field the vector field and the density And in fact, I will take them on this Newtonian form So so I will have one term corresponding to to the kinetic energy which has in the in the inertia tensor And then another one corresponding to the potential energy. So that's the setup and Once you have that you can derive the equations of motion for this so the governing equation will look like this So M is the momentum here, which you get by applying the inertia tensor to the velocity field But now of course this this operator here may depend to believe that's the point. It depends on row, right? That's the difference from the fully right invariant setting and If many of you here are used to the pedification which corresponds to the case when you have a fully right invariant metric and what is the difference? Well, it's essentially these three things the first is that the a here depends on row That's the first difference And the other difference is you get this extra term here, which is row times the gradient of The derivative of your Lagrangian with respect to the density So so notice this is This is non-zero even when you're potentially zero because because you also have the fact that the metric And then you have the transport equation for your role so now there's a coupling between the two because M depends on this fashion. So that's the difference from from you Okay, so we want to say something about short first local results for these type of equations And of course in this generality, we can't say anything. We have to specify What our inertia tensor is right? So let's discuss some natural assumptions on this So here are that the three natural assumptions we came up with And I will try to sort of explain how this is different from From what you get in the fully right invariant case. So for the first one is that This operator that takes your vector field and your row and assigns a Pytode to the vector field This is a smooth differential operator in both row and V Right so and in row it can be nonlinear typically this but of course linear in V And it should be of order 2k minus 2 K is an integer in row and 2k in V And then we call this metric of order K So that's the first assumption the second is that for any smooth density And I sort of now fix the density actually the inertia operator should be elliptic We need to be able to invert it in a nice way And then the third one which is sort of The one that's a little bit more complicated here. So we need to say something also about derivative with respect to a row so if we take a We take this inertia tensor and we differentiate it with respect to a row we get something that That takes a V in a row and it gives you a map from C infinity to the space of vector fields, right and If we can take the L2 adjoint of that mapping and we require that the L2 adjoint of that mapping is also a smooth differential operator of order 2k minus 2 in the row variable and 2k minus 1 in V and U Okay, so when you foresee this it looks a bit strange, but actually let's say that these conditions are Sharp in the sense that if you take away any of them for sure you will find metrics where you don't have Where you don't get local results, but at least not in a good is having horseman technique and if you if you relax any of these conditions here 2k the orders of the differential operators. It also doesn't work. You can find the counter example So so let's be more specific and let me give you One case where this works so one metric for which in which this works and one case So the first example is I take so I specify my metric It's enough to do it in just row V and the velocity variable V, right? And it's defined through these some here where AI are just functions from our From you know takes a positive number returns a positive number and then and then I construct it using covariance like this So this will be a metric of order k And if I do it like this and I Because the following conditions either if k is one I require that a zero is positive and a zero is positive And a one is Completely constant uniformly, right? So it cannot depend on the row at all or if k is a greater than or equal to two the only condition is that a zero and a k are strictly positive Then it will fulfill that these assumptions So notice that something you have to treat the case k equal one so h1 matrix has to be treated Especially this is one of the first Let's say major differences from the standard setting that that there is something funny happening even for h1 We're fully writing variants. Normally when you have h1 you expect things to work out So in that example the green equation show what the expansion you'd better to take it up to the second term Yes good observation So the observation is that that board is made is that that equation does not fulfill this So in fact, we cannot use for green I think we cannot use the second term We can yeah if you include more yes, we can because then it's enough for justice Yeah, that's something that comes out of when you do these calculations of the analysis So that was a good example. Let's now take the counter example So now let's L be any nice. Let's say elliptic invertible differential operator And you construct your metric just like this So okay, why would you want to do it like this? There is actually a reason for doing it like this and it goes back to some work So because this metric although it doesn't have these nice smoothness properties that other very good convexity properties That's something we're working with now But but anyway, this is one that does not fulfill those conditions Yeah, let's see First result is the following so we take M to be compact I think for simplicity with me without the boundary in the paper and then we we take the metrics that fulfill all the assumptions I stated some slides ago and then we let the Sobolev index S be strictly larger than D over 2d is the dimension of the manifold plus 2k Okay, then we get local well-posted as free the sick on the sobolev completion diff s So notice here 2k Right, this is different from from what you have in the fully right in my case and this is this is necessary So that's that's another the pitfalls in the analysis that you really need More smoothness in order for this to work Okay And this is for the geodesic equations this is when you have no potential function But in fact if we go go through the proof it is easy to see that it's not a problem to add the potential As long as it doesn't differentiate row too much and and there were what this means here is that if you take the The variational derivative of your potential it should be a differential operator order maximum 2k minus 2 If that's true, then you also get all the local well-postedness when you have a potential Okay So so that that's I would say maybe the main result. That's the local well-postedness result. Let's now talk about Going to global results for these type of metrics and the first thing is that the standard let's say Arguments when you're on a league group or some some group structure and you have a metric that is strong But normally you can you can get a global existence of your desks and this is essentially how it works you have Let's say we start at the identity And we have some some local results saying that as long as your initial condition you're not initial father velocity Is in this ball here? There's a domain such that we have existence here right so then we can move along this curve and end up somewhere Right and then the argument is that if the metric is strong Then the velocity at the end point here will still remain in this ball Oh, they're epsilon ball of the same size so then you can just take that velocity translate it back again to the identity Computer geodesic as far as it goes and then copy back right so then you can iterate this process and you get global existence, but now Of course, this doesn't work when things are not fully right invariant because your epsilon Here will depend on row and row. It's not right. I mean it's it's move is changing right? So so so you cannot use this simple technique and actually you can ask yourself can we even expect? global existence So so the question is if you have Any same invariant metric of the type I just Showed you and it's the order is high enough. Will you always have a global existence of your basics? And the question is can you can you expect that I don't really know I don't have a mathematical argument But I have a physical one and it comes from shallow water equation So really in shallow water models you expect that you get some way breaking So so maybe you cannot even expect to get the global existence But at least existence up to way breaking or something like that So this is something to study what we did obtain however was let's say like a cheap global result and what we did is we take the Dementric of the form that I specified as the first example and then we make some assumptions on on these coefficient functions So the first one is that a zero is great strictly greater than some positive constant and the same for the last one a k and then Things work out we get that the diffeomorph is Def k this the case of the completion with this metric is complete metric space and it's just equally complete You have global geosics and the proof is essentially just that this metric is uniformly stronger than a fully right invariant metric that is strong So so there's no deep thing going on here. It's just comparing it with it with another metric Okay That's what I wanted to say about the global results now There is one more key thing in the ebb and Mars in paper And it's the fact that you don't lock yourself at just the specific s You want to to prove that this works when the sublum index s also goes to infinity so that this in fact works In the fresh air setting of smooth diffeomorphism So this result is sometimes called no loss no gain it means that if you have some existence results and you start with initial data that is Great which has more smoothness than what is required from your existence result Then in fact the smoothness is retained in the solution So if you start with hs plus one the solution would be hs plus one So so the question is does this work and and yeah, this works Absolutely the same for semi invariant metrics. So the statement is the following you take just any vector field on the tangent bundle Which extends to a smooth vector field on the on the sublum completion for every s greater than some s not and Then if you have Initial conditions with higher Regularities and required then what so so this is the exist the maximum existence interval it is similar to the existence in for the lower for the smaller s and Then well, what is the proof the proof is essentially to carefully read what is in the Evan Marston paper? Sorry, oh, yeah, yeah, okay, that's important. So the vector field has to be as the active variance That's the whole otherwise this will not work I'm explaining what is j also the day existence interval yes Okay, so can you also? increase s very light in this space with which you're working then Exponential map of our solution operator I think the most important thing is the Set open set of the same size or whether the size shrinks for example the size of that set depends on No I Don't you have to go to I think the way we stated it is that we get the same result in every Marston but Probably not more than that Yeah, okay, but anyway, it's essentially containing that Just that they didn't state it And of course we if you let s go into infinity you don't get result in the corresponding for the corresponding for shaley Where it's truly actually? Okay So yeah, that's essentially what I want to say what is the outlook well we want to maybe Do more sophisticated global analysis? And study things like yeah, of course you can also ask questions about friend on this You can study the sectional curvature for these kind of matrix So maybe that's something interesting the other thing is what all told it what she studied I gradient flows with respect to this remanus structure. So because our metrics are writing variant you get the big class of remanium metrics on On probability densities so you can do the same thing as study gradient flows and other things that might lead to something interesting Okay, thank you Moving on spreading So two questions Impact of this in this context here, so we have this money narrative H3 and different steps to be higher So so if I understand your question so correct you want to add some viscosity to this And see what happens and I think at least So so for me this would be like adding a force that is not from a potential So so now so now the question is if is that force going to be a smooth when you translate it, you know to to the whole bundle on the group and the result Like this is already given. I think in the Evan Marston paper for some type of Dispation correct me if I'm wrong there are many experts here, but we haven't looked into that but possibly I don't know Number of derivatives For example, you refer to that is so yes And that's why we get some nice results Then related thing is These are home Tonya so equations so here Not the fusion but dispersion may help So I'm not even sure if say green nag me has a dispersion dispersion relation for free There may be some week dispersion like a massive week dispersion, but KTV another shadow what equation this strong dispersion And then you can use for example completely different techniques. You can tell your space function space in which you are standing The question problem to the queen of the linear term of the equation. This is what we're getting in this series of papers in the 90s He came out with a so-called light spaces or space time space time spaces where he got global existence even for a to initial data and used of course the strong And in analytic fallacies using so-called spear But if in your context if you get these some of these equations with strong dispersion then maybe this is the methods to Not so They don't distinguish between the actions Slavery they're good for certain types of equations, but may not be good for others So yeah, so once I one thing I forgot to mention is for the green nag the equation first like Boris pointed out our Existence result doesn't work for green because of this. We need this extra condition on the coefficient function and the other thing is that people have already of course Analyze this equation. So so so local existence of solutions is well known for green It's well known for green magma already Other questions? So what about the noro's no-gain results when you look at the boundary? Same as for the different one, for fully writing the right matrix is also problematic. You're not better. What do you mean by problematic? Yes, you mean like convexity or something like this? No, I mean If you start which are given initial and final configurations which are smooth enough, let's say hn and greater than s You have any results? Every jubilance It's the same problem as in the fully writing by itself As long as there are no conjugate points along the jubilance You can go higher in its days. Yes, but there's no global results in it. There is something from very special cases. I think on Toros by No capitalizing. We take something which But I think that's the way Toros by Not in full generality as long as they're the best thing that I know is from you effects There you have something. Oh, is it only by markings? Okay, yeah, so and that is the result with no conjugate points Yes, we don't expect the first batch I I'll comment about what you guys talking so the finite dimension of this is like different product Problem What you're doing, right? Yeah, so so that and and then I want you to say that the potential that you have that this electric operator, right? It could be like the equivalent of the moments of inertia in the rigid body being shifted So in this way one can control a rigid body certain positions, right The tumbling where you can move the rigid body by controlling let's say two beats a lot of Along one of these axes of moment of inertia And what I want to say related to your question is that the shooting right would be affected by the way you change the role So for different roles, you will get you so I'm trying to say that you can rock and control your flow In a way that in a rigid body moving the beats you're going to control the position of the pipe Yes, that's that's Okay, but my question was a little bit off right I was wondering are there any peak on solutions like the Kamasa hole You know that kind of You know this thing We could look at them I mean they go through and body problems So, I mean if you take if you take the Compressible or an equation that's the simplest the simplest example right it's barotropic equation the one Boris was talking about It could be a first place to start Yeah, you don't have What do you have So In my dimension maybe to the No So Yes We yes, they started because for Number of different inertial operators that are given by maybe the pseudo differential Yeah, but still right So this My intuition on this would be something like this so because how do you construct the peak on solutions in the fully writing Variant in case one way to look at this just take take the action Of the diffio on some finite dimensional space right so the points just moving around some points, right? And then you get the momentum up and the units the inertia turns to invert The rock pulses and you get something right so but now you also have the density So I would expect that you get the movement of some densities plus some point vortices that are moving around Yes, so there seems to be But the density changes are going to change the weight But it's not even clear that I think Laurent you mentioned that it's possible that The direct delta dissipates And leaves the space immediate So if that happens then there's not conserved anymore you need to show that you are Delta the concern no, but look it's like this right Disappear Formally it's like this right you start with you start with the action of your diffium group on something and now the action will be on On some on some finite dimensional space corresponding to the point vortices times Let's say the densities right and then you lift that action to to a Hamiltonian action And you compute the momentum up and and that should be at least formally that should be some sort of Simplactic leaf where things are moving around right so so I expect that At least formally this is something we're drawing it and I'm just trying to say you can write It sounds For the discussion for the long break and we'll take it right now. Thanks class So let me just mention that there's a guided tour at one p.m. After lunch
 And I would like to thank the organizers for inviting me here. It's very nice  so the first two talks were  About the few more face groups of the few more face I'm going to continue  But but now I will change the metric structure, so I'm not going to use the L2 metric  Well, I see you missing. And what I present here is joint work with  Start with a little outline  So first some background and motivation for what I want to do  I will present some some results  And essentially this is related to the analysis that Gerard was talking about  We want to study what happens  And the metric is no longer fully writing variables  So there are some pitfalls when you try to emulate this a bit more student analysis  So we'll talk about that and then how you can move from global to from local to global results  We also heard a little bit about this  And also what happens when you go to Banach from Banach to a for shade quality so from from so many spaces to  Okay, so the background was presented very well by Boris  Many of us consider this paper by Arnault from 1966 as a foundation of this field of geometric hydrodynamics  So you know what it's about so the Euler equations describing an incompressible fluid evolves on the space of  vector field, let's say  So there's an interpretation of this as instead a geodesic flow on the group of  Sorry, I should say divergence field and there's an interpretation of this as a flow on volume preserving the homomorphism  And  As Gerard presented the the relation is just take either the Lagrangian or the Ilarian point of view  So so after an old paper the  Evident Marson did analysis on this based on this and they came up with these local post in this results  That's the paper contains a lot of things actually  And since then many authors many were here today have been working in  in this fashion, let's say and also trying to do the same thing not just on the volume preserving the homomorphism  but on the whole  and  Normally then you work with higher order right invariant sub-left-right  and the key point I guess is that  Gerard also said is that the  Geodesic equation that you end up with  It's a smooth you obtain a smooth second order order differential equation of on a sub-left completion of your different morphism  So here is the basic question that I will discuss today  So what happens if you have a metric on the space of all the few morphism of some very compact manifold?  Where the metric is not fully invariant it's just invariant with the subgroup of all in preserving the  So an example of that metric is exactly the fluid metric that Boris was talking about this morning  But we want to do higher order versions of this and and  What happens then?  So so we assume that we have some some metric structure or some remanus structure on the group of the few morphism  And we assume also that it's invariant with respect to  From the right with respect to elements living in the volume preserve  What can we do?  And before I tell you what we can do why why are we doing this at all?  What is the the sort of underlying questions or where does this come from and of course?  I think it's sort of interesting in itself to see just see how much of this seven Marston  Analysis goes through if you drop  if you drop the  full writing variant assumption  So that's one motivation the other one  That's how we sort of started with this comes from this auto calculus framework, which Boris also told you about  So let me summarize that maybe my point of view is the same picture we had before  So this is the manifold of the few morphism, right? It has a group structure. So at some point there is the identity  And then if you take the projection  So you fix some reference density  And here I actually think of density is not as volume forms, but as functions  Relative to some fixed volume form your manifold  So you take the projection and which I called the x here  So you take the projection given by the the action the left action of the diffio on that volume form  corresponding volume form you get the projection down to these  densities  probability densities, so we assume they integrate up to one and  And this gives a result by most sir, but this is in fact the principal bundle  So this this gives us a vibration of the space of the morphism and this works  This works in the for Seattle Polydus in the case when you have smooth diffium or face  It also works in the in the solar  Okay  Well, yeah some restrictions, maybe it's not a smooth principal bundle  Okay  But anyway  So so what is the autocadabulous? Well you take this?  L2 metric on this M  Given by the expression here. That's the same one that was talked about earlier  And because the metric is right invariant it induces  metric on this quotient here, so on the probability densities and  This is called sometimes the automatric and this the distance of this metric generates the L2 washerstein distance  So that's the point and then  The open problem here, which already has been addressed by some authors here  Is how can you generalize this to to higher order?  So that's that's one problem stated in the 800 pages book by a villainy to sort of try to find  generalizations of this  So that's one motivation the other one comes from our shallow water equations  so  in fact, there is a very  closed relationship between shallow water equations and compressible oiler equations, so in fact and this equation up here which sort of the simplest  Shallow water equation H describes the depth or the height of the wave if you like and V is the horizontal velocity on the surface  this and  Then the simplest model looks like this  So you have the evolution of a vector field to say one or two dimensional typically  and  Then you also have the evolution of the height function  Which is transported like a density?  Okay, and the energy functional here contains  Two terms and the first one is just H times V squared  So so this is like a kinetic energy term and then you have a potential as well, which is just a square  So this is the simplest possible one and if you like you can also think of this as a simple  Compressible fluid where so this is the internal energy that Boris was talking about before and then H would be the density  So H would be row  And the corresponding so you can view this of course as a  Newton system so the under Lagrangian on  On the tangent bundle would look something like this where fee and fee dot are elements in the tangent bundle of the diffumes  Okay  So here again you have this standard metric plus some potential that only depends on how the diffused acting on the reference  But this is only the simplest model what people tend to do in the shallow water in the shallow water field is that they find more  More refined models  So so one of them is the green 90 equations  Which is sort of if you remember this first part here is just a standard one and then you add  these terms here  To sort of correct this to make a better model and did the transport of the height function or H is still the same  if you if you look at this you see that  From the point of view of energy the energy functional is you have still this term, but then you have a correction terms that depends on the  derivatives of your vector field P  So now the first two terms this is still quadratic in P right the first two terms that is positive can be interpreted as the kinetic energy  And this is still the same potential you had before  So this somehow if you think of this as a Lagrangian on the tangent bundle  of the diffused this would then correspond to an  H1 metric with  just a potential and  In fact if you look at this you see that it comes from just a  Tailor expansion so you if you want even more refined models can continue this and get even more derivatives on your  Yes  Oh, so it's just the adjoint of the knowledge with respect to F2.  So yeah, so not yes divergence generalized I guess because  Well, let's see. Not exactly right depends on the curvature. It's a flat if it's flat  Okay  So so you can generate even more refined models and that corresponds to actually having higher order metric  So that's one motivation  Yes, so that's what I said this just corresponds to an H1  Okay, so let's get into business  And see what we want to do now which kind of metrics structures do we want to study?  So the first thing which is sort of implicitly why I already stated here, but let's state it more formally is that  So you're on T  The tangent bundle of the your different morphism you have a symmetry given by the volume preserving the morphism and then if you sort of look at  The reduced phase space it is it consists of just a vector field and a density. That's what you have seen  so so all of this  would be equations evolving a vector in involving a vector field and a density and the  Lagrangians that I would consider on the tangent bundle will always come from let's say a small Lagrangian  depending just on on the  Velocity field the vector field and the density  And in fact, I will take them on this Newtonian form  So so I will have one term corresponding to to the kinetic energy which has in the in the inertia tensor  And then another one corresponding to the potential energy. So that's the setup and  Once you have that you can derive the equations of motion for this so the governing equation will look like this  So M is the momentum here, which you get by  applying the inertia tensor to the velocity field  But now of course this this operator here may depend to believe that's the point. It depends on row, right?  That's the difference from the fully right invariant setting  and  If many of you here are used to the pedification which corresponds to the case when you have a fully right invariant metric and what is the difference?  Well, it's essentially these three things the first is that the a here depends on row  That's the first difference  And the other difference is you get this extra term here, which is row times the gradient of  The derivative of your Lagrangian with respect to the density  So so notice this is  This is non-zero even when you're potentially zero because because you also have the fact that the metric  And then you have the transport equation for your role so now there's a coupling between the two because  M depends on this fashion. So that's the difference from from you  Okay, so we want to say something about short first local results for these type of equations  And of course in this generality, we can't say anything. We have to specify  What our inertia tensor is right? So let's discuss  some natural assumptions on this  So here are that the three natural assumptions we came up with  And I will try to sort of explain how this is different from  From what you get in the fully right invariant case. So for the first one is that  This operator that takes your vector field and your row and assigns a  Pytode to the vector field  This is a smooth differential operator  in both row and V  Right so and in row it can be nonlinear typically this but of course linear in V  And it should be of order 2k minus 2  K is an integer in row and 2k in V  And then we call this metric of order K  So that's the first assumption the second is that for any smooth  density  And I sort of now fix the density actually the inertia operator should be elliptic  We need to be able to invert it in a nice way  And then the third one which is sort of  The one that's a little bit more complicated here. So we need to say something also about  derivative  with respect to a row  so if we take a  We take this inertia tensor and we differentiate it with respect to a row  we get something that  That takes a V in a row and it gives you a map from C infinity to the space of vector fields, right and  If we can take the L2 adjoint of that mapping and we require that the  L2 adjoint of that mapping is also a smooth differential operator of order 2k minus 2 in the row variable and 2k minus 1 in V and U  Okay, so when you foresee this it looks a bit strange, but actually let's say that these conditions are  Sharp in the sense that if you take away any of them for sure you will find metrics where you don't have  Where you don't get local results, but at least not in a good is having horseman technique and if you if you relax any of these conditions here  2k the orders of the differential operators. It also doesn't work. You can find the counter example  So so let's be more specific and let me give you  One case where this works so one metric for which in which this works and one case  So the first example is I take so I specify my metric  It's enough to do it in just row V and the velocity variable V, right?  And it's defined through these some here where AI are  just functions from our  From you know takes a positive number returns a positive  number  and then and then I construct it using covariance  like this  So this will be a metric of order k  And if I do it like this  and I  Because the following conditions either if k is one I require that a zero is positive and a zero is positive  And a one is  Completely constant uniformly, right? So it cannot depend on the row at all  or if k is a greater than or equal to two the only condition is that a zero and a k are strictly positive  Then it will fulfill that these assumptions  So notice that something you have to treat the case k equal one so h1 matrix has to be treated  Especially this is one of the first  Let's say major differences from the standard setting that that there is something funny happening even for h1  We're fully writing variants. Normally when you have h1 you expect things to work out  So in that example the green equation show what the expansion you'd better to take it up to the second term  Yes  good observation  So the observation is that that board is made is that that equation does not fulfill this  So in fact, we cannot use for green I think we cannot use the second term  We can yeah if you include more yes, we can because then it's enough for justice  Yeah, that's something that comes out of when you do these calculations of the analysis  So that was a good example. Let's now take the counter example  So now let's L be any nice. Let's say elliptic invertible differential operator  And you construct your metric just like this  So okay, why would you want to do it like this?  There is actually a reason for doing it like this and it goes back to some work  So because this metric although it doesn't have these nice smoothness properties that other very good convexity properties  That's something we're working with now  But but anyway, this is one that does not fulfill those conditions  Yeah, let's see  First result is the following so we take M to be compact  I think for simplicity with me without  the boundary in the paper and then we we take the metrics that fulfill all the assumptions I stated  some slides ago and then we let the  Sobolev index S be strictly larger than D over 2d is the dimension of the manifold plus 2k  Okay, then we get local well-posted as free the sick on the sobolev completion diff s  So notice here 2k  Right, this is different from from what you have in the fully right in my case and this is this is necessary  So that's that's another the pitfalls in the analysis that you really need  More smoothness in order for this to work  Okay  And this is for the geodesic equations this is when you have no potential function  But in fact if we go go through the proof it is easy to see that it's not a problem to add the potential  As long as it doesn't differentiate row too much and and there were what this means here is that  if you take the  The variational derivative of your potential it should be a differential operator order maximum 2k minus 2  If that's true, then you also get all the local well-postedness when you have a potential  Okay  So so that that's I would say maybe the main result. That's the local well-postedness result. Let's now talk about  Going to global results for these type of metrics  and the first thing is that the standard let's say  Arguments when you're on a league group or some some group structure and you have a metric that is strong  But normally you can you can get a global existence of your desks and this is essentially how it works you have  Let's say we start at the identity  And we have some some local results saying that as long as your initial condition you're not initial father velocity  Is in this ball here?  There's a domain such that we have existence here right so then we can move along this curve and end up somewhere  Right and then the argument is that if the metric is strong  Then the velocity at the end point here will still remain in this ball  Oh, they're epsilon ball of the same size so then you can just take that velocity translate it back again to the identity  Computer geodesic as far as it goes and then copy back right so then you can iterate this process and you get  global existence, but now  Of course, this doesn't work when things are not fully right invariant because your epsilon  Here will depend on row and row. It's not right. I mean it's it's move is changing right?  So so so you cannot use this simple technique and actually you can ask yourself can we even expect?  global existence  So so the question is if you have  Any same invariant metric of the type I just  Showed you and it's the order is high enough. Will you always have a global existence of your basics?  And the question is can you can you expect that I don't really know I don't have a mathematical argument  But I have a physical one and it comes from shallow water equation  So really in shallow water models you expect that you get some way breaking  So so maybe you cannot even expect to get the global existence  But at least existence up to way breaking or something like that  So this is something to study what we did obtain however was let's say like a cheap global result and  what we did is we take the  Dementric of the form that I specified as the first example and then we make some assumptions on on these coefficient functions  So the first one is that a zero is great strictly greater than some positive constant and the same for the last one a k and then  Things work out we get that the diffeomorph is  Def k this the case of the completion with this metric is complete metric space and it's just equally complete  You have global geosics and the proof is essentially just that this metric is uniformly stronger than a fully right invariant metric that is strong  So so there's no deep thing going on here. It's just comparing it with it with another metric  Okay  That's what I wanted to say about the global results now  There is one more key thing in the ebb and Mars in paper  And it's the fact that you don't lock yourself at just the specific s  You want to to prove that this works when the sublum index s also goes to infinity so that this in fact works  In the fresh air setting of smooth diffeomorphism  So this result is sometimes called no loss no gain it means that  if you have some existence results and you start with initial data that is  Great which has more smoothness than what is required from your existence result  Then in fact the smoothness is retained in the solution  So if you start with hs plus one the solution would be hs plus one  So so the question is does this work and and yeah, this works  Absolutely the same for semi invariant metrics. So the statement is the following you take just any vector field  on the tangent bundle  Which extends to a smooth vector field on the on the sublum completion for every s greater than some s  not and  Then if you have  Initial conditions with higher  Regularities and required then what so so this is the exist the maximum existence interval it is  similar to the  existence in for the lower  for the smaller s and  Then well, what is the proof the proof is essentially to carefully read what is in the Evan Marston paper?  Sorry, oh, yeah, yeah, okay, that's important. So the vector field has to be as the active variance  That's the whole otherwise this will not work  I'm explaining what is j also the day existence interval yes  Okay, so can you also?  increase s  very light in this space with which you're working then  Exponential map of our solution operator  I think the most important thing is the  Set open set of the same size or whether the size shrinks for example the size of that set depends on  No  I  Don't you have to go to  I think the way we stated it is that we get the same result in every Marston but  Probably not more than that  Yeah, okay, but anyway, it's essentially containing that  Just that they didn't state it  And of course we if you let s go into infinity you don't get result in the corresponding for the corresponding for shaley  Where it's truly actually?  Okay  So yeah, that's essentially what I want to say what is the outlook well we want to maybe  Do more sophisticated global analysis?  And study things like yeah, of course you can also ask questions about friend on this  You can study the sectional curvature for these kind of matrix  So maybe that's something interesting the other thing is what all told it what she studied  I gradient flows with respect to this remanus structure. So because our metrics are writing variant you get the big class of  remanium metrics on  On probability densities so you can do the same thing as study gradient flows and other things that might lead to something interesting  Okay, thank you  Moving on spreading  So two questions  Impact of this in this context here, so we have this money narrative H3 and  different steps to be higher  So so if I understand your question so correct you want to add some viscosity to this  And see what happens and I think  at least  So so for me this would be like adding a force that is not from a potential  So so now so now the question is if is that force going to be a smooth when you translate it, you know to  to the whole  bundle on the group and  the result  Like this is already given. I think in the Evan Marston paper for some type of  Dispation correct me if I'm wrong there are many experts here, but we haven't looked into that but possibly  I don't know  Number of derivatives  For example, you refer to that is so yes  And that's why we get some nice results  Then related thing is  These are home Tonya so equations so here  Not the fusion but dispersion may help  So I'm not even sure if say green nag me has a dispersion dispersion relation for free  There may be some week dispersion like a massive week dispersion, but KTV another shadow what equation this strong dispersion  And then you can use for example completely different techniques. You can tell your space  function space in which you are standing  The question problem to the queen of the linear term of the equation. This is what we're getting in this series of papers in the 90s  He came out with a so-called light spaces or space time space time spaces  where he got  global existence even for a to  initial data and used of course the  strong  And in analytic fallacies using so-called spear  But if in your context if you get these some of these equations with strong dispersion then maybe this is the methods to  Not so  They don't distinguish between the actions  Slavery they're good for certain types of equations, but may not be good for others  So yeah, so once I one thing I forgot to mention is for the green nag the equation first like Boris pointed out our  Existence result doesn't work for green because of this. We need this extra  condition on  the  coefficient function and the other thing is that people have already of course  Analyze this equation. So so so local existence of solutions is well known for green  It's well known for green magma already  Other questions?  So what about the noro's no-gain results when you look at the boundary?  Same as for the different one, for fully writing the right matrix is also problematic. You're not better.  What do you mean by problematic? Yes, you mean like  convexity or something like this? No, I mean  If you start which are given initial and final  configurations which are smooth enough, let's say hn and  greater than s  You have any results?  Every jubilance  It's the same problem as in the fully writing by itself  As long as there are no conjugate points along the jubilance  You can go higher in its days. Yes, but there's no global results in it. There is something from very special cases. I think on  Toros by  No capitalizing. We take something which  But  I think that's the way  Toros by  Not in full generality as long as they're the best thing that I know is from you effects  There you have something. Oh, is it only by markings?  Okay, yeah, so and that is the result with no conjugate points  Yes, we don't expect the first batch  I  I'll comment about what you guys talking so the finite dimension of this is like  different product  Problem  What you're doing, right? Yeah, so so that and and then I want you to say that the potential that you have that this electric operator, right?  It could be like the equivalent of the moments of inertia in the rigid body  being  shifted  So in this way one can control a rigid body  certain  positions, right  The tumbling where you can move the rigid body by controlling let's say two beats a lot of  Along one of these axes of moment of inertia  And what I want to say related to your question is that the shooting right would be affected by the way you change the role  So for different roles, you will get you so I'm trying to say that you can rock and control your flow  In a way that in a rigid body moving the beats you're going to control the position of the pipe  Yes, that's that's  Okay, but my question was a little bit off right I was wondering are there any peak on solutions like the Kamasa hole  You know that kind of  You know this thing  We could look at them I mean they go through and body problems  So, I mean if you take if you take the  Compressible or an equation that's the simplest the simplest example right it's barotropic equation the one Boris was talking about  It could be a first place to start  Yeah, you don't have  What do you have  So  In my dimension maybe to the  No  So  Yes  We yes, they started because for  Number of different inertial operators that are given by maybe the pseudo differential  Yeah, but still right  So this  My intuition on this would be something like this so because how do you construct the peak on solutions in the fully writing  Variant in case one way to look at this just take take the action  Of the diffio on some finite dimensional space right so the points just moving around some points, right?  And then you get the momentum up and the units the inertia turns to invert  The rock pulses and you get something right so but now you also have the density  So I would expect that you get the movement of some densities plus some point vortices that are moving around  Yes, so there seems to be  But the density changes are going to change the weight  But it's not even clear that I think  Laurent you mentioned that it's possible that  The direct delta dissipates  And leaves the space  immediate  So if that happens then there's not conserved anymore you need to show that you are  Delta the concern no, but look it's like this right  Disappear  Formally it's like this right you start with you start with the action of your diffium group on something and now the action will be on  On some on some finite dimensional space corresponding to the point vortices times  Let's say the densities right and then you lift that action to to a Hamiltonian action  And you compute the momentum up and and that should be at least formally that should be some sort of  Simplactic leaf where things are moving around right so so I expect that  At least formally this is something we're drawing it and I'm just trying to say you can write  It sounds  For the discussion for the long break and we'll take it right now. Thanks class  So let me just mention that there's a guided tour at one p.m. After lunch
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