 Tukaj da bo se počičati, da je tko je vzpešnja plaj, in da se ne vzeni, da se ne vzpešnja plaj. Zelo se, da smo se načali, se da se načali na projekte na sodaj Chidi Shanga, ki je však však v Šanghaji. Parti površt, parti v projekti. Vsoj je dobro tega pa vse. Seltim se, da bomo nekaj nekaj na odliček. Kako je to problem, kako je vseč izrečo? Zelo nekaj nekaj najsakimi generali poslusti, ker v seče, da se nekaj nekaj nekaj prezipovati, tako vseč nekaj, da nekaj mače pred dva naši medifoldi. Tarket menifold je remanjan menifold. Na zelo, basen menifold je komplet Lorenzian menifold, če so vsega zelo vsega zelo. To je vsega nočnja, nekaj napoč. To je kretik pojel na lagrangian. Lagrangian densitiv je vsega, Zelo je tako halti v hrojnji zrani z vsego setjju V. Vsega način, je zelo izvaj, da se to vsega kodrata vistsi mokriči, nrej tukaj. Znamen jaz, da ta tukaj hrojnji je hrboliče, bo, da je vsega skrboliče hrboliča. One, the easiest way to feel, the equation is to think of it as a hyperbolic version of the harmonic map equation. So this is a very natural generalization of the harmonic map equation, but of course rather different. Maybe it's not necessary to advertise for them because they are well known, but they pop out in several different theories, including general relativity under the assumption of the existence of some kind of fields you get with map equations in some dimensions. So let us consider more concretely what kind of equation we have to deal with. So the standard situation in which maps are studied is starting from flat Minkowski space, of course. So in this case M, well, usually one should write RT times M. This is the simplest and best understood case, but let's say flat Minkowski space, dimension N in space. If you expand and coordinate, you get a system of nonlinear wave equations, quasi-linear wave equations with a very nice structure. So gamma are the Christoffel symbols, and you get a quadratic form in the derivatives, which is a null form. This is sometimes called the extrinsic formulation, why the extrinsic formulation is nicer from a geometric point of view, from a formal point of view, but it doesn't lead to... I mean, it's more difficult to work on it using this formulation, of course, but it's nice to know. So if you embed isometrically the target manifold into Rk, then you can just say that the wave maps are the maps such that box U, the standard box U, is a vector always orthogonal to N, and U takes values in the target N. So this is the nicest way to look at the wave map equation, but not the most useful. So if the target manifold is realized as the zero locus of some function, then you get some very nice expressions, global nice expressions for the wave map equations. So for instance, if the target is a sphere, you get a very symmetric equation, of course, and here you see very clearly the null form structure of the nonlinear part. And the similar, more general expression you get if S, the target N, is the zero set of some smooth function. So you get always the same kind of structure. So some functional U times null form. In this case, when the base manifold is flat, you have a lot of results now. The theories, some parts of the theory are really well understood. There are some corner cases, which are still under intense study, but especially two-dimensional case. But of course, the natural setting is the Cauchy problem with the initial data. Here it's not written in a very precise way. It should take values in the tangent space, the second data, but you understand. It's hs, the usual scale of several spaces. And you look for functions which are continuous in hs and c1 in hs minus 1. And you immediately check that you have a nice scaling, which is given by this. So the critical space is hn over 2, the critical index is n over 2. And you have immediately the natural expectations on the situation for this problem. This is completely standard. So the problem is critical when s is equal to n over 2. The problem is super critical when s is lower than n over 2. You expect something bad to occur. And if s is larger than n over 2, you expect well-posedness in some sense. In the critical case, maybe one should put a question mark here, because it is clear that in that case the geometry of a target and base manifold, in this case a target, should play some more important role. But basically this is the picture you expect. And this is confirmed from the local theory. So the local theory behaves nicely. So essentially if s is strictly larger than n over 2, you get local well-posedness. This was proved by probably the first application of the bilinear method in the context of wave equations, to my knowledge, by Klenderman de Mercedon, and then perfected with Runjanski. On the other hand, the situation for the s smaller than n over 2 did not receive a lot of attention. Anyway, it's not too difficult to construct very strong examples of ill-posedness. This is written like this, but actually our counter examples are in dimension 2, so target is the sphere. Myself, I didn't think too much after that paper, but I think it shouldn't be difficult to check that also in higher dimension you get essentially the same. Maybe not that strong, because here you can write two different weak solutions, but some pretty strong ill-posedness results. In the critical case, you see that not everything is nice. Following work by Tatarro showed that actually it's a good space, so you can construct solutions which are limit of sequences of smooth approximating solutions. But at the same time you can prove that the solution flow is not well behaved. So you have some troubles, because you can do this very explicitly. Anyway, so let's go on. The global theory is the nice problem, the interesting problem for the web map equation. So you have a good result for small data, and this is also expected. So if you have small data in hn over 2 and targets are reasonable, so you don't really need, of course, any special assumption on the target, apart from trivial ones completeness, because small data, they will only notice a small region of the target. So essentially this is what is expected. On the other hand, for large data, and this problem becomes already rather difficult, you can prove blow-up in funny time. So if similar blow-up when the target is a sphere in dimension larger or equal in 3, if the target has an open geometry, then it's more difficult, because it still can produce blow-up. By the way, let me notice that low-dimensional blow-up, I mean, if the target has low dimension, I don't think anybody has yet proved blow-up for this problem. So this is interesting. Okay, anyway. So this is the general picture. There are many results, but these are the general results, so maybe something for large classes of equation. The 2D case is especially interesting, because the critical space is also the energy space, so you have the question of maybe the local, you expect the local solution to be continued or not, so this is really the geometry of the target comes into play here. So in group results of several people, some are here, this is still undergoing, the complete study. Anyway, for instance, if the target is the two-dimensional sphere, you have blow-up, if the target is the real hyperbolic space, you have global existence, and there is some going blow-up classification and code proved a nice solid resolution in the case of a sphere. Okay, this is a related result, but maybe not relevant here. Now, let us focus on what is the corresponding to radial solutions. Let's say that the equivalent web maps are the corresponding to the radial solutions for the web equation, and of course it's a more general situation. Actually, from an historical point of view, this was the first case studied, but this is the subject of this talk, so let us consider the special case now. Now, the target is a rotationally symmetric manifold, so you have a radial coordinate, and then the k2 is just the standard metric on the sphere, as l-1, and you have this coefficient, so essentially the metric is completely determined by this function here, of course. So if you pick a different function, you get different geometries, and, okay, you do some standard computations for the Christoffel symbol, so in this, for this kind of target, remember that the base manifold is always Flatminkowski, for this kind of target, the equation undergoes a dramatic simplification, of course. Let me just show you how to think about this coefficient, sorry, this function g. So, for instance, if you have a sinus, so a function with another zero, you have a closed manifold, like a sphere. If you have a function growing, you get an open geometry, like the hyperbolic space, so hyperbolic sign, okay. So, the equivalent web maps are maps which are completely the couple, so the radial component of u and the angular component satisfy two different sets of equations. So, you assume that phi, the radial component of u, depends only on, of course, time and on radial variable on the Minkowski space, and chi depends only on the angular variable on the space component of Minkowski space, and you get that chi has to satisfy just the harmonic map equation from two spheres, and so which gives some rigidity to the dimensions, of course, then you have to pick L, given by this formula, of course, and phi satisfies a nice equation, which is just a radial, semi-linear wave equation, but with some singularity here, which makes this third critical, okay. And the critical space, of course, is still h n over 2. So, this is the equivalent web map equation, and this was the first model considered, of course, already in the 80s. Of course, the previous results, the result I mentioned previously applied to this equation, still have global existence for small data, and actually the blow-up results are known for precisely this model, the results I mentioned before, and you also have a global existence of large solutions in which the target is sufficiently open. So, this is the relaxing condition on the metric of the manifold then, and it can be relaxed to this one, okay. So, now, this is a picture, a very spotty picture, in the sense that I didn't mention the many, many results. There is a lot of ongoing work on understanding the blow-up mechanism and classifying the possible rates of blow-up, and there are some more precise results, even for the small data existence, but, okay, this was just a very general introduction. So, now, the problem I'd like to address is what can one say at the same level of generality, so the basic results, when the base manifold is curved, you have geometry also in the base manifold, no flat geometry also in the base manifold. This is a partial list of results. Maybe there are others, but, I mean, there are not many results in this case, attacking in generality the problem. For instance, there is stability of special solutions in old result, but, okay, this probably the results by Lore and others are the only ones that try to prove some result in more generality. So, the first goal here would be to re-prove the global system of small equivariant windmaps in this setting, okay, in the equivariant case. Of course, this is just the first step, but, I mean, already, there are some nice... there is some nice mathematical impulse. Okay, so, first of all, we would like to see precisely what kind of equation we have to deal with. So, now we have two manifolds which both have rotational symmetry. The first one I call the function, the radial component of the metric H, and the second one is before G. And the equation becomes this one, okay? Of course, the R is replaced by the function H. This is the radial box H on the manifold M. Okay, so, you see, the structure is very similar, of course, but you have this difficulty because of the presence of this... The presence of this operator with the variable coefficients here, and so what can we prove for this equation? This is the list of what we prove. I shall show later some explicit statements, but essentially this is... these are all natural results. There are no surprises involved here. So, what we prove is the global existence of a small equivariant windmaps in the critical space, the target manifold is arbitrary, the interesting part is that we outline a class of manifolds, base manifolds on this, the computation is not especially difficult to... I mean, we can do this, we can reproduce the procedure on flat space. You have the usual version of the theorem with local existence with large data. This is a non-difficult modification. You have to work a bit harder to prove a higher regularity result because everything... There are some parts of the equation which are singular, so you don't get actually arbitrary high regularity but up to some degree. And this is useful because it gives you an unconditional uniqueness result by just improving a bit regularity of the initial data. I will show you what we can do concretely. Let me also mention that we have some work in progress, but I'm not confident enough to announce anything. So the 2D case can be done. I mean, it was just a technical problem. I think we removed it. And also, we have strict estimates. Also, the manifolds should be rotationally symmetric, but the data could be allowed to be non-radial. So you have to work harder, much harder for this, but it's possible to do essentially just with strict estimates. And this is still to be checked, so I hope in the next two months. OK, so this is the definition of the missing manifolds. So try to get the biggest possible generality. It's a list of conditions, but you will see quickly that there is essentially a couple of conditions which are important. So first of all, OK, I recall you is just the function which drives the metric of the base manifold. So I form this quantity here, and this is second derivative in radial, of course, second radial derivative. OK, these three should probably decrease to two. But anyway, so first requirement is asymptotic. So I require that this function tends to a constant for R large, with a certain rate, a non-negative constant, OK? This says that there is some convexity at infinity for this function. For instance, this is for the hyperbolic manifolds, this is n minus 1 over 2, to say, something. And this is really not, this is typically you check just by looking at the function, OK? So this says that for large R you have a certain number of derivatives decay like 1 over R. So typically this is trivial to satisfy the second condition. This one is, of course, very restrictive, and here is another restrictive condition. So first of all, the metric must grow linearly, so you don't have closer manifolds certainly. And this function must be positive and decreasing, OK? So this is a global condition, which is harder to satisfy, but you should notice that, OK, you should see with the examples that actually this is really effective only on, say, some bonnet set. So this is an asymptotic condition and this is a condition on some compact set, because it becomes very easy to satisfy for large R, as you will see, OK? I don't know if this is very convincing, this list of conditions, but let me show you some nice examples, OK? So you can appreciate that there are a lot of manifolds in this category, some of which don't trivial. So first of all, the parabolic space is included. It's easy to check. So you see the function h is a constant, h infinity, plus something which decays extremely fast, and so all those conditions are easy to satisfy. The only one which is restrictive concerns this, you have to form this function p and check that it is positive trivial and decreasing almost trivial, OK? So this is easy. But as I will mention later, once we have one manifold which satisfies this condition, we have an open set of manifolds, OK? Our condition is open and we can quantify this, OK? So let me show you what we get immediately. You can perturb a chain. So instead of just a parabolic sign, you consider any perturbation mu, which satisfies this set of conditions. So some derivatives must decay, like epsilon, epsilon should be sufficiently small. So notice the following fact. This is, of course, increasing exponentially. So you see that this condition is, OK, is effective for r small, let's say, but for r large, you have a lot of manifolds, a lot of geometries are included. OK, this one is really not strong condition, OK? So we have a whole class of perturbations, but I don't know if perturbation is the corrector, because at infinity this can be quite far from a chain. Of course, the other test is if we can include asymptotically flat spaces, and this works easily. I mean, you perturb the flat spaces, h is precisely r, and you can perturb with something which decays, like a simple, and this is actually small, a small perturbation, OK? The reason why this is so dramatically different from the other case is that what is important is the ratio between this mu and h. So if h grows a lot, this is automatically small, OK? But, of course, these are just the first two cases which come to mind, but it's quite easy to construct a lot of examples which enter this set of assumptions. For instance, if you want to construct an arbitrarily growing exponentially growing metric you can do, if you want some polynomially growing metrics you can do, and once you find, it's nice because once you just find one which satisfies the constraints then immediately you have a lot of examples by perturbation, OK? OK. So this is the type of conditions which we get when I say perturbation. If h is an admissible metric, then any type which satisfies set of conditions like this, actually you also need another set of similar conditions, but without epsilon up to n minus 1 over 2. But essentially this is the restricted condition, OK? So you see, it's not really necessary that this is small, but this must be small compared with h, which can be big, of course. OK, anyway, so this is the general picture. We have three different perturbation results which allow some freedom to produce examples. OK, now let me show you the precise statement, so which is, I mean, what you completely expect at this point, I think. So this is, we consider the Cauchy problem for the equivalent with map equation between two rotationally symmetric manifolds with this coefficient coming from the harmonic map equation. H infinity plays a role in the statement and this is the limit which is assumed to exist of this quantity. OK, so this is not a notation, this is a standard definition of subalive space, sorry, shouldn't have written notation, this is a standard definition of subalive space on the manifold dam. And I also used this notation of type Q, the weight is this quantity, OK, this in some set of coordinates that should probably easy to write a simpler expression, but, OK, it's not a point of I would get rid of this quantity as much as possible. So there are two statements, slightly different statements, but essentially they say exactly the same thing, so depending on if H infinity is strictly positive of zero and so you have go by systems with sufficient small data in the critical space, OK, that's it. And of course you have the usual appendix, so since we use strickers estimates, you get an additional property of the solutions, and also an additional constraint for uniqueness, OK, so you can prove uniqueness only in this restricted space. OK, in the case H infinity equals zero, you just have to slightly modify the definition of the Hs, so meaning that there is a small part of the derivative, which is homogeneous and the rest is non-homogeneous. This is technical and could probably completely be changed, but I mean, this is essentially the same result. And this is our choice. There is not much freedom in the choice of the strickers respondents, I mean, which appear here, but anyway, probably there are other exponents that could also work. OK, so this is the main result. Of course, as I was just mentioning, you have this annoying problem that you need an additional condition for uniqueness, so this additional condition in some strickers space and you have the natural standard question about unconditional uniqueness, so if it is sufficient, it can do there exist other solutions with the same initial data in Hs in the space at infinity Hn over 2. Note that, I mean, this problem was solved not too long ago by Prasmodi Pranshon in the flat case, I mean, when the baseman was flat, and the trick is just to go below with regularity. I mean, so you work in some space of negative regularity. I mean, it's just technical and it's a nice idea, but the idea is simple. The problem is that, I mean, you then have to estimate some linear quantities in some norms, which are not easy to handle, and this is more complicated in our situation, so I guess that it should be possible to do the standard conditional uniqueness result also in this setting, but it's technically slightly more difficult, so we didn't yet do it. What you can do is, I mean, and this is just by implementing the same techniques carefully, if just first of all you need a higher regularity result but this is not difficult, so if the initial data are slightly more regular up to certain order, which is determined by the singularity in the linear term, then the regularity is preserved and if as is, if the initial data are slightly more regular than the critical space, then you have unconditional uniqueness in this space, and this is not very large but not arbitrarily small either, so not larger than 1,6, but ok, so this is also more or less basic result and ok, let me just describe how it works since the situation is really simple, because the equation is just I mean, and radial wave equation with some funny coefficient I mean, so it's not too difficult to reduce to non-techniques, you don't use hard techniques but still you have to work a little so the first step is I mean, this is our approach the first step is proving directly on the manifold proving estimates so this is a well-known way to prove Strikert's estimates so first we prove splitting estimates by multiplier techniques then we change coordinates so the equation is reduced to a wave equation with singular potential so this is the only annoying part of this proof so then you have to prove estimates for this equation but since you have already splitting estimates this can be done nicely at this point you have the necessary tools which are more than sufficient I mean, you don't even need the full strength of Strikert's estimates just some point to prove, to implement the fixed point method for the critical semi-linear wave equation you have some troubles in the coefficient of the inside and outside the non-linear thread but this is just, I mean, technical you have to prove some radial estimates with singular coefficients it's not really a big problem you have to work a bit harder for the higher regularity and and then you have to read the result in the original coordinates, okay so you have to go back to the old sobo-lab nodes and then, I mean the rest of the work is essentially to construct manifolds to see how how large the class is let me show you so this is the change of coordinates that you use to reduce the equation to this form so essentially you get standard wave equation you increase the dimension by 2k which is nice because gives you some better in some sense better properties the problem is that this potential V is singular and critical decay and you know that if you go if the negative part of this V is too bad, too big you lose everything there is some part of risk here and also here you see it's a essentially you have just a singularity here but nothing intractable gamma is so there is a linear part and sorry which has been taken here and then you have a cubic term okay so this is a cubic similar wave equation with singular coefficients which has the same critical space as the as usual n over 2 and essentially this is what I just said I can probably skip this okay this is just to see the estimate we can prove the full set of estimates with the exception of the end point also in this case these are the standard homogenous trickers estimates with the usual conditional admissibility on the pq and this is the in homogeneous version this is precisely the set of estimates we can prove okay in some coordinates it's a way okay let me briefly recall you that there is now an excessive literature on strickers estimates and dispersive estimates and smooting estimates for various perturbations of the standard equations for the wave equation we need something which is a bit borderline because we as a critical decay there are many results this is probably the first one so then the first group attacking the case of critical decay was work plancheon starker and then we did some work on small and large magnetic potentials there are many results and maybe I should also mention that for genetic second order small perturbation of constant coefficient case but small tataro is probably the best result okay so the first step is to prove a smooting estimate for the resolvent operator so this is the plus bell tram operator on the manifold dam and this must be very precise I mean it's easy to prove smooting estimates with worse coefficients but this is actually sharp so here you have to work a little once you have this by the suitable change of variables you get also the estimate for the resolvent of your perturbed Laplace operator now at this point this gives you immediately a smooting estimate of the following form this is a standard kato teori but notice this is of course a shreddinger operator the shreddinger flow but this is for free from this you get this so typically you should work a bit harder to transform this in corresponding smooting estimate for the wave equation so after maybe the fifth or sixth time I had to do this in general so you can patch kato teori so consider any self-adjoint or negative operator on a Hilbert space H any closer densely defined A if you have a smooting estimate for shreddinger you get this smooting estimate for the wave equation so anytime you need this you just can apply the kato improved teori of course in the case of the wave equation you get some worse estimate so this is one half derivative typically A contains one half derivative it's a weight the typical estimate here is a weight and one half derivative ok, let's forget about this ok, once you have this estimate for the wave equation then the potential can be put on the right hand side it's just perturbations ok, you have to work a little because of this annoying coefficients which give you weights which are singular but essentially it's simple ok, once you have a stricar estimate this is just to recap the space in which you get contraction this is more or less standard you have to pick carefully the p and q but you can do triz and you have to work in different spaces according to the value of h infinity ok I think ok, let me just mention that the only annoying part in this computation is that you need to be careful when you need new nonlinear estimates for this quantity in fractional space this is the only online part where you have to work a bit hard to produce the suitable estimate with Renato Lucca we developed some theory for fractional integrals to handle which handle also this case ok ok, I think I can stop here thank you for your attention thank you it's time for questions this is more a remark than a question if you want to play with domains which are rotation is symmetric you can consider for example if you replace sinc function you have for hyperbolic space by a cos function you would get the domain for which you have the regularity because there is no sample so this would be cylindrically symmetric I need some input on the interesting examples of course otherwise you can just go in any direction how big is the step to get from radiality to general ok, if you mean same geometric setting for non radial data this should be I think contained essentially in the method of shot times true where from 94 should work we prepared the stricars estimate in this case for non radial data so this should be the only essential tool but we still have to check non radial geometry this is of course another story then you need completely different techniques this is a simple approach but I mean it's surprisingly effective because you get really a lot of spaces just with not too much effort there seems to be no more question then you have time before dinner thank you