 Prišli, da je božem, da sem tko veliko našel. Mi smo se vse izgledačil, tako da je to vse vse, zato je to zelo, da bi bom izgledačil. Prišli, da sem tko našel, da so prišli, da sem prišli našel v ključnih sku, če so nekaj dodajte, da se prišli. Početno, ta božem je na PDI, ali način je tko, ko je zelo, počelti teori, božem, da sem izgledačil, o če so taj začali. Všelo tukaj je to tako. So ta je in zame konlaboracija Assembly by daj student Eilio Marconi, ki je tukaj naya. Imamo to, da smo p fermenteda so. Znamo. Srečo je? The problem is if you have a conservation law, well, it's a textbook result to show that you don't have uniqueness and you need some additional constraints and the typical constraints is kako je vse entropij, kaj je zelo vsega, vsak je vsega. Na skala, na skala vsega, da je vsega, in vsega, da je učin, da je unik. V matematiku, ko je prefes, konvex, konvex, nekaj. Vsega entropija je konvex, in entropija nekaj nekaj, je vsega entropija, Enthropij is something that when applied to the solution of this PDE and using the chain rule, which is given here for at least the most solution, you have zero but for solution, which are whatever you want, distributional solution, you get that this distribution as assigned. So in other words, because as assigned this is a negative measure. So the entropy is dissipating. And this is the dissipation of entropy. So the idea is that there are several reasons why you need a sign here, or you prefer to have a sign here. And typically one of the main argument is that if you assume that your solution here comes from some approximation. So what some more complicated system which has some microscopy structures. Leto, da je bilo vsočen, prej zelo, da je zelo. Haj vseh jezak, ko je zelo, da je zelo, načal se pri infotru ljud vse vse vsezak. Vsezak je zelo, da je vsezak, kaj je zelo, načal se pri infotru ljud vsezak. Tudi načal se pri infotru ljud vsezak, nekaj ne zelo, ko je zelo, ko je zelo da je zelo, zero, ki je ta vsega zelo dzivutnega, ali je ta dobrojevačne, ne možem spraveni, ali zelo način, when you apply eta prime of u to the equation ut plus f prime of u, ux minus uxx, this guy here is just the chain rule, eta prime f prime is equal to q prime, so it's zero, well, the distribution sense is okay, In je prejko vsega zelo vsega delovina, ali druga delovina je vsega delovina, kaj je za tukaj uvratil. Zato, da je tukaj, ta je začin. To so tega pomembna, ker je tukaj za tukaj, ko je vsega delovina, je negativna. I potrebno, že vsega delovina zelo vsega delovina je zelo na tega delovina nekako to vsega delovina. In vsega... zelo, da smo ispeži vsega, še je aj načo vsi sprem svojo vsega. To je načo, da jim ti vsega, ko mi je vsega. Očeš, zato prijeva, je, da jim ti je vsega, je je počet, je je vsega, prijeva, ki bo oččila, jezelo, da je začeljim začetjamo, da je zelo, da je zelo, and you get again the equation, which is zero by definition, being a solution. Now starts something more complicated, which is B-V, so for B-V, B-V you have still a chain rule, which is the Volper rule, here it is written in a little bit more complicated setting, but essentially the Volper rule tells you that the continuous part of the derivative, so for B-V you know that if you is B-V exactly equal da se da vse začustimo, da je bilo nijak konfinist, zelo počo na delivat. Vseh razvajstva vzelo, je to zazavljeno, vzelo je zazavljeno, da je od vrštih, in tudi, tudi razvajstva tega, da je na prejzovat, zato je Vita, je v Vitah vzelo v Vitah, in zato srečno, da je delivat v Vita, in tudi je srečno, da je tukaj. In, OK, formula je, že eta je eta prime of u times the continuous part of u. So, being a measure, we have some part, which does not give any measure to sets, which are small, and then you have some the jump part, which is the, in some sense, the most, is the discontinuity part of u. Plus, and then is the jump, let's say, eta u plus minus eta u minus sum over all jumps. So, this is in 1D, in multi D, in our case it's in 2D. Well, the jump is controlled by a sum over lepšice curves. So, you have the two traces, the plus and the minus, and it turns out that when you apply the rule here to this distributional form, you get this guy over here, which is zero, simply because this equation implies that the continuous part is zero. The jump part, you get this relation, which is the rankinug on your condition, so it means the conservation. So, whatever transit from one side should exit from the other side, and instead for the entropy you get another relation, but in any case the important thing is that the term, which is not zero, is not this guy, is just the jump part. So, essentially this is what I mean by concentration. So, we will say that the dissipation of entropy is concentrated if it is concentrated on countably many lepšice curves. So, let me make a picture here. So, a priori, if you take the plane, you know just this distributional form less or equal than zero, so it is a measure that you don't know where it is, it can be wherever you want at a priori. In fact, the problem is that you have to add also the other equation, and I say that it has concentrated it instead of having this picture, you have a much nicer picture, so you have some curves, and the entropy is just concentrated on those curves. Let me say that this is really, it's not the worst case, it's the best case, because the equation is invariant by the solid, this is a similar scaling, and this condition is exactly invariant for the same, the similar scaling. So, all the continuous part of the divergence will disappear if you just scale t to alpha t and x to alpha x, and then just this part survives. So, it's the natural scale of the equation. OK, I hope this is clear what I mean by concentration. And so, now why the conjecture comes from? Well, there are several, I'm telling just how they tell me the conjecture, but in the lepšice or in several form is present. So, the conjecture is the following. So, it is related to a model for, you need to know the speed of convergence of a kinetic model with a stochastic model to one dimensional Borgere equation, which is this special case. And you know just that one entropy, which is this special entropy, is assigned measure, not, you don't require to be negative, just as a measure. And now, they want to know if it is concentrated, because in that case they can compute, they can prove what is the gamma limit of some functional, and this functional gives you, OK, they know better than me, but it gives you the speed of convergence of these approximations. OK, so concentration means, as I said, that you can find, can't tell you many cores, the purple here, and you know that this distribution is just a measure along the curve. And the measure is exactly, well, h1, so the length. And this is because the vector field is in L infinity, and it cannot see anything sharper than h1. OK, so this is the conjecture. However, I cannot solve it, so we prove something else, of course. So, let me start before, what was known in the literature. So, the first case is in the case when the flux is convex and concave. So, it's, and it's exactly this case, but the only requirement is that now the measure is negative. So, in that case it's much easier, because you know by several works that essentially the solution is bv for any positive time, because you have tk estimate. So, the positive part of the derivative is bounded by a measure, and this case is 1 over tL1, and then it means that this guy here is a measure, and then you deduce that u is bv, and then you apply the same chain rule. In multid is clearly more complicated, and as you see that the conjecture is presented in set, also in multid, can be stated independent of the dimension. So, in this case what we know is essentially that you can select a rectifiable set, which replace the core with some leap sheets, the dimensional surface, and it turns out that at least one part of the dissipation is over there. But, so I'm saying that the jump part of the dissipation, so the sharpest one, the less regular one, is along surfaces, but you don't say anything about the rest. So, it can be some other part. And finally there is a result in the case when f is, as some regular, in this case there are finitely many inflection points. So, it's like something like close to this guy, but you don't have this estimate here in that case, but still something similar can apply. And in fact what you obtain is that not u is bv, but essentially the characteristic speed which is the derivative of the flux is bv. And there are some additional conditions, but in any case the result is that the jump where the characteristic speed jumps, you select those curves. Being bv, you have at least a way to find the curves. Ok, so the problem is that, ok, you can forget about the picture, but the problem is that in general is not true, and this was an example. Ok, so you have infinitely many inflection points, clearly, otherwise the result would be correct. And, ok, you just take a model set, you prove that for this model set, the parameter, one parameter converges the other, not you put, you pile, you piece them together, at the end you get that f prime is not bv. Ok, this is not interesting. So, ok, so I think this part is better to understand with the picture. So one of the main things that you have in 1D, I think, is the fact that you can use the method of characteristics. However for the method of characteristics is ok, and so the method of characteristics means that so you take again your equation, which is ut f of u x equal to 0, but then, ok, you write in a cos linear form, so f prime u x equal to 0, and this is known, it means that u is constant along the curves, so u of ut, let's say x plus f prime of u in the same point t, ok, ut plus f prime of ut, t, sorry, t is close u f prime t is equal to 0. So u is constant along straight segments, and the thing is ok until as the standard example, what happen is that, well, when the characteristic cross you have to continue in some way the solution, you know that the solution can be continued, but clearly this representation is false. And at the end, when the method of characteristics is used is in the case of Burger, so let's say the f second greater than 0, because in that case what you can prove is that f prime, ok, is a semi convex. So it's monotone, let's say monotone. In that case there is a good notion of solution of anode of a differential inclusion with the right side, the monotone operator. And it turns out that essentially what comes out when they meet, you just follow the discontinuity of f prime, and essentially this discontinuity is measure 0, so you recover the solution by just continuing the characteristic from t equal to 0. But clearly in general this drawing is false, because this condition is not, as I said is already solved. Because you know that you don't need anything about the method of characteristics. So what I want to present with this picture is how the method of characteristics can be extended. So the idea is that instead of having, so instead of, sorry, instead of taking the initial data, you forget about the initial data, and you take just the flow x, and all the thing is that this flow generated by, in this case, is monotone, so they cannot cross each other, so you require this flow to this monotone, and then you associate to the flow a value, which is not necessarily the initial data. And you say that this value is a good value until some time, which you decide, for example, which you have to choose at some point, so let's say this value, so this curve here stays on the graph of u. Clearly, being a solution u, not, let me say, not a continuous function, so the graph is not a closed set, you have to specify in which sense is valid, and so we can come back. And the sense is the following, it's written here. So at the level, so the value of u can be just, or the derivative of u is just the image of the derivative of w, where w is this value here. So you associate the value w, and then this value w is okay until some time t. So essentially what you say is that I have a graph and a flow, or I have a graph w, which depends only on some parameter y, and the parameter y is not related to x, just in r. And at any time t, I have the map x to y, so these maps, which is a monotone, so w changes side, changes shape, and I say that the graph of this guy over here exactly contains the graph of my original u. So this was done at the beginning, at least for bv by my former student, Stefano Modena, the definition is very complicated. Another important thing is that actually you can prove that if the solution is bv, the derivative of the flow, so this flow x solves the characteristic equation. So when u is continuous is the derivative, otherwise if you have a jump, you take the speed of the discontinuity. Okay, so at the end, this is the extension of the method of characteristic, at least for bv. So the problem is that for l infinity you have a big problem, because if we come back to the formula here, I have to specify w in the set where you see x, now the derivative of x is a measure, but depends on t. And so w should be defined, almost heavy, we expect to the family of measure, the yxt, which is not a priori, but can be very bad. I don't know a priori if this, I mean at the end could be, maybe you need to have it, they define for any time, can happen that, and that clearly is not possible to do. I mean you can show that. So the idea is to change interpretation. So the idea is to interpret this curve here in another way, and let me take the picture, so, sorry, let's forget about it. So the picture is the following. So this was the original idea, this is just the value of u, on the other hand I may think of following. So let's cut the solution only on one side of the curve x, and I give you the value w. So I'm saying I want to solve the, now it's a boundary initial problem, where here I take the initial data, and here I take this value w, which was my, in the Lagrangian representation before was the value that I choose. I can do in one side, I can do in the other side, and it turns out the following, that if you take u plus, define u plus as the solution on the right side with that boundary data, or you take u minus with the solution the left side, and you piece them together, so essentially I'm saying that you can do the following operation. So I have this, I have this curve x, and I have a solution, so ut solves, ut plus f of u, x equal to zero. Then what I can do, on one hand I can solve u plus minus t, this solves ut plus f of u, x equal to zero with the value on gamma is, let's say t is w, gamma plus minus, this solves in x greater than gamma t, or less than gamma t. And if I cut to let's say x greater than gamma t, or less than gamma t, here I obtain u plus or minus, let's say t is with the tilde, and they are the same. So essentially I can first, I solve the whole of pd, and then I cut it, or first I cut and solve, and I get the same thing. So this is a nice structure, because it turns out that, sorry, it turns out that the following holds that this notion of boundary is stable for a lot of converges, so in particular, is stable when it is the solution converges in L1, the curves converging C0, and the boundary data converges, well, in R, essentially. So it turns out that all these notions of convergence, they leave this picture invariant, essentially the convergence in the product space. In the end I can just take, so once I construct this value for a bv, I can take the limit in Kuratovski, or the uniform limit, and what I get is a close composite of all the values and all the curves, such that this picture holds. So gamma is some sort of generalized characteristic and I don't know yet. Yeah, at the end would be. But at this point, no, because it just requires convergence. It's just a boundary value, such that this picture holds. OK. So in fact, what you get is you get a compact set, which is made by three components, so you need the curve gamma, the value w and the time t, and t is less or equal to some function, and this function is t-scorp as a boundary value. And well, as again, the monotonism is preserved by uniform convergence. This is preserved because of compactness of boundary values, admissive or binary values, and because of this compactness also the function will be upper semi-continuous, because any limit is inside your set. And now it comes something more delicate, which is related to the PVE. So one relation is that essentially all your values, well, essentially they have to, they contain all the, well, in PVE I would say these contain the jump set. In general what I can say is just if I fix some two values, y, so two curves, the set of admissible values is connected. And the only relation to the PVE surprisingly is this relation here, which tells you that the speed in any ball is given by, well, all the possible speed you can have. So the value in some point is the intersection of all the possible speed you have in the nearby balls. So this is just because they come from a situation where this holds at the beginning. So the picture is like this. One essentially you have the set y and w, which are the values, and for any y, w I say, OK, this is a good value, so y is, we miss a curve gamma y up to some time t, y. OK, this is a compact set. This is the typograph of this upper set be continuous. So the problem is what I do with this thing over here. OK, this is the picture essentially in the tx plane of the meaning. So essentially fix y, along this curve I can have several values, no, because I'm not saying that it's just one value. In fact, it's not just one value. And for example, the red value, w, y is OK up to this time here. Then something happens. Here is for bv clear, I cannot draw some L infinity function, so t is just a bv function. And t is canceled, these values not needed anymore is canceled. But the green value survives bit farther. The thing is that if I cut this time with the green value, I recover the solution on both sides. If I cut up to this time even also with the red value I recover the solution. OK, so now once we have the characteristic in some time, let's call it the characteristic. One of the methods here for the monotone is the maximal and minimal characteristics. If you want to know the value in this point essentially the domain of dependence. So you say, OK, let me take the maximal characteristic coming from this side and the minimal characteristic coming from that side. And then I say, OK, this is the domain of dependence essentially. And to compute the solution here only to study the initial data here or the equation here and do my business depending on the question. Now I can do it because X is monotone. If you give me a point I can find what is the last characteristic on the left passing through the point. The maximal characteristic on the right passing through the point and this is the positive. So the infimum of all characteristic greater than that guy and the minimal also on the other side. So now I have just to consider the cases. Here is very easy. It happened that you have just one characteristic and in general you can see for the monotone the picture is like this. You cannot do anything better. Actually it's very nice. But while here clearly situation is more complicated. So the first thing is there may be in that point two characteristics at least two characteristics. The number of characteristics passing through X at time t is greater than one. Well, and now I know that maybe I'm confusing you but let me recall that for bv is very easy to say what is a jump. Because you have the right and left traces. You said the right left traces strong right and left traces they are different and then you recover this jump set. For L infinity is not clear at all what is a jump set. So how do you define a jump set? It can be this continuous everywhere. Ok, so but the natural thing is if you look at the picture here the fact that this is a discontinuity is exactly that I'm coming from two sides and since the speed are different the value of u cannot be the same because the speed is f prime of u. Ok, so I just say and this means that two characteristics they join in that point so I say ok, my shop set is the set of points such that at least two characteristics pass through that point. The fact that the map is monotone means that if you have two characteristics there is a small interval and then it means that if you take a dense set of y it is countably curves countably many curves they cover all this set. So we recover at least we found countably many curves. Well another thing is that while you are approaching from the left at some point one characteristic touches before. In the case where it is convex this cannot happen that's why f prime is monotone but in the general case this happens and in fact all the troubles are exactly there. So it touches before and then well it changes. And now the thing is that inside here recall that the values along this curve blue and the green curves are some boundaries. So essentially in some here the solution is given by solving my PDE but with this boundary data and there is no initial data because at some point they join and in that case you can prove that the solution is basically explicit it is like a Riemann problem so the building block and actually is monotone is BV as all the good thing you expect particular for us means that is BV it is an open BV region and for BV I know already that the entropy is concentrated. So these regions are I don't need that. And finally the last case is when I'm approaching and I never touch and this happens the magic is at this point because it turns out that in that case is a straight line. We can forget the proof because is a little bit technical but let's see the implication this was if you remember is the original picture it seems nice but at the end it can be very complicated because you have the flow is not unique in the future and not in the past and you have by forkation you don't know what to do. However the picture, the true picture so you have this countally many curves these BV regions and the rest are straight lines so the picture is much nicer than what one can expect. So far so good, so at the end how to prove the conjecture countally many curves I'm okay in the conjecture if the entropy is over there is okay. BV regions, well BV I can apply the chain rule so here is still okay but the thing is to analyze is what happen on these segments. Okay the fact is that I can take the balance now it turns out that what is the play between the dissipation and the equation so I can make balance along regions of this form for the two equations so first I have ut here so it's here that the magic happen so let's start with the balance of the second equation so I have the flow on the top minus the flow on the bottom this epsilon is just because I want the segment to be Lipschitz and is equal to the dissipation inside this is exactly what the second equation tells us and what is the result the fact is that since this segment they exist from let's say 0 to 1 they cannot the dependence of the speed is Lipschitz otherwise they will cross no and so I have that the density of y so let's say is the length of this segment as t changes is a Lipschitz function can be computer over here okay so we apply this balance now we pass to the limit to the first equation and again here you have 0 now you pass to the limit you replace this guy with the density so change of variable the derivative of the value of u times the change in density is equal to the derivative of the flow okay this is exactly so this is the derivative of u times d sorry u times d the value and this will be the derivative of the flow and now you have to compute okay just you prove first of all the derivative in respect to t of this guy is 0 and there are two cases okay one case is that well we can forget it in any case is that you can show that this guy is derivative 0 and you recover the most important formula which is this chain rule so let me say what this chain rule is important because if f is smooth that is trivial no f u if u is smooth lambda u lambda is f prime of u u y you differentiate by standard calculus get minus f second so u u y u the only thing is that for u l infinity this is completely nonsense no because I am computing l infinity times the distribution and okay nonsense however this guy was lambda so my characteristic speed depends on y and this guy is lambda the derivative respect to y of lambda y and now you see the formula is completely meaningful and this is exactly the chain rule so usually in this problem what is missing is the chain rule so you need the fact that you solve a PD tells you that u is not any l infinity function but satisfies some additional regularity and actually it turns out that this is the additional regularity okay once you have the chain rule well now you reapply to the entropy okay let me forget and essentially you get the second chain rule which is this guy over here and this okay up to some computation um sorry okay this is a little bit complicated but if you want I can explain in the case it turns out that the second chain rule tells you that the measure is just okay it's just the jump part of q you don't have cantor part and the fact that is the jump part is in some sense the the sorry the conjecture will tell you that this value only jumps and the jumps of q is exactly since u is continue is exactly the measure over here okay and then okay you can count them is not important so at the end of the basic rule the basic fact is that you are able to prove an additional chain rule and as for the bvk you can use the chain rule to cancel the term you don't need okay so thank you for the attention so context related question you said that if you take an infinity initial data then you generally don't expect the primal you to get into bv and that's part of the difficulty is there any smooting effect in here any regularizing effect that applies to fm but it's very subtle it's subtle because you have to state in the language of measure you cannot say immediately so essentially it turns out that well there are several cases the smooting is that once you select these curves so you have cantalumene curves which are the curves where something happens okay and now you ask what how is the solution in these curves so essentially in bv what you know is that if you take a point outside this, along each curve have the left and right trace in l1 well here you cannot say that but you can say that there exist the left and right trace in each point up to the linear energy component of f prime so let's say these are the components where f second of u is equal to zero so the connected components of this set so what happen is that well you cannot say what is the limit but you can say that the limit exist in the quotient topology and the u is not bv but the structure up to this quotient topology which means that outside this cantalumene set you have here is a continuity point in this sense of course and here you have the jump part in that sense but if you know the limit in this topology you are able to reconstruct the speed so like for bv you can obtain the Rankino-Gonier condition and essentially is you recover the formula very close to the bv setting so that to state it is complicated infact I prefer not to I just that is more complicated it is not that complicated it requires some tool but yes there are also additional things for example that well you can find in the representation of this boundary more regularity ok maybe just a question how is this related to kinetic formulations ah well the problem was if one open question is whether the right hand side of the kinetic formulations is not just the derivative of a measure but it is a measure and we don't know we try to prove that it is a measure that is not so so clear because well clearly you have once you have this guy over here you know that the kinetic formulation because you have the continuity so let's take the t-set of just points so it's a weakly generally no linear so the limits are taken strongly which is outside is continuous and then you have the strong traces so you know where this measure is going to be concentrated however so everything is explicit in some sense but your bound is just the projection is a measure not that the derivative of that thing is a measure you know that I think you know what I am talking that when you have the derivative of u the dissipation of eta derivative over k the dissipation of eta k and we don't know so we know the set we know everything we can compute everything that is bound