 Allora, devo cambiare l'ultima trasparente per dire cose fantastiche che lo faccio, siccome non sarai. Non ci sarò io. Foi anche parlare maledì? No, ma le no. Ma le no, sciocchetti. Ok, so, welcome to the afternoon session. As you already know, Professor Cabret, flight was cancelled so there will not be stopped this afternoon Enrico Valdinossi will do the second of five lecture after Professor Vascats, who is going to talk about non-linear fractional parabolic equations, problems in bounded domains. Please, please. First of all, let me thank the organizers for the kind invitation to this meeting. I come to Italy, which is like my second home country now after so many years. I was here many years ago, I will not tell you where, because it's embarrassing for me. One of the first activities I organized abroad from Spain was here. And also I'm very happy to come back to place which has this international favor. Let me just tell you that my name is Juan Luis Vascue, so I am not Cabret. And this is bad for you, but there is one point. I come from Madrid and Madrid is probably the best football in Europe. So it is not so bad. Ok, this is more or less the outline of my lecture. This is a panorama of what I am doing with collaborators. In fact, I have prepared a lecture with the last development and I think that this is not suitable. I just changed a bit because the title was just the last line. But since it's easy today to get information once you get to know what is going on. Let me just give you out of five points four which are not operators in bounded domains. And the idea is very simple to start with. This lecture is based on a summer course I gave in Chetraro, Italy last year. And it contains what we did after this last year lecture series. So the idea is that the diffusion equation is a motivation that has been driving me in the world in my research in mathematics since you need a life motive. Partial differential equations are big endeavours. There are so many aspects and you need some driving force. Mine is diffusion. And diffusion describes how a continuous medium, for instance a population, spreads out to occupy the available space. And then this is a physical concept and mathematicians put into it mathematics in the sense of models, equations and results. And in fact the influence of mathematics is that we call now in the applied sciences diffusion things that can be modeled by mathematical diffusion. So there is something called mathematical diffusion. And it applies to all kinds of things, fluids, chemicals, bacteria, animal populations, of course you know that part of these diffusion models are applied in the finance market. This is where many people get their jobs now. And the main origin of these things which makes it very strong is that the basis is a beautiful equation that everybody knows if you are in the business. And it has incredibly many properties and you never stop learning about grid equation and what the grid equation does for you. And so this equation has had enormous influence in science, not only in PDEs. It's an equation that has strong influence in stochastic processes. Has strong influence now in geometry, in the study of the evolution of manifolds. Has an influence in physics, an influence in the social sciences. And there are many things that you learn because the grid equation has taught you how things work. For instance, if you don't know mathematics, you don't understand the grid equation and the Gaussian profile are brother and sister. And then once you know the Gaussian is the grid equation, then your life is different. And there are many other things like separation variables, Dirichlet forms, spectral analysis. Ok, so the idea is the grid equation in itself produces lots of mathematics. But are there any other equations? That's a point. There's this big community of people working on equations of the family. And this family is called in code, mathematical code, parabolic equations. So there is a huge kind of parabolic equations. Now parabolic equations come in two main stripes. A linear and non linear. The linear theory studied in master courses. And what was very interesting for people in my generation, is that after the work on fluid, particularly in calculus variations, in the beginning of the 20th century, the mathematics were ready to tackle non linear diffusion problems modeled by non linear parabolic equations. The beast that comes out of these investigations is this kind of general equation that if you are good enough, you can study and you write a book and that's it. And this was more or less proposed in 1963, that maybe the general equation is instead of dEU with respect to d, take the dT of some function called the enthalpy. Instead of the laplation, separate the gradient here from the divergence here and put some non linear action once you have split the laplation in two. This laplation splitting is one of the most beautiful things that happen to non linear signs. When you split the laplation in two, you can play with the two different things and put things in the middle. And this is the basic of much of progress in non linear diffusion. And then of course you can put all over order terms that I collect here as b of x dEU and dEU, one derivative. Now this is the general problem and the general problem as Jim Serring who was my friend for years told me, telling the old days, they had a hope that maybe with good luck they could write the book. The book is to be written which is good news for you. So there is lots of progress done in this field and they have the progress splits into non linear diffusion and reaction diffusion. If you are working on non linearities that are basically here, you are working on non linear diffusion. If you accept here that this is the laplation and you put the non linearities here, then you are working on reaction diffusion. Remember that reaction diffusion includes never stocks. So it's not so trivial. Ok, so next is that what you do, since there are so many specific examples, you need to concentrate on details. And this was something that took like 10 years to people to realize that work on the general models was not very fruitful because you were saying, let's say in America you say nonsense. Let's say blah blah blah, it's ok, you get general spaces, general results, but no real progress about the dynamics or the basic models. It's not nonsense, it's a way of talking. I mean it's not the respect for the people who are very abstract. It's the fact that if you want to know what really happens in the real models, you need to go to details. And once it was understood that this was not risk doing details, people when I came to the field were working on four important problems where the only difficulty was that instead of the laplation they were using some non-linear thing. And the more models were the Stefan model, the Hinshaw model, the Poros Medium model and the evolution of the laplation. And at the time I met Luis Caffrelli and this is more or less the story of my life in the neighborhood of this wonderful personality that you met last week. So he was doing essentially Stefan problem, he did incredible contributions and also the non-parabolic problem that is called the obstacle problem, very closely related. And I was in one of these planetary systems doing Poros Medium and evolution of the laplation. And we were like close planets under closer values like him, but we were doing well. So, and then there was another idea that I will not explore here is that if you want to put the non-linearity on the lower order term, there's this beautiful world of blow-up that did incredible progress by exploring a very particular model which is called the Fujita model, a scalar. There are still many problems open in vector, but if you go to the systems, please go to Navistokes. Ok, so next is fractional diffusion. So for many years we were happy doing that. And then in 2007 I was spending some time with Luis in Texas and he said, well, Luis, this time you are going to do fractional laplations. I said no. And he said yes. And this is how I said yes. It was so very easy. The conversation was relatively easy. Well, come for some a southern home and I will explain to you why you say yes. And the idea is saying yes to a new object that came part of the family. In fact, the object is in most elementary form the fractional heat equation which is not new, it was known by people in probability because there is a connection that probably you know that if you work on PDEs, the evolution of a certain distribution of probability according to Brownian motion is represented by a master equation, blah, blah, blah, blah, blah, and then you get the heat equation. Now if you are now working on stochastic process that has certain properties of similarity like Brownian motion, the other option is leviflights, alpha stable processes. And they come in a family with the parameter. You go to the master equation, you make your calculations and you get it instead of the heat equation, you get this equation. So people in probability knew that. So what do you have to do? Essentially, let's say if you are a reasonable person you say nothing, these people in probability are strong enough, they are doing it, that's okay. Infatti, it's not so easy to read if you are in PDEs, great probability, stochastic processes, it's a complicated language, so you let it go. Now Louis told me this always, probability is very strong in doing linear problems. But there are these problems coming from geometry and physics which are heavily non linear. And PDEs is not very strong in doing these things. So there is a chance that PDEs will be stronger than probability in tackling some of the problems. Let's do it. And if they are strong enough, it's okay, we are friends. So the idea is doing these things. Now the first thing you want to know, we are trying to do parabolic equations. What is the good definition of the fractional application? Probably you know that, but she said this is a rather, I decided to do it very elementary. I will not probably collide with Enrico, so let me repeat all by definitions. First definition is going to the people in harmonic analysis. They tell you that you do Fourier in the whole space and there is a symbol. And the symbol of the laplation is c squared, oh, the minus laplation. The operator that has a nice symbol is minus laplation. Everybody knows that the positive operator in analysis is not the laplation. It's the minus laplation. Okay. And then you say the fractional operator should be go to the symbol and take a power between zero and one. Now you can take any power, so in principle the fractional operator can be defined not for fractions, for more than fractions, extrapolating to the whole range of exponents, which is very important to do equations of third or third and a half order. Because the order of the number of derivatives of virtual derivatives is reflected on this exponent. Now this is very good for linear analysis, but the point is that we didn't want to do the linear analysis. So you say listen, do you have any other thing in your storage? And the same people in harmonic analysis, there's a very strong group in my university related to Stein and to Pheferman. They knew that you can invert the product in the Fourier space and make a convolution with the Fourier inverse of the symbol. Now the Fourier inverse of the symbol is a tricky thing because it has a derivative that is n plus 2s. So it's not integrable. So you get a singular integral that doesn't make any sense and then you have to do something and they knew what to do. They do, instead of a convolution, a modified convolution where you put ux minus uy upstairs. This is called a hypersingual integral. They knew how to do that. In physics it's called renormalization so the integral can be computed and it really corresponds. So it's okay. So you now know how to do these things integrating in the whole space. And this is a beautiful formula because it has, as well as you say, the kernel. So next thing for us is that this is to linear. And what these people probably say is that they were not doing any of these things. They know that because, of course, they know everything about Fourier analysis. But they wanted to do Markov processes with transition probabilities. How do you model a discrete version of what I want to do? The idea is that you replace next-neighbor interaction on the computation of the laplation by interaction all over the place. And this is where you begin to learn what is a living process. There are several ways of looking at it but one of them that is very important for me is that the interaction of a particle with a particle far away depends on the inverse power of the distance. Why an inverse power? Because it has the properties that I know I need to do symmetric and highly invariant operators. And when you do the powers and you know how to do it, you tend to do the non-powers. The student has to be good enough, of course. We are still looking for a student. So the idea is to do the powers. Now, how do you go from this stochastic process to the operator? Oh, this is very easy. It's the infinitesimal generator of the semi-glue. So it's okay. But this is not good enough because this is linear. And then there is an idea that came from geometry because Luis Caffrelli is a very good expert in minimal surfaces. So he knew what these people in geometry were doing and he said, oh, there is an extension version of this thing that comes from geometry that says that your manifold where you are working are to the end is the border manifold of a certain space that lives upstairs from you. This space is really the space where the state, the virtual state works. And people who know hyperbolic geometry recognize immediately that the upper part is where the hyperbolic laplation sits and this is the infinite horizon. And there is a boundary operator corresponding to this study. So there are manifolds that have a certain hyperbolic structure. Okay, forget hyperbolic. This idea from geometry. Now how do you define the laplation, fractional laplation with extension? The first attempt is terrible because you go to these grahams working whatever things and this is what Alice chance knows and you make no progress. You have to start the geometry and it's too late for you. But there is another version for people in PDEs. You write the equation upstairs in the virtual space the half space upstairs and it is a field equation that is curiously enough it's a laplation with weight and the weight is vertical distance which in hyperbolic space is the geodextrin distance. So with respect to the hyperbolic structure you get the distance here that comes from wherever. It's a distance. Then if you are in analysis for the geometry. So there is a weight and you solve the equation there and then to define the laplation so you first take a function downstairs you extend it normally normal extension but in the way that it has to satisfy this the elliptic way to understand and then you take this thing you take this Neumann boundary condition with the weight and then you get a vector from the letter Neumann satisfies the laplation not only for alpha equals 1 there is a problem that there are two versions the people who write minus laplation alpha halves and the people who write minus laplation S so I use S between 0, 1 and alpha halves between 0 and 2 So we have been introducing the notations that people respect but people do whatever they want so Enrico do better ok I'm using alpha in S and so this is a way now this is very good because solving equations does not have to be linear and taking boundary operator is ok and maybe no linearity will sit downstairs so this is something I can do so much of the initial progress was done by doing extensions and this is a very strong theory now the construction of generalize of this extension was the thesis of Luis Silvestre a guy who is very intelligent and he was very lucky because he came at the right time to the right place and he did a wonderful thesis and then there are two people in my department who did these extensions and constructed the same group for general operators more general than the laplation and these people I mentioned them Stinga from Argentina and Torrea from Madrid and now there is another version that comes immediately if you are doing operators that are positive that is the fractional operators related to some Laplace transform with weight so if you integrate from 0 to infinity the solution of the heat equation laplation minus F and you put this singular weight you get the fractional aplation and this is the same group approach so now you have so many objects to play my point in this talk is which one of them is good to do boundary problems now let's go back ok doing Fourier analysis in boundary domains looks a bit hopeless unless you don't have anything else in totale puoi farlo ma non si ha lo di sperare secondo oh doing integrals you can do the problem is but this is in the talk by Enrico is where do you integrate if you are in a boundary domain the x belongs to the boundary domain in the y what do you do with people who live out of the border so this is a problem that is very well known even in politics la laplation is satisfied inside will you take into account the border or not a standard diffusion does not jump the probability says you never jump you touch your next neighbor and push him so in the end you go to Trieste and there is a border there and there is no way you can go to Slovenia without passing the border now in non linear then you do something like drugs you jump you find yourself in Ljubljana what do you do if you are in Ljubljana you come back, do you report back and this is what happens in information theory every moment when you do work on the telephone you are in Trieste but for internet there is no borders the information for this talk came from talking to my friends in Guaric, Madrid and United States so how much is the influence of people outside of my close community this is what you have to decide so there is a problem about if you are in a question in a domain will you take into account what happens outside or will you really solve the question in the whole domain and this is what we have been discussing for a long time and this is a discussion that we did in the parabolic setting and not in the lifting in the lifting the talk let me see about this the lever process was a big motivation not to get lost in the analysis because the people in probability know what you do with the particles and jump and they know how to decide what to do with the particles so different definitions will depend on the probabilistic interpretation now there is another way if you get another client that gets you a fractional appellation from another motivation will this new client have a new appellation and the answer is yes last week in Warwick I was discussing with Christian Smaitha from Vienna and he was telling me that they do fractional by a limits of kinetic and they find a different definition of what to do with points beyond the border so every client has a right to influence our mathematics and now let me see the extension and this is very interesting because there are two options for an extension for an operator in a banded domain and this was not understood by very intelligent people who were thinking another way suppose your banded domain sits here and you want to have an extension you see this idea that my boundary behavior is like the life on the infinite horizon downstairs of a certain hyperbolic movement in the new dimension so where is this hyperbolic new and people in geometry knew that the best idea was to consider this part of the horizon as the infinite horizon of a certain manifold that sits here there is a manifold here and this is the horizon so the problem is which kind of manifold do you think of using and there are two options you already know that because you are pretty smart you are not doing siesta for people who are not doing siesta the first option is taking the whole space I know this is the second first option is taking a vertical cylinder and then you say oh my god and what are the boundary conditions since I am a simple person coming from a simple place I take boundary conditions zero here and zero here so my extended boundary condition on the elliptic cylinder will be zero all the time the first people who do it did that in this way I am writing clearly was Chaby Cabret speaking at this moment here you see here hello Chaby Chaby Cabret did that now there is another option which is for many people in analysis more natural but it takes time if you are not in analysis I mean you are in geometry you don't see that we are in analysis you see that very clearly you have a function here and by some method you construct the laplation by a formula how do you do it when the function is defined here put u equals zero here and of course there are infinitely many possibilities of taking your manifold like this and putting here zero here, zero here and here is omega depending on the goal you are using for your definition I don't know if these goals have any interest but in principle for people in geometry they are completely natural why not so this is another definition ok and this is what we try to do in fact there are two basic definitions of extended of fractional applications of abandoned manifolds abandoned domains and this is probably I will not say much I only need this for this talk but since Enrico will be talking about that let me say that we discussed a bit and I don't know Enrico if we got the right names people complain all the time because they are impossible so this thing corresponds to what people in semi groups call the semi group definition and this is the spectral fractional application this other version flattening this extending to zero and applying standard formula apply the integral formula which is really very much harmonic analysis some people say that we should call this the standard fractional application abandoned domain but for people in geometry it's not very natural so we have this client complaining the name we put is restricted and people have been complaining ever since ok so let me tell you now what are the results this is what we have to do let me tell you something about the models we have been working since 2007 so in 10 years with many people around friendly people working a lot there are lots of progress in this thing I don't know how important it is but there is progress in several directions instead of describing the theorems which are well documented online I will try to describe the directions the first thing I will tell is the idea of the linear equation where in principle we didn't do anything because it was supposed to be something standard known by people in probability so Valdinoci write a version of the fractional application good for students and he published this in bulletin of the Spanish Society for Applied Mathematics I don't know why and then after some years they said well he went back to this problem and said listen if this is a linear equation and this is a Markov process there is a transition probability you get the fundamental solution to integrate and if you know the properties of the fundamental solution you know everything as he said the problem of these fractional operators is that they don't get exponential decays they get power decays and this is what Blumenthal and Gettur in 1960 published a very beautiful paper very short telling that this generator of the fractional heat semi group had a certain a priori estimate from above and below that was very nice and it was true for every S from 0 to 1 in fact if you put S equals 1 which is the one-half laplation the formula is exact but if not they can do it and it relies on beautiful theorems on complex variables by Polia so it's very old stuff recovered by them and then the idea is that in a paper with Peral Sorian Barrios from my university they proved something that was interesting for people in analysis suppose that you take data that are not integrable but maybe constants o maybe increasing how many data can you solve by this fractional semi group the idea proved by Professor Widder many years ago was it in the heat equation you can get all the functions that grow exponentially and the theorem for them was that they can grow only like a power there is a power because there is a difference in the parabolic case at least that should be in the list the powers of the growth power maybe it was in your list but for us it was very good no we took this paper with my friends just to read because we like reading these papers that are well written and we discovered with Matteo Bonfort and Yannick Ciro one year ago we were in Baltimore doing that we discovered that in fact we could give if and only if characterization of the extended semi group for the fractional heat equation support the data are no negative if and only if the measure has this estimate the growth estimate there is a solution the solution is unique in the sense that it has to be precise certain strong weak solutions and then every solution has an initial datum of this type every initial datum this type solution in particular measures are included for all effects and this is a very interesting contribution that says that you can contribute something to the weather theory so we are happy with that this was one of this one month work after starting the paper by Amaldinoci ok so next thing is that oh this is in archive oh I think it's already published when I wrote this transparency it was in archive but people keep asking you the paper and listen it's in archive and now let me tell you the talk containing the results until summer of last year is documented in this thing that you can read I put it today on my web page easy to access you go to Chime course and where it says lectures click I did my clicking because I'm very slow and it works so let me tell you what is the non linear fractional diffusion the point of these models is that there is a beautiful coincidence between the motivation that comes from people who know what they are doing and the theory that comes later all of this is like the Georgie and this is beautiful because I mean this kind of miracle happens in the best of worlds we live in a very nice world where the motivation goes along with lots of work in the middle and a beautiful theory in the end so the first model we did this is what I did in 2007 invited by Luis Caffarelli I'm not going to do it yes you are going to do it so the model is the solution there is this conservation law that probably you remember because we have been working for 10 years and then in porous medium you is the density and V is the velocity and you are transporting with the velocity that depends is a gradient of a potential so it's a viscous flow and the potential this law is called dorsilow and the potential is called in dorsilow pressure but the name is not important so now you have a potential flow and the only point you have to put into this model is what is the choice of potential in the typical motivation for porous medium flows the potential is a function of the pressure because at the beginning it was a model for gases incompressible layers or for gas layers underground it was the oil industry the simplest model was for gases not for NAFTA so there was a pleasure isobaric approximation and you get the model and then it was also got by Busineschi in the study of underground infiltration which is very important applications so this is it and we are not going to do that because if you put V gradient P and P more or less you you get the porous medium and we already have done that believe me I'm not going to tell you but it was a very successful life no, there was an idea the idea is adapting this modelling from there was people in the statistical mechanics community that came to Lewis with this idea in some of the models in statistical mechanics and now in information theory long range interactions are very important because of whatever intermediate asymptotes and you cannot say that the pressure that is the agent of your movement is just proportional to you in fact people from very far away are pressuring you this guy who calls from Seoul South Korea telling you Juan Luis are you going to mention me this is pressure right it's only one pressure from very far away maybe he doesn't listen but he can make a pressure on me by just sending me an email this is very important information theory so people knew that there is a presence of interaction where the pressure is the agent but the pressure comes from very far away now and then the simplest idea it was the simplest idea was taken from chemotaxis the agents of pressure are a certain risk potentials or Newton potentials instead of Newton we take risk and in this case risk operator is the inverse of the laplation and you get fractional risk operators which are the laplation minus S are beautiful kernels and all of a sudden you realize so these brothers this knew in 1930 everything in fact they didn't know because they knew the risk operator but they didn't know the fractional heat equation probability was not developed but they do these kernels and they studied them so in some sense the sum of the spaces were more or less in the beginning doing they cut this operator's motivation ehh so you have this equation this is my equation it looks a bit like a chemotaxis equation right with only one term my term is diffusive you can take aggregation pressures or diffusive pressures my is in the way of diffusing can you change it? of course it becomes imposed unless you put another diffusion this is coming in Jose Antonio's talks but in 2007 we didn't know how to manipulate the attractions with the diffusions and we prepared a bit the machinery ok now you put oh if you want to impress people you write the equation in this way really very impressive but it has to be for impressionable people not for you so you put initial conditions and you go and the idea is that you know that there are many many works on fractional appellations for elliptic equations and I'm not going to say that because they talks by Baltinoce here and then I also say that what we were doing about banded domains but in regularity have been done recently some previous results that were needed for us by Shavia Rosso Tong who is now in Zurich and so let me see ok and the modeling you are probably not very interested in modeling I only say about the modeling because people want to be sure there was a modeling dislocation by Biller Carr's model that works in one dimension and they did a very careful analysis of the equation in one dimension or the equation that I call kafarelli equation but in several dimensions dislocation is not that you cannot go from one dimension to three dimensions and you say the physical model is the same, normally it is not so dislocation in three dimensions has nothing to do and they didn't do the theory in several dimensions so it is saying in one dimension it was done and the model from the people in statistical mechanics where these people are coming and they propose this very general model this is my you this is you again this is a coefficient of diffusivity and this is as you say if you define the free energy the operator the inverse operator is the functional derivative in the sense of calculus of variations and you put it here, whatever so then all of a sudden you realize that you are doing calculus of variations which is amazing you don't sleep for several nights I talked to Luigi Ambrosio he said be quiet calculus of variations is ok so can you use operators that don't use operators why not use special operators which are the inverses of this thing and this is an interesting news for the students because the solutions in some sense are the same or better in some other senses are not the same because the operator doesn't have the scaling so you have to do things to recover the scaling in the limit and let me jump on this line and let me tell you what we did as a program and let me give you an idea what was the approach of our work that we did of four years with Luis to try to settle mathematically this question problem first thing you want to do is getting weak solutions in some sense that can be integrated because you want to do some calculus of variations so they want to have an energy if they are too weak you will not be able to do any computation of no energies no nothing so we constructed weak energy solutions and since we were I mean I was in Madrid, he is in Texas he is busy, it took four years to publish this thing and then the journal was terrible to us the second thing is that we produced the asymptotic analysis of what the solutions do and they converge to a certain statistical equilibrium that is not the Gaussian and it's not the P that we found in the linear case it's a new thing that is the solution how do you do these things I mean you have to do some calculus of variations and you find that the profile of this evolution in time tends to a certain a certain thing like it looks in first approximation like Gaussian but it has come back support and this is what we call bare and blood fractional solution because of our friend bare and blood and then once more or less the space was clear he says ok let's stop monkeying around let's prove regularity and they will say come do we do and this is the georgia the georgia do everything and if you know according to lucif you know the georgia then you know that solutions in L1 are in L2 and solutions in L2 are in L infinity and solutions in L infinity are c'alpha and solutions in c'alpha then you come to the talk by Enrico are in c'alpha they refuse to be better because they have an obstruction c'alpha regularity of even the best of the profiles of these harmonic functions try to be c'alpha in the border there's nothing you can do I mean if you are very naive and you are in your country alone you prove c'alpha in place c1 c2 and you try to publish it be careful we didn't do that because we knew that there was this thing ok so now let me tell you you take the you can take the limits non zero or when s tends to one remainder in my equation is u t equals divergence of u gradient of minus laplation minus s of u if s goes to zero this is the porous medium equation if s tends to one it's a funny equation that says gradient of u gradient of new term potential of u and this goes P-u², perché l'applicazione s è l'applicazione minus, quindi è la potenza neutra, la potenza neutra è l'applicazione con l'applicazione minus. E questa è una questione molto strana in principio, se sei in p-sci, non è molto strana perché questo è il modello per la fluidità superfluida che Luigi, Savare e Serfati stavano lavorando. Quando Serfati arriva a Barcelona,が era l'un giorno che stavano boccando. She said, I problemes of this problem, they are very difficult. I say, no no come on, my problem is very difficult. Look at that, it looks like the same. So we combine, this is a curious equation for A is equals one which is going really back no further continuity, Quindi il limite delle equazioni formali che sono paraboliche, il grande parabolico è un'equazione che non è tutto parabolico e produce i vortici, è un'equazione di vortici. È una situazione molto bella e coincidenza di parlare con le persone di cose che non sono freddiche. Poi ho scritto questo tema che potete trovare nella questa abel, infatti ho scritto questo in 2010 perché i risultati erano già là. Poi ho scritto questo tema che José Antonio ha scritto molto, che è provare una convergenza a profilo con risultati esponenti, che imparano a sapere molto di energete e entropi. E in una dimensione non conosciamo come farlo in più di una dimensione. Quindi la connessione di questi problemi vengono da meccaniche statistiche. Con l'analisi vengono, a volte, attraverte le qualità funcionali. Le qualità funcionali sono, in alcuni casi, nuove. E questo è nuo perché è interessante. Siete quali sono le qualità funcionali che volete essere certi. Quindi provate a provare. Amande di millioni di possibili qualità funcionali, c'è uno che volete davvero provare. E questa è una motivazione bella perché normalmente sono certi. E si trova interessante. Siamo chiamati interessante, se vogliamo sapere. E non siamo interessati. Ok, quindi questo è tutto. Voglio solo dare un'idea di come abbiamo fatto queste cose. Perché è interessante. Se la questione non è lineare, non possiamo usare l'analisi harmonica. La rappresentazione non è vera. Può essere vera in Elema. Con la presa del resto dell'operatore. Ma i basici idei hanno a priori estimazioni di compagnia. La presa di compagnia che viene da fisica è la conservazione di massima, la conservazione di alcune potenze, e il discorso di queste potenze che producono qualcosa che è chiamato il rito di discorso, o la dissipazione. E la sua dissipazione ha derivativi. Quindi è il modo in cui controlla le derivativi e la compagnia. Quindi se non hai la dissipazione, il tuo gioco non è molto interessante. Ora questa è la calcolazione che ho sempre fatto, perché... No, perché mi piace fare queste cose. Questa è la compagnia che ho fatto nel Blackboard con Louis quando abbiamo scoperto che la entropia di Boltzmann è molto buona per noi. Allora, facciamo questo. You know, u t love u plus u u t over u dx. Ma questo in questo go e questo è zero, perché di conservazione di massima. E allora dici che la dissipazione di la entropia di Boltzmann è lo u gradiente di u gradiente. Mi chiamo K, il gris curde. Ma questo è minus gradiente u... No, sorry, gradiente u over u, u gradiente K of u. E questo è dove facciamo questa cosa che facciamo in Spagna. Giac, giac, molto importante. In un paese giocando le nostre livelli. Giac, giac. E poi hai minus gradiente u, gradiente K of u. E credo che questo è un operatore molto bello, perché K è un operatore gris che è K ½ square. E questo è simetrico. Quindi potete mettere gradiente di K ½, gradiente di K ½ e questo è la dissipazione. Gradiente K ½ u square. Quindi si dissipa. E la dissipazione ha derivativi, quindi è comparto. E si dice, wow, questo è tutto. Quindi questa è una competizione bellissima. Ora, come sapevamo che questo K ½ che chiamo H nel transparizio, chiamo H, solo per mettere il nome. H è il potenziale di K ½. Plays a role. E si dice, chiamo dissipare il potenziale di K ½. Perché quando stiamo parlando di questo obiettivo, le persone in porosmedia iniziano che questo obiettivo è una dissipazione bellissima per la entropia di porosmedia, la famosa entropia di Renny. Quindi se si dissipa il potenziale di K ½, fai la cosa bellissima con derivativi anche. E queste due cose donano compagnia e poi vogliamo provare la resta di teoria, che è l'aproximazione, l'existenza, la compagnia. E tu hai bisogno di spazioni funcionali per controllare questi prodotti, queste dissipazioni. Queste dissipazioni sono divertenti perché sono aspettati. Questa cosa è aspettata da te, che è buona. Buona, non è... Quindi devi fare qualcosa con le equazioni funcionali. E poi questo è quello che dobbiamo. Letto mi dico... Questa è la calculazione. In modo di capire queste integrali tu fai cosa? Tu fai una equazione, multipliati da f di u e integrati f di u l'ut è capital F di u derivato di rispetto a tempo. Quindi se multipliati con una certa funzione, f di u, f, small f di u e integrati f di u f di u, tu fai questo tipo derivato tra l'ultimo e l'ultimo. Questa è la tipica di equazione quando puoi mettere la potenza. Nessun nuovo. Il problema è che c'è il caos nel secondo... nel secondo mezzo. Tu hai un minus, gradiente di f di u, lì ho messo l'ultimo. Ora ho messo f di u. gradiente di f di u, u gradiente P. Ma poi questo u e l'ultimo. Quindi, io faccio questo strumento l'ultimo è uguale f di u e f di u quindi questo è un gradiente di f di u. E poi ho questa cosa. Ma io ricordo che faccio strumenti con alcune applicazioni. Questa cosa sembra terribile, ma se lo isolato qui questa cosa è una forma bilinea su alcun spazio spazio ed è scoperta. Ma questo spazio bilinea dopo diversi giorni di computazioni è una cosa. Ma questa cosa se computa l'ultimo che è un spazio qui è una cosa con queste derivazioni se le puoi mettere qui e usate la rinormalizzazione e questo è il spazio grigio fraccio quindi si dissipa nel spazio grigio e sono in business. La tua vita è più bella e il pappio finiscesolà. Se non fai queste belle coincidenze e sei il tuo spazio funzionale stai ancora vivendo. Parti delle computazioni fanno in spazio che non sono così evidenti. Ne butte un esperto per guida, per esempio il miglior spazio. Perché sei integrato e è complicato. In questo caso è così bello e semplice. E questo è tutto. Quindi poi puoi provare questa bilinea. La dissipazione di una certa energia è in bilinea formazione che rappresenta l'energia del gallardo e quindi hai una compagnia. E poi come usate il metodo Giorgio? Non lo dirò perché ho 10 minuti per produrre un po' più evidente di quello che abbiamo fatto. Questo è addizionale nel lavoro di recenti in cui ho guadato diversi personaggi che hanno lavorato con noi. E non so il nome perché sono un po' tardi. La informazione sarà disponibile e si può parlare di loro. Stanghi in tesoro sono i miei studenti. Rizzini, Maenini e Segatti sono in Pavia. E poi ci sono molte questioni open per esempio qual è l'uniquità? Abbiamo metodi che sono molto buoni in entropi. Ma in unice unice non è semplice non so come fare. In una dimensione c'è unica dimensione che permette l'uniquità. Perché si può usare le risoluzioni di fiscossi in diversi dimensioni che non sono abili. Quindi c'è un problema open ma c'è un gruppo di persone cittadini su Xiaoyun Cheng. C'è un'uniquità di piccolo tempo per soluzioni che sono molto belle fino a iniziare a essere c'è alfa. Per queste persone c'è alfa è una singularità. Quindi un piccolo tempo fino a finire l'unice. Però questo è ok. E le spazie sono le migliori spazie. E siamo lavorando su studi numerici. Quindi ve lo chiediamo il modello 2. E questo è... il modello 2 è un'altra possibilità di fare un modello in poros medio non lineari con le plazioni fraccionali. E non avevamo muchi perché l'altro modello era un lavoro intensivo. E l'altro non avevamo molte persone motivandoci a farlo. E questo è un'analisi che sai che hai a farlo. Ma non c'è motivazione e sei un membro di un gruppo che non c'è. Non hai rischio. Quindi abbiamo fatto alcune calcolazioni sulle spazioni e stiamo continuando. E poi, quando ero in Paris ho scoperto che Stefano Olla e il suo gruppo Milton Jara dall'impia in Brasil stanno facendo queste meccaniche statistiche per ammettere le emozioni e hanno trovato le plazioni fraccionali in un lineari e in un non lineari in caso di un processo di pranzo. E poi diciamo ok, facciamo le emozioni. Abbiamo fatto un progetto con i miei collaboratori, Pablo Quirose e Anna Rodriguez. E abbiamo prodotto in questo caso le emozioni non hanno difficoltà di trovare le emozioni fraccionali. Ma questa emozione che è il secondo modello ha un'uniquità, la contrazione, tutto che vuoi in questo mondo, il principale massimo, quindi è davvero in questo libro. È molto lavoro e lo facciamo per le emozioni, le emozioni per le emozioni sempre. E abbiamo pubblicato due paper che erano molto accerti ad avanzi e comunicazioni allo stesso tempo, perché abbiamo fatto le computazioni. E poi c'era un grande problema nel modello di ottenere la teoria di regolazione. E questo è quando prima della teoria di regolazione ho fatto l'analisi sintotica perché non conosciamo come costruire le soluzioni di bar e blad e ora le soluzioni di bar e blad nel modello 1 sono come questo. Quindi uno dei modelli per le emozioni di bar e un altro modello di bar e blad e una altra questione per te se hai visto Enrico è come è questa distanza? Beh, va bene, e cosa è l'esercito qui? E poi lo scopriamo è una per una per una per una per una e poi lo dico è l'esercito? E poi guarda questo numero è è ok? Non non vedo la non linearità La non linearità è così fredda è influenzata nella formazione del paese ma davvero fredda nel freddo quando quando la diffusione fraccionale è già scoperta scopriamo l'U.M. reale città è veramente è veramente una vittoria reale una vittoria quindi questo è quello che abbiamo provato ai miei studenti e questo è importante perché abbiamo settilizzato la discussione di supporti compatibili situazioni di paese con la diffusione fraccionale produzione, supporti compatibili modello 1 sì modello 2, no ok? E questo è importante e hai bisogno di usare le tecniche di l'analisi sintotica e questo è un buon punto per finire il mio paese perché nel momento quando abbiamo fatto l'analisi sintotica abbiamo scoperto che è stato relato ad estimazioni eliptiche e ho fatto con il mio collaboratore Matteo Bonforte in Italia che adesso è più o meno spagnolo è stato qui più di 10 anni e ha lavorato con me su queste cose e abbiamo iniziato a fare quantitativi diciamo cosa chiamiamo gli italiani gli italiani quantitativi, estimazioni e qualità quantitativi e quantità quantitativi è il mio stile e poi con le inequalities basi sulle idee questa situazione è positiva ha una struttura e ricorda che quando si fa ha un decay la struttura deve essere relata a un certo peso se non puoi mettere la struttura non puoi avere una struttura, ovviamente perché devi fare in un concorso in un concorso che è molto fuori e poi abbiamo fatto questo e poi abbiamo iniziato la collaborazione di fare estimazioni quindi abbiamo fatto questo paese con Bonforte con quantitativi e positività e struttura quantitativi per questa equazione di paese con nessun frivanderi in più grande di 1 e in meno di 1 poi abbiamo deciso e magari persone vogliono sapere cosa succede se non ci sono potenze quindi ci sono due mani uno di loro è cambiare la collaborazione a più generale operatore come Faldinocchi ha detto un altro è cambiare l'U.M. per un altro paese l'idea di cambiare per un altro paese per queste persone Pablo, Kiros e Rodíguez e c'è un paese in Giorno di Mathematica Europea e è già on-line ha preso due anni per poserlo on-line e poi quando abbiamo una regolarità per questo problema Bonforte Figali e Rosso Tom provviano una regolarità di soluzioni in domani e up to the boundary e questo mi ha fatto molto felice perché questi ragazzi in Texas lavoravano in nostro paese oh my god, è bello ok, e poi con questa simmetrizia siamo completamente crazy KPP, questo lavoro numerica non è così difficile e poi abbiamo discutito la questione di mettere le potenze qui per vedere a qualsiasi momento a qualsiasi critical moment passate dalla propagazione finita alla propagazione infinita ho messo questo perché questo è il lavoro di Dianastà e Nteso che hanno preso due anni con la mia domanda, ovviamente e poi abbiamo già settilizzato la questione, sappiamo quando la propagazione vede e quando la propagazione infinita vede e... mi dico solo che puoi fare altre cose come piloplasi in cui le persone sapevano che sono questo è una cosa... è molto flexibile infatti la terza cosa che ho scoperto è che una volta che puoi fare questo le persone in chemotaxis stanno cercando di usare queste fraccionali e potenze di le nuove e potenze in porosemidio per fare diversi approcchi per le chemotaxis quindi la machine è molto flexibile e poi mi dico... oh, cosa sta passando? oh no, no, è ok se mi dico... se ho due minuti mi dico che cosa sta passando ricordando l'ultimo che abbiamo fatto con Matteo, Giannico e poi con Alessio quindi abbiamo questi operatori dei domani e poi l'idea è che c'è una funzione verde e lo constrazione le soluzioni ma il punto è che c'è la fraccionale dei domani quindi è solo la memoria di cosa... oh, abbiamo l'ultimo l'ultimo è che prendete cosa chiamiamo restricti è un domani semplice è estenduto da zero e poi prendete la fraccionale per la fraccionale non importa cosa e non calcolate l'ultimo il senso di non farlo è che l'ultimo è soddisfiato l'ultimo non ha detto l'ultimo di soddisfiare niente quindi può essere molto bianca per le persone che parlano di te di Corea o di Spagna a volte impossibili non hanno alcune questioni per soddisfiare con te perché sono nel loro paese quindi questo è un problema noi pensiamo di controllare un'equazione in tuo domenio non fuori il prossimo in questo caso questa fraccionale è un obiettivo diverso nel senso che ha alcune funzioni e alcune valori che sono discritti ma non sono the same as the laplation e hanno questa c-alpha regularity on the boundary che è l'ultimo l'ultimo e poi hanno una funzione grida per l'interso operatore che ha questo curioso tema che le persone in probabilità hanno i power della distanza che rendono alcune soluzioni in soluzioni nella boundary le persone probabilmente hanno saputo molto che ci sono soluzioni in soluzioni il secondo tema l'espectro l'espectro in cui definisci per l'analisi spettrali della questione di giocazione prende i valori eigen mantenete i valori eigen e mettete i valori eigen a un power quindi i valori eigen sono the same e i valori eigen sono i power quindi l'operatore non è the same e la c-alpha regularity è già in Enrico Stock è linear, non è come noi quindi c'è uno operatore che è buono e un altro operatore che è nuovo c'è molti altri operatori si può trovare molti altri operatori quando si scopre che c'è due possibilità che si può giocare e abbiamo pubblicato infatti, questo è l'espectro di funzione grida c'è un altro che viene un mix di due che viene in probabilità è chiamato il senso refractionale e vi dico che abbiamo tre modelli la riferenza nella literatura sono qui Bogdan, Berti, Ceng, Kim e ci sono i nostri documenti il suo ultimo lavoro è fatto da noi Bomforte, Figali e Vasquez e è stato esplorato a fridayo da Matteo Bomforte nella simposione che abbiamo organizzato in Warwick dove sono ora che è chiamato non locale quindi se vuoi sapere quali sono i teori che abbiamo provato per l'evoluzione non linea semigravole per questi operatori o sul domenio puoi sapere questo e ripetere ho una coppia per me ma hai solo chiesto il mio collaboratore Matteo Bomforte e grazie molto