Perspectives in nonlinear diffusion: between analysis, physics and geometry
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Different (Kate Ryan album)RenormalizationAlpha (investment)Exponential functionScaling (geometry)InfinityEntropyNormal (geometry)Goodness of fitMortality rateCalculationL1-NormMass flow rateCentral limit theoremDimensional analysisFunction (mathematics)Cumulative distribution function1 (number)GeometryTerm (mathematics)Hessian matrixPhysical quantityDerivation (linguistics)Algebraic functionDissipationMoment (mathematics)VarianceDiagonal matrixPower (physics)Multiplication signBoltzmann constantDiffusionsgleichungNichtlineares GleichungssystemDirection (geometry)Profil (magazine)ParabolaPressureMatrix (mathematics)Group action2 (number)ResultantThermodynamicsFinitismusTheoremSet theorySocial classRight angleDistanceRule of inferenceTime zoneSpacetimeSinc functionGreatest elementCharge carrierAreaGraph (mathematics)Meeting/Interview
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Transcript: English(auto-generated)
00:00
OK, good morning. It's first with great pleasure and then with a real emotion that I have learned that I was proposed to be the chairman of the Juan Luis Vasquez. He is one of the extraordinary persons that I know in the community, both by his scientific quality and by his personal charism, his warm heartness,
00:22
and continuous cheerfulness. As you may have seen in the ICM daily news of the 24th, Juan Luis Vasquez is the first Spanish mathematician invited to give a plenary talk at an ICM meeting. Actually, for many in the community,
00:40
that should have been acknowledged already during the last Congress in Beijing. This is just the expression of the excellent health of Spanish mathematics in large and impartial differential equation, more specifically. This community has recovered in spectacular way during the last 25 or 30 years.
01:01
Juan Luis, as many of us of his generation, had to go abroad in France or the United States of America in order to come into contact with high level mathematics because in those days in Spain, the university was far from the level it has today.
01:21
After some time abroad, he came back in Madrid where he got his position at the Universitat Autonoma de Madrid. And with Adil Fonsodias, who is professor in Computanza and member of the Spanish Academy, Juan Luis played an important part in the extraordinary times where the level of Spanish mathematics
01:42
rose at an incredible speed. In some sense, they are both the architects of what we can see today. I think we should associate also the name of Jacques Williams and Aime Brezis, with whom Juan Luis worked during his thesis and after, for the important support they gave, starting at this time, to support the partial
02:02
differential equation school. Nowadays, luckily, the young generation doesn't need to go abroad to find very good advisors or excellent PhD problem. The Spanish school has indeed flourished in quality and in quantity. Juan Luis is an outstanding specialist
02:21
of non-linear partial differential equation, and more precisely, is very well known for his important contribution in understanding the non-linear partial differential equation, singular and irregular, and more precisely, the diffusion phenomenon, which played an important role,
02:41
for instance, in the study of porous media. Juan Luis has collaborated with a large number of mathematicians in different countries, like USA, France, Italy, Russia, and in 2003, he was awarded the very prestigious National Research Prize in Mathematics.
03:04
Juan Luis was also president of the CEMA, the Spanish Society for Applied Mathematics, from September 96 to September 98, where he's transferred a lot this association, and in particular, with Enrique Fernandez-Carrar,
03:23
he invented two prizes of the CEMA, the one for the young, the Jovei investigator, and the prize for the Difelcacao de Mathematica Applicata. During all this time, he didn't neglect it
03:41
to contact the contact with the media, as you may have seen in the different newspaper nowadays, but this is something he has been doing for a long time, and I think the whole Spanish community has benefited from his scientificalism and his fantastic personal qualities,
04:00
so it's an honor for me to introduce Juan Luis Vazquez on his talk on non-linear diffusion, the porous medium equation, from analysis to physics and geometry.
04:20
Thank you very much, Timo, good morning. First of all, I would like to thank the organizers of this event for the invitation. It's for me a great honor to be given the opportunity of addressing these distinguished audience.
04:42
It's a merited honor for me, but it's probably the merit of my country. My talk will be divided into several sections. One of them will be, let me put my glasses,
05:01
even if I don't like to. Non-linear diffusion as a subject, and I will try to be understandable by a broad audience of people with scientific education, and then I will try to concentrate on what mathematicians do, which is doing mathematical theories.
05:21
And my favorite one is porous medium and fast diffusion flows. And the aim of this theory is to develop a mathematical theory that involves all the machinery and tries to be complete and have a physical basis
05:41
which is sound enough. And in doing that, we have discovered this combination, which is so beautiful of PDEs, functional analogies, and infinite dimensional dynamical systems. And recently we have been fortunate to find also in our way differential geometry and probability.
06:00
Before I tell you my story, let me mention the names of these giants of science, which I met in my life, and I was really a very fortunate person. First of all, I met Chaim Brezis and Felipe Nelangui who introduced me to the subject of partial differential equations in my doctoral thesis. And then I went to the United States
06:21
and I was very happy with Don Aronson and Luis Caffarelli, who knew everything about controlling three boundaries, which is geometry. And in the nineties, I collaborated with Bert Pelletier, Shoshana Kamin, Grisha Barenblatt, and Viktor Galaktionov. And the last three explained to me the beauties
06:45
of physical mathematical formation in the former Soviet Union. And this was so beneficial to us. These people are the people I collaborated with, and there are many others, but these are the people who really made an influence.
07:01
And there are other giants who collaborate in the theory that I will explain later. And some of them must be mentioned because they are really prominent people and they are not unfortunate here, like Mike Randall, Craig Evans, Ahmed Friedman, and Carlos Kenich. Okay, so what is diffusion?
07:21
Populations diffuse substances like particles in a solvent diffuse, like medicines, for instance. Heat propagates, electrons and ions diffuse. The momentum of a viscous fluid diffuses, and if the fluid is Newtonian, it diffuses linearly. And there is diffusion in the markets and new spread.
07:42
And you want to know what's diffusion anyway, and you want to know how to explain it with mathematics. And if you are a technical person, you want to know if it's linear or not. So my tale starts with a heat equation, which is one of the three sisters of PDEs,
08:01
the equations you find in all elementary presentations of PDEs and the analysis was proposed by Fourier in 1807 and published in 1822, which introduces these beautiful concepts of Fourier decomposition spectrum and so many others.
08:22
And these mathematical models of heat propagation and more generally of diffusion have made a great progress since then. And they have had a strong influence in all five areas of mathematics that we mentioned before. And if you want to know what the linear theory has us
08:44
of key features, you discover this heat representation by an interval that is a convolution with the Gaussian kernel. And also when you work in bounded domain, you get this separation into a countable number of modes.
09:02
If you are an engineer, as I was many years ago, you say the countable number is like three, but if you are a mathematician, it goes up to infinity, which is very strange. In any case, you have to calculate the modes and they have a time part that is easy and a space part that is a solution of an elliptic problem
09:24
which is the famous eigenvalue problem. So this is a whole work for years and years for mathematicians. But then the direction from 1822 until 1950 of the theory had to like a diversion into ways.
09:45
One of them is doing Fourier analysis, the composition and learning about functions and set theory and capacities in potentials and anything else. And this is what my department does day and night. And then the Mavericks decided to do the linear
10:01
theory of parabolic equations. And if you want to know what's a parabolic equation, it's a little monster that is written over there that replaces the replacement by a general second order operation. And then it puts more terms that are called lower order. And this term represents convection
10:20
and this reaction and this forcing. And if you do that, then you say, can you do mathematics with that? And the parabolic theory says that if the coefficients are regular, and you are in the beginning of the 20th century, you learn maximum principles, shelter estimates, hallmark inequalities, and you give yourself a functional framework.
10:42
You cannot do anything without a functional framework. And the framework is the famous folder space is C alpha and Ck alpha. So sometimes the derivatives are continuous in the C alpha way. And sometimes the derivatives are also to a certain extent. You do the theory and you discover that
11:01
these combines with potential theory and with generation of semi-groups. Now, this is okay. And this covers, for instance, the famous books of introduction to PDEs at the second level. But then when you try to solve the non-linear equations that I will try to solve in the next slide,
11:22
you use linearization. And in doing that, the coefficients are frozen. Values of the non-linearities. And you don't know a priori if those coefficients are good or not. So you are assumed that they are only continuous or maybe only bounded. And then you have to do a theory that is more difficult
11:40
and was done decades later. And your spaces are W2P spaces. So this is functional analysis again. Sobolev spaces. And your theory has names like Calderon-Zygmunt and your solutions are weak. So this is like complicating your life. And let me tell you, this is not the end of the story.
12:00
In parallel with that, in the 20th century, the people in probability developed the probabilistic theory of diffusion, which started as, I mean, this is a popular topic in this conference because of the prize given to Professor Ito. And well, you know the names of Bachelier, Einstein, Smoluchowski.
12:22
Many people try to overlook the work of Smoluchowski, which is so important today. And then you know that the Brownian motion was put on firm mathematical foundation by Wiener and Levy. And the stochastic calculus with this beautiful equation was done by Ito in the beginning.
12:42
And now there is a curious duality between this equation and that equation. According to the people in probability and diffusion, they are the same, which is difficult to tell, but this is so. So one of the things of this part of mathematics
13:00
is that the coding of the information changes from slide to slide because you are talking to different people in different ways. Now, nonlinear heat flows started in the 50s of last year, seriously, last century, and the idea is that mathematics are more difficult, probably.
13:22
They are more complex, and they are more realistic for the interpretation of physical phenomena. And let me tell you that the group of research that I have been working with in Spain and abroad is busy with the areas of nonlinear diffusion and reaction diffusion.
13:42
And I will present here mainly an overview of nonlinear parabolic equations of the diffusion type. Now, the general monster that we deal with replaces the box in the previous slide by a new box. And this box is the box I copied from Jim Sarring,
14:04
one of our friends and mentors, that in the 1960s wrote several papers telling that you should consider general equations and try to solve all of them by putting conditions on this A and this B. So this A is the nonlinearity
14:22
that replaces the coefficients of the parabolic matrix. So this is nonlinear parabolic. And this B replaces all of the rest of what we call junk, which is not junk, first-order terms, convection, reaction, and forcing. Can you do a theory for that? In fact, Jim Sarring started with trying to prove
14:44
all the machinery that was proved by the linear theory by Schauder. And he had already a good starting point with the Jordi Mosher-Nash. But the problem is not finished.
15:01
And very soon, people realize that you have to concentrate on the particular models, because in the nonlinear world, complexity is always lurking. And if you do not concentrate on the models one by one, you have the tendency not to say very deep things. So I will talk about, well, our work concerns
15:22
these two model equations mainly. This is the porous medium, and this is the typical reaction diffusion. So let's concentrate on these nonlinear diffusion models. Nonlinear diffusion models are many,
15:41
because in the microscopic world, the laws of nature seem to be not one, two, three, or four like in the microscopic world, but many of them. This is the realm of complexity. And the most popular of our models in the evolution case is the Stefan equation model.
16:02
The Stefan problem was posed by Lame and Clapeyron in 1833, and they posed the problem as an important physical problem, and they could not solve it. At the time, Fourier analysis was already working in the beginning of the rigorous proofs
16:20
of convergence of the Fourier series by Dirichlet and so on. And then Joseph Stefan in Vienna took the problem in 1880, and he could not solve it. Now, is the problem so difficult? And the idea, no. In fact, there is a combination of two phases of the same medium, like ice and water.
16:42
And in the water, the temperature u is positive, and it obeys the heat equation, which is a linear equation, and we know how to solve it. And in the ice, you suppose that you have still the heat equation with a different conductivity, but temperature is negative, so it's still linear, so we can solve it.
17:01
So where's the big thing? And the big thing is because there are transition conditions on the interface separating the two media. And one of the conditions is very natural. It says that the two media are in contact with continuous temperature.
17:21
U equals zero makes the transition. There are more sophisticated models, but continuity of temperature of energy is a main thing to put as the first condition. But the free boundary moves, and this is the main point. The geometry is not stationary, and you have to tell how it moves.
17:41
And this is the famous law that is used here, Stefan law, that says that this term is the flux, heat flux in one medium. This term is the heat flux in the other medium. The difference of the two terms is the available energy
18:00
on the interface, and you have a multiplying factor that converts this energy into movement. Essentially, the physical process is that you use this energy to destroy the crystal structure of ice and convert it into water. And so there you have the geometry. There is a beautiful problem where
18:21
the partial differential equations are easy, and the geometry is not easy. And it was solved in 1958 on the theorem of existence in uniqueness using weak solutions by Shoshana Kamin in Moscow. So it took 130 something years to have the mathematics
18:40
that tell the mathematicians that this is a well-posed problem. And now all the qualitative theory is still not understood completely, and this is the lifetime occupation of my friend and mentor Louis Caffarelli, who had made these incredible contributions in the 80s about Stefan problem and obstacle problems,
19:03
and they're still giving trouble. So if you want to simplify your problem, there is a beautiful model that I have to tell here, because, well, I have a friend in physics who's in the audience, and he says that heat shock cells are beautiful. So the heat shock cell is in some sense
19:21
a simplification of the model there, where instead of the heat equation, you have a potential equation. The equation equals zero. So your functions are harmonic functions, which are the real parts of analytic functions, and we are talking about conformal maps. And you eliminate the second phase,
19:40
and you put the same transition conditions, and then you say, what is the big deal about the conformal map? If you ask the people in Moscow, like Sam Harrison, they will tell you, because the boundary moves, and you don't know how it moves, and you have to be careful, because it develops singularities. The fact is that the singularities of these heat shock cells
20:00
are still a very important open problem, because nobody knows very well how to continue from singularity in a physical way. In fact, nature knows how to. We don't know how to, but nature knows, which is curious. And I will talk about porous medium.
20:21
Now, porous medium is curious guy, because this is just a non-linear heat equation. You will ask yourself, where's the free boundary? And there is no free boundary in first sight, but there will be a free boundary later, which is hidden there. And in fact, the model is more or less the same model as the Stefan problem, but it has the benefit of a scaling variance.
20:42
People who know about the scalings will realize that this is an equation and you can do wonders with the scaling groups. So we like groups, like any mathematician, and we will do some of that. And then there is the pin operation equation, where you can do these things also.
21:02
So my group is working also on reaction diffusion models, and the typical equation is this one. We have a satellite conference next week in West Korea, and Professor Fujita will be there. And the idea is that this equation has a Laplacian in U P, eliminate the Laplacian, and you discover U T equals U P,
21:23
with P larger than one, has blow up in finite time. Everybody knows that U T equals U square has as a solution one over one minus T. So at T equals one, it blows up. What do you do afterwards? Probably you say,
21:41
the problem is badly posed and it has no physical meaning. And the people in chemistry explained in the 1930s, like Seldovich and Frank Kamenetsky, that these problems are very important in reaction, in combustion, as first approximations.
22:00
So the problem is, if you put the Laplacian, do you avoid explosion? And Fujita explained in this beautiful paper in 1966 that it depends on the exponent, and you have to know mathematics to explain why for certain P it happens, and for certain P it doesn't happen.
22:21
So then this is a scalar equation, which is the toy model, let's say, and you can get another scalar equation where Laplacian is replaced by a general parabolic operator, and U P is replaced by F of U, and this is more realistic. But let me tell you that the real thing nowadays, that we have made a lot of progress,
22:41
is that you work with systems. And Juan Jose Velasquez of University Complutense will talk this afternoon about chemotaxis, I think. And he did beautiful contributions to the chemotactic system, where there is a singularity that can be continued.
23:00
And again, the physical world knows how to continue, and mathematicians are having big trouble to know how the world does. I finish this presentation by telling that the links of non-linear diffusion and reaction diffusion with other people are now fantastic,
23:20
because the fluid flow models, like Navier-Stokes and Euler, take the Navier-Stokes. It has viscosity terms that for us is diffusion. It's diffusion of momentum. And do you get solutions that are C infinity? Nobody knows.
23:41
So this is a clay problem. And the difficulty is that maybe there is blow up in finite time. And so this Navier-Stokes problem is closely related to the reaction diffusion problems, but these models seem to be easier for mathematics. And the last thing I want to say at this moment is that
24:01
the geometrical models have been very popular in this conference because of Perelman, of course, and his beautiful work. And if you look at the equation, our dt of the metric, which is a tensor, equals another tensor. And the second tensor is a second derivative operator on g.
24:24
And if you linearize, the first part is Laplacian. And the rest is quadratic. So it's a reaction diffusion and you have singularities. And this is what Perelman did. So I will forget this thing. And let me tell you,
24:41
to finish this block of presentation of nonlinear diffusion, an opinion of John Nash. He said that the open problems in the area of nonlinear PDEs are very relevant to applied mathematics and science as a whole, perhaps more so than the open problems in any other area of mathematics. And the field seems poised
25:02
for rapid development. It seems clear, however, that fresh methods must be employed. He said that in 1958. And then he went on in the next sentence to say that little is known about the existence, uniqueness, and smoothness of solutions of the general equations of flow for viscous, compressible,
25:20
and heat-conducting fluid. And if you ask yourself about smoothness, there is still the problem on the clay foundation. So we still have the need for new methods to address the nonlinear world. In a characteristic, humble mode of mathematicians, he started investigation immediately in this paper by working on the model
25:42
of the linear parabolic equation without lower order terms. And he proved a beautiful theorem about continuity of solutions of elliptic and parabolic equations, which with De Georgi and Moser's work, is the foundation stone of the theory. So we have to be humble
26:02
if we have to follow John Nash. And I will be much humbler than him, of course. And my idea is that, let me tell you the story of U T equals to Laplacian U M, which is the Paul's medium equation. And let me tell you how many mathematics you can find in doing the theory. And the equation,
26:21
remember, is this equation. But let us be a bit physical. If you take the Laplacian as divergence of a gradient, and you make the first gradient, the gradient of U M is gradient U times M U M minus one. So the coefficient here, the diffusion coefficient,
26:40
is the power of U. So it's density dependent diffusion. This is it. And the point that is very crucial for mathematics is that this degenerates when U equals zero if M is not one. If M is one, you have the heat equation, and we are supposed to know what happens. In the rest of the talk,
27:00
I will try to do three things. First, I will try to tell you that the equation has not been cooked up. It represents sound physical theory. Second, I will tell you that in the last decades, we have been able to produce a complete mathematical theory in, well, complete.
27:22
And third, I will tell you that we have been lucky. And now we have beautiful new problems. So let me start with applied motivation. You take the flow of a gas in a porous medium, which was proposed by the people
27:40
in the engineering industry in both the Soviet Union at the time and United States, Leibnizan and Muscat. And the idea is that you take this equation, which is the conservation of mass. This is the basic transport equation.
28:00
Now this equation, if rho is variable and V is variable, is already non-linear. And as Pierre-Louis-Léon would say to you, this is a basic non-linearity that is difficult to tackle. And then you have to close this thing because rho and V are in two variables. And you say that V does not obey the Navier-Stokes equation,
28:22
but in a porous medium, you can use an average law that is Darcy's law. And this says that V is a gradient of, oh my God, I went further, we go back. Okay, that's it. V is a gradient of a certain potential
28:41
that happens to be the pressure. And the pressure is related to the density by the state law. And of course, then you have the porous medium flows are potential flows due to the averaging of Navier-Stokes equations at the pore scales. And if you take this last law,
29:00
which is the state law, then you have a power that can be gamma equals one if it is isothermal, or larger than one if it's adiabatic. You plug the last into the former and this into the first, and you find in easy computation the porous medium with m equals gamma plus one,
29:21
which is equal or larger than two. Now these people pose the problem and work with it, and they discover that formally the solutions were very different from linear flows, but they could not solve it. And since they were engineers, they hoped for better times in the future. And then underground water filtration
29:41
has to be recalled because the famous Boussinesq had posed the problem in 1903 in explaining the filtration of water underground, something that is nowadays a big business because of water problems in all countries, in particular in Spain.
30:01
And he discovered the porous medium with m equals two. Since these are equations in complicated media, let me tell you that this is a simplified model and we are taking one equation of a big system. But the main star of these derivations is the derivation by Zeldovich and Reiser in Moscow around 1950
30:21
when they were studying the plasma radiation for, let's say, obvious reasons in the two superpowers. And they discussed the derivation of the heat equation. This is a very beautiful page of rewriting mathematics. This is what everybody knows as the balance law for heat. And what they say is that
30:42
this k of t is almost constant for heat propagation by conduction, which is exchange of momentum by colliding. At high temperatures, this is not the phenomenon of heat propagation. Heat propagates by radiation and there are photons
31:01
transmitting this energy. So they had to do radiation theory and they decided that it was too complicated. They showed average part of it and they found at the end of the analysis that you could put k of t equals a power of the temperature.
31:20
And if you apply the, as always, the Gauss theorem, then you find this theorem. And then you find that this is the operation of t to the n plus one. Now, according to these models in plasma physics, n can be from four to six. So you get a transmit integration with a very high exponent. And according to engineers,
31:41
we talk two, six is almost infinity for these matters. So you are thinking about the operation with p infinity. Now, if k is not a power, which happens, for instance, in many cases, that martian waves, then you integrate this thing and you call the primitive phi
32:01
and you get the filtration equation, which is the same, more or less, but doesn't have a scaling variance because it doesn't have powers. And this really complicates because you lose group theory. And this is complicated life. And now, once the equation is founded, it is solidly derived,
32:22
you discover that the people who study biology of diffusion of species, like not populations of people, like amoeba or protozoa, they found that many of these species tried to avoid crowding.
32:42
And if they avoid crowding, they try to diffuse more when u is larger, u is now density. And this is the porous medium again. And there is a beautiful theory for thin films. And the people in kinetic limits came into the picture.
33:00
Carleman was the first who derived the diffusive limits of these particle models. And there was a beautiful paper by McKean many years ago and recently by Pierre-Louis-Léon and Toscani. And this is something that I did recently, some contribution with my students, Alvarani. And there are many others.
33:22
So let me tell you the basics. Okay. The basics of what is different is that if you take the Laplacian of u square, you have two terms. And one of the terms is u Laplacian u, that for u larger than zero looks like higher order.
33:41
So it's a parabolic equation. But for u equal zero, it's singular perturbation. And that's a problem because the remaining part becomes the important part, the principal part. And we say kernel equation. So there is the combination of parabolic and hyperbolic.
34:00
Inside the medium, it is a diffusive process. On the boundary, it will develop three boundaries. And you say, but this is because you took the Laplacian of u square. Can you do that for m less than two or larger than two? And there is a beautiful trick that says that if you take v equals u m minus one,
34:20
and you change the variables, then you get the splitting again. And all of these tricks of changing variables are very prominent in applied mathematics like Louis Vega said yesterday with his beautiful transformations. So the planning of the theory now from now on, let's forget the physics now because we know now that the equation comes from
34:41
things that have to be known and described. And we are now becoming pure mathematicians and try to do a precise meaning of what is the solution. What is the non-linear approach which says that you get estimates and you look for functional spaces and you get solutions.
35:00
And then do you have a theorem of resistance or not? Because in some cases, solutions do not exist. There are obstructions. And do you have uniqueness or not? And once you have solutions, you prove regularity. And this is the basic problem that was addressed by the George Moser and Knox.
35:20
Are solutions ck? And what is k? And these solutions have interfaces. And this is where Louis Caffarelli and the rest of the group where I was working wanted to know are the interfaces ck surfaces or they are terrible monsters. And for evolution people,
35:41
you want to do asymptotics. Does it converge to a beautiful pattern? And does it have a rate? Is this pattern universal? And of course, you want to do probabilistic approach since asymptotics is related to probability. And when you fish from the people in probability techniques, you discover things like
36:02
the Wasserstein distance and Wasserstein estimate which are very popular today. And there are lots of generalizations. So let me start with applied mathematics which says that before you do any theory, you want to know if there is a sensible solution around.
36:22
And in heat equation, there are several solutions but the main solution is the Gaussian kernel. So do we have a solution that represents the Gaussian kernel here? If you want to do Gaussian kernel as a kernel where you can do convolutions, the answer is no. You will never do
36:40
internal representations here. But if you want to find the source solution, yes. Now, a source solution is a solution that begins with Dirac delta and represents the evolution of a lump of mass that diffuses in space. Suppose that you have the plasma problem. There is a heat release with high energy and it goes around.
37:00
So can you find this solution? And this is what the people in Moscow did. And this was done by Seldovit, Kompanyelis, and Barenblatt. And we usually know it from the name of Grisha Barenblatt who came from the east to visit us and explain to us these beautiful things he had written very beautiful papers before.
37:20
Now, the solutions are explicit and they are similar. Now, people here in this audience will know what is a similarity but let me tell you so similarity means that if t is equals one you have to calculate f as a function of x and if t is not one you have to rescale.
37:40
So the t dependence is what we say trivial with respect to the x dependence and it is not really trivial at all because it has two exponents which are the similarity exponents or the group theoretic exponents. You calculate them and for these people they were algebraic exponents alpha and beta the similarity exponents
38:01
and then they discovered that luckily enough they were genius they discovered that there is a solution of this thing which is not an explanation like in the Gaussian corner it's a parabola and it looks like this now you see the parabola goes down with time it spreads in space
38:21
and since it is a parabola it has a finite support with a very clear front so there is a place where it's hot and a place where there is nothing the wave didn't come to you and this is important in physics of plasmas because it gives you a hope that the plasma will not blow you up
38:42
so you calculate the distance of this plasma and you find that it is t to the beta because this the variable here is c equals x over t to the beta so this gives you the spread exponent and beta is less than one half
39:01
so you have anomalous diffusion this is for us in non-linear theory the explanation of the anomalous exponents in the diffusion process now this is not the only explanation you can have fractional operators in anomalous Brownian motion but they are not the same thing so this is a new thing
39:21
and now you have a scaling law you will not have the very general equation but you have this beautiful equation so this is it now there was a big problem as for the theory that I will tell you now a parabola is an algebraic object so it tends to live forever
39:40
so the parabola goes down and goes down, down, down so it becomes negative why did they cut it at the level u equals zero they did it because they wanted to and the explanation was never clear they say something like this should be physical
40:00
now this sentence means nothing for mathematicians so we have to introduce the concept of solution that tells you that they've had a right to cut the parabola but if you cut the parabola like they did in this corner the function is not any more differentiable so how you justify that
40:21
you are taking two derivatives in space okay the next thing you do is to think about the case m less than one which was not considered by the people in Moscow but was considered much before and it's very popular nowadays because this is one of the advanced things that we do now
40:42
if m is less than one you get something called fast diffusion and if you do the algebra the algebra is also similar there is almost no news in that derivation but there is a term that goes downstairs and has a plus sign so the profiles are like this the exponents are the same
41:01
but now m is less than one so there is a minus here and there is a minus here and these solutions have potential decay which is completely abnormal also for a diffusion process it is not the central limit law and now beta is larger than one half and the tails are potential
41:21
so they are called fat tails fat tails are investigated by people in statistical mechanics and in economy and they are more realistic than the typical random noise tails now the big problem is that if you make this analysis with algebra you find that this exponent of similarity collapses when this is zero
41:47
and the critical value of collapse is distinct so what is the long time behavior or the representative solutions of fast diffusions where the exponent is so small
42:01
and this is what we call the subcritical case which is a big box of surprises and this is what we are investigating now and i have written a book i will tell you about later okay so the concept of solution is something that we are worried about the classical solution doesn't work because of what i will tell now
42:22
the limit solution because of Seldovich and Barenblatt and Compagniec you have these parabolas they are not classical solutions and they say that this was the limit of some numerical computations and it was something that had physical sense
42:40
but mathematicians are never satisfied so all our lady introduced eight years later in moscow weak solutions which was a novelty in 1985 not for not for lere but for the rest of the world and you integrate by parts with respect to test function if you integrate the space derivatives once
43:01
you get the weak solution if you integrate twice you get the very weak solution so you have a choice but in the weak solutions you don't get a sense of derivatives and then you consider strong solutions where with some extra work
43:21
your derivatives are not distributions but they are functions so you can do generalized calculus in the sense of stampakia which was invented in the 60s so you can do calculus and strong solutions are what we do but then the people in semi-group theory came along and said listen guys
43:40
we can solve the equation in a very beautiful way you discretize in time you write this equation let me replace the power u up to the n by phi of u you make implicit discretization in time and you write u to n minus one is the previous value that is known
44:02
u sub n is the new value that is not known and the v is phi of u and inverted to call u beta v if you make these transformations this equation is the same as this equation and phi of u is v is the same as this equation
44:22
so this is non-linear elliptic theory which was really a big thing in the 70s and 80s and this is the famous krundle ligand approach that tells you that all of these evolution problems are in fact problems in non-linear elliptic equations plus semi-group theory in the non-linear way
44:42
and there is a big bunch of investigation by Ambrosio, Savare, Nocetto and other people in this issue and there are many other concepts of solution in the non-linear world now you say why people in porous media worry if they can prove that they have weak solutions and strong solutions
45:02
because in the rest of the people in the family of non-linear diffusion these weak solutions do not work all the time so the fully non-linear theory asks from you that you prove that there are weak solutions and Luis Caffarelli and myself tried in 1999
45:22
to prove weak solutions for degenerate parabolic equations and we did it but it was much work and we could not make it general for other non-linear diffusion equations with degeneracies so it seems that the viscosity solutions are difficult to tackle for this thing now the people who do viscous fluids in this diffusion case
45:48
with convection like entropy solutions which were as we say credited to Grushkov but this is just written to impress you with the idea that if you want to be general you have lots of solutions once you get solutions you get estimates
46:04
now estimates are for the experts in Paris something so elementary that they have to jump on that slide you estimate different norms and they evolve in a good way with time some of them and the rest do not evolve well in time
46:22
so you keep your list of the good ones and you produce some existence of solutions by using this estimating approximation method so the first estimate is that you can control for this diffusion processes the LP norm
46:41
and this decreases in time but this doesn't give you compactness so you try to prove control of derivatives and this is what you do you multiply by u to the m and this integral decreases in time because of a certain dissipation
47:01
and this is the typical trick conservation plus dissipation now there is another trick that is use the multiplier UMT and then what is conserved dissipated is this gradient so you see now the work with energies
47:21
which tells you that these people have learned their physics well and there are lots of energies but these three energies allow you to prove boundedness and compactness and to solve the problem and if you are numerical you want to solve also stability
47:41
and these typical estimates do not prove stability so there was this question of finding which is the space in which you can find the stability for these flows and it was found that it was L1 and this was a contribution main contribution of Philippe Benilang
48:02
our great friend who has left us L1 is a bad space it's not a reflexive space it has bad compactness properties people hated L1 but there was the only way to prove at a certain time that you have contraction
48:21
and contraction gives you stability the derivative in time of this difference goes down in time and so people were desperate for any better space and Britishies found H minus one which is probably not your piece of cake and lately the people in probability supplied us
48:42
well in fact Felix Soto supplied us with Wasserstein spaces which is probably not so much better than L1 so you get the standard concept of solution this is an impressive slide that you can forget what a person in partial differential equations called solution
49:03
is not the nicest thing because you have to tell lots of details about functional spaces but the main point is that there is something that is controlled in some spaces and the derivatives that appear in the equation are really functions not only distributions
49:22
this is satisfied almost everywhere and the whole thing is continuous in the L1 space and this continuity is what allows you to look at the flow as a curve in infinite dimensions and this is what tells you that you can use infinite dimensional dynamical systems
49:40
as your friend and then you get this beautiful dissipation estimate that we call ultra contractivity and this is really what the functional analyst like the story you for positive times is bounded no matter what you do at t equals zero you only need to get a distribution in L1
50:02
and the decay is power-like which is a scaling law so the work with the scaling laws and the relation between these scaling laws and functional analysis sovereign inequalities Gallardo inequalities logarithmic sovereign inequalities is one of the driving forces of this theory
50:22
and let me tell you now with regularity results we know now that the second derivative is an L1 function but this is not so good can you get something better we were trying to get the second derivative estimate which is point wise and the only thing that was obtained
50:43
was this curious estimate which is a universal estimate if you take the pressure which is u to the n minus one and you take the laplacian it's bounded below now this believe it or not is a manifestation of the second law of thermodynamics for porous medium
51:00
and it's mysterious because it was found more or less by chance but it has allowed Caffarelli and Friedman to prove one of the main results in 82 which is that solutions of this problem are really c alpha the Georgi Moser Nash for this equation and the interface is also a c alpha curve
51:23
which is beautiful and then we try to go further and I proved at the beginning of my stay in the United States that you can have metastability I will show you immediately what it is it is that the interface does not move at the beginning
51:40
it tends to be stationary for a while but inside the fluid is moving and all of a sudden it moves and then I proved with Caffarelli and Friedman that at the beginning of the movement of metastable states you can have corner points and the interfaces even in one dimension are not C1
52:01
so the regularity of the interfaces was destroyed even in one dimension and then we proved oh this is the beautiful part of the theory the geometric theory I will not have much time to talk about it but let me tell you that in one dimension you can have for instance a right interface with a metastable stretch and then it moves
52:23
and this is the corner point and you can have a left interface which is very nice and you can have inner interfaces which collide here with a certain point and you can have even an isolated point which is metastable and then disappears
52:42
and what we proved in 87 with Noemi Volanski of Buenos Aires is that after a finite time solutions are C1 alpha even in several dimensions this is the picture in several dimensions and we were stalled
53:00
for eight for ten years we didn't know how to do better and then Herbert Koch wrote his thesis in Heidelberg and it was long 160 pages and he used non-standard harmonic analysis to prove that the interfaces are in fact C infinity
53:22
the stumbling block was going from C1 alpha to C2 alpha because the geometry is in C2 the curvature is already in C2 so that's it he went over C2 alpha and then you get the rest for free and Herbert is an incredible person
53:40
he never published his thesis because he said that it was too long it was already internet so that's a good example of a person who does wonderful work and doesn't care very much about publishing in 100 pieces I will jump on the difficulties of the solutions with holes
54:05
because I want to tell you the non-linear central limit theorem which is something I have to tell before I finish choice the domain Rn choose as data a L1 distribution probably with two signs
54:21
take the equation that you have to write in this way when you have sign solutions put a forcing term that can be a L1 function next in time and call m the combined mass due to the initial data and the forcing term and here's the theorem that was proved
54:43
first by Friedman and Kamin partially and then the whole version by myself and says that if Barenblatt solution is considered with this asymptotic mass and your solution is U and the Barenblatt is B the difference goes to zero after renormalization
55:03
and alpha is the Barenblatt exponent so this is the real scale at which you see that the zoom goes to zero and then you can do the same computation in all Lp spaces in particular if p is one
55:21
p prime is infinity and this is zero so it says that the L1 norm goes to zero you are proving that the two functions are distributions of probability that look asymptotically the same in L1 and they look very much the same also in L infinity the exponents are the algebraic exponents of Barenblatt
55:46
and let me tell you that what Friedman and Kamin did was they found this idea and they proved the case where U0 has compact support and f is zero and the solutions are positive
56:00
and in our investigation of the general result I proved that the rate cannot be improved this is the rate that you can get it goes to zero and I also proved that you can allow m less than one but remember the collapse of the exponents has to be super critical
56:22
so you get the non-linear central limit theorem but only if m is super critical and if m is less than this curious exponent it is very fast diffusion then you enter a different thing that is relatively new world
56:41
we know now how it works but it's a very anomalous central limit theorem and for m equals zero this is the logarithmic diffusion and this is the Ricci flow in two dimensions so let me now try to tell you how this thing evolves because people like pictures and my collaborator Raul Ferreira allowed me to have this
57:08
this is it if we talk for still one minute I will do it again okay oh my god this is great
57:24
so thanks to Raul who's doing good work in reaction diffusion okay so let me tell you now that so after this non-linear central limit theorem you want to know more can you get weights
57:41
and then Carillo, Jose Antonio Carillo from Barcelona wrote a beautiful paper with Giuseppe Toscani in year 2000 proving that there are weights, potential rates if you assume that your distribution has moments has bounded variance
58:04
I have proved that if you don't get bounce on the moment you cannot get rates and then we have been working for years on the issue of rates and it's not still complete and then my friend Kiam Lee from Seoul, Korea and myself
58:21
wanted to know something about eventual geometry and concavity and we proved this result that I find beautiful if you take your pressure which is u to the n minus one which is the relevant physical quantity and you take the hessian matrix and then you compare this matrix with a certain diagonal matrix
58:41
and you renormalize it by multiplying by t it goes to zero and since t is the correct scale what you say is not only you converge to the bar and black profile you become really parabolic so this is the really strong version of the second principle of thermodynamics in this case
59:01
now I didn't give details of all of these things and my time is almost over so let me tell you now two slides by telling you the calculations of entropy rates which are a beautiful piece of what has been done in the last years you have to rescale your function which is like this
59:25
by renormalizing it to size unity and this is the renormalized equation which has an extra term due to renormalization this is like the coriolis force and then you define a certain function which is again some integral of something
59:42
this is the Boltzmann idea and Villani gave a beautiful lecture about how these functionals can be difficult but this is just a standard functional in the non-linear case and then you make two calculations the derivative of this functional is never
01:00:00
and equals some dissipation. And the dissipation goes down with a certain rate. And finally the rate is proportional to dissipation. So you integrate this equation, it gives you exponential decay. You put exponential decay here and you go to exponential decay. So E decays exponentially in S. S is a log. So E decays power time in T. And that's
01:00:28
beautiful details take like 30 pages. And so let's give some pictures. This is the computations we did for the paper with Kiam Lee. It's again courtesy of Raul Ferreira.
01:00:43
You see in time, time goes in this direction. And this is the initial profile in 1D and in 3D. And it has two humps. And after some while, it's a parabola. And in three dimensions, it goes down because of the effect of dimensionality. Now you can do the same
01:01:04
thing for the heat equation. But in the heat equation, the pressure, the corresponding pressure is the log. So this proves log-concavity for the heat equation after finite time. And this is a new result we proved, Kiam Lee and myself, for heat equation.
01:01:22
Now in the first diffusion equation, we did the calculations and you see that this bad monster doesn't like to be concave. You know, it is resisting us. And this is true. We proved that. It is becoming worse. Now since our talk that tries to be general
01:01:44
cannot enter technicalities, but technicalities are the core of pure mathematics. I have taken the trouble of writing the story of my life among these people with beautiful minds in a book that will appear soon in Oxford University Press. And it's 600 pages of
01:02:04
calculations. But if you don't want to go for that and you want to go for more advanced topics of research, I wrote another book that has already appeared and I will thank Oxford University Press for the extraordinary work they did to have the book ready. And this tells
01:02:20
you everything about the scaling laws, smoothing and decay estimates. If you only want to know the tale about the nonlinear central limit theorem, this is written in a survey that I wrote three years ago that is called asymptotic behaviour for the porous medium equation in the whole space. How much time do I have? Okay, so let me tell you that
01:02:47
there is a whole set of things that people are doing now. And just in one minute I will tell you that one of the vistas into the future is becoming probabilistic and doing Wasserstein techniques. And we proved there is a beautiful book by Villani on topics in
01:03:07
optimal transportation, which is very good for mathematicians, and he's probably here around. And Carrillo, McCann and myself and other people have been working in proving contraction for the porous medium in different Wasserstein norms. But let me jump on that. And I will
01:03:26
tell you that there are new fields, for instance, fast diffusion with m less than one, but much less than one, so that this central limit theorem is not the one I told you but much more difficult. And this is contained in my book Smoothing. But you want to do systems,
01:03:45
for instance. And let me tell you again about the work of Arreiro Velázquez. Or you want to general parabolic, hyperbolic equations. And there is work by Jose Carrillo, which is a different one. We have lots of careers in this country. And Benny Lang and Bitvolt. And you want to do
01:04:03
probably non-linear diffusion image processing. But then you have to go to the porous medium equation now, to the gradient diffusion, which is the pill operation. And in the limit you have the total variation equation, which was described by Vincent Caseljes. And Andrés and Mazón are
01:04:23
working with him. Now, I don't have, of course, time to tell you the beauties of the logarithmic diffusion. But just as last slide, I will tell you that the application to geometry says that if you write in two dimensions this evolution equation for a conformal representation of a surface,
01:04:45
ds squared u dr squared, which is the standard metric, according to Hamilton in 88, you get the log diffusion equation, which is completely describing my book, what we did. And there is a beautiful thing that took some time for us to discover. The first thing is that this seems to
01:05:06
be a diffusion equation, but it has a curious property. You take an initial distribution, you make it evolve in time in the whole plane, and it loses four pi units of mass per unit time.
01:05:20
And there is no way you can avoid it. So there is some something happening at infinity that you don't discover doing physics. If you do geometry, this is the Gauss-Bonnet theorem. So the connections between geometry and non-linear diffusion are not only the free boundaries.
01:05:40
In the fast diffusion case, it is connected with this log diffusion, which is the Ricci flow, and also with the Yamabe flow. And of course, I will only put to you a picture of how this mass is lost. This is the picture. If you take the diffusion equation for m less than one,
01:06:02
but not near zero, it goes down more or less with a fat tail. But if you approach m equals zero, the fat tail becomes so thin that it looks like a direct going down, and this is the four pi t that goes down. And this is what we are doing now. So a summary of my talk.
01:06:27
The non-linear diffusion problems are based on physics and represent a working pure mathematics, essentially partial differential equations and functional analysis,
01:06:43
with deep connections now with geometry through the free boundaries and the geometric flows, and also with probability through the Wasserstein techniques and the non-linear central limit theorems. And I think I will stop here. I thank you very much for your attention.
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