The mathematical theories of diffusion. Nonlinear and fractional diffusion
TThe mathematical theories of diffusion.Nonlinear and fractional diffusion ∗ Juan Luis V´azquez
Universidad Aut´onoma de Madrid, Spain
Abstract
We describe the mathematical theory of diffusion and heat transport with a viewto including some of the main directions of recent research. The linear heat equationis the basic mathematical model that has been thoroughly studied in the last twocenturies. It was followed by the theory of parabolic equations of different types.In a parallel development, the theory of stochastic differential equations gives afoundation to the probabilistic study of diffusion.Nonlinear diffusion equations have played an important role not only in theorybut also in physics and engineering, and we focus on a relevant aspect, the existenceand propagation of free boundaries. Due to our research experience, we use theporous medium and fast diffusion equations as case examples.A large part of the paper is devoted to diffusion driven by fractional Laplacianoperators and other nonlocal integro-differential operators representing nonlocal,long-range diffusion effects. Three main models are examined (one linear, two non-linear), and we report on recent progress in which the author is involved.
Contents Introduction to Diffusion Fractional diffusion ∗ To appear in Springer Lecture Notes in Mathematics, C.I.M.E. Subseries, 2017. a r X i v : . [ m a t h . A P ] J un Nonlinear Diffusion PME: degenerate diffusion and free boundaries The Fast Diffusion Equation Nonlinear fractional diffusion. Potential model The FPME model and the mixed models Operators and Equations in Bounded Domains p -Laplacian type . . . . . . . . . . . . . . 51 Further work on related topics Introduction to Diffusion
There are a number of phenomena in the physical sciences that we associate with theidea of diffusion. Thus, populations of different kinds diffuse; particles in a solventand other substances diffuse. Besides, heat propagates according to a process that ismathematically similar, and this is a major topic in applied science. We find many otherinstances of diffusion: electrons and ions diffuse; the momentum of a viscous fluid diffuses(in a linear way, if we are dealing with a Newtonian fluid). More recently, we even talkabout diffusion in the financial markets.The word diffusion derives from the Latin diffundere , which means “to spread out”. Asubstance spreads out by moving from an area of high concentration to an area of lowconcentration. This mixing behaviour does not require any bulk motion, a feature thatseparates diffusion from other transport phenomena like convection, or advection.This description can be found in
Wikipedia [257], where we can also find a longer listingof more than 20 diffusive items, that are then developed as separate subjects. It includesall classical topics we mention here.
In this survey paper we want to present different topics of current interest in the math-ematical theory of diffusion in a historical context. To begin with, we may ask if mathe-matics is really relevant in the study of diffusion process? The answer is that diffusion is3 topic that enjoys superb mathematical modelling. It is a branch of the natural sciencesthat is now firmly tied to a number of mathematical theories that explain its workingmechanism in a quite successful way. The quantity that diffuses can be a concentra-tion, heat, momentum, information, ideas, a price,... every such process can be called adiffusion, and its evolution is governed by mathematical analysis.Going into the details of how we actually explain diffusion with mathematics, it sohappens that we may do it in a twofold way: roughly speaking, by means of the diffusionequation and its relatives, or by a random walk model and its relatives.The older work concerns the description of heat propagation and mass diffusion bymeans of partial differential equations (PDEs), and this is the view that we are going tofavor here. The type of PDEs used is the so-called parabolic equations, a family basedon the properties of the most classical model, the linear Heat Equation (HE), which iscalled in this context the diffusion equation.On the other hand, according to probabilists diffusion is described by random walks,Brownian motion, and more generally, by stochastic processes, and this is a long andsuccessful story in 20th century mathematics, culminated by It¯o’s calculus. Let us recallthat the connection between the two visions owes much to A. N. Kolmogorov.Actually, an interesting question for the reader or the expert is ‘How much of themathematics of diffusion can be explained with linear models, how much is essentiallynonlinear ?’ Linear models have priority when applicable by virtue of their rich theory andeasier computation. But nonlinear models are absolutely necessary in many real-worldcontexts and most of our personal research has been based on them. Diffusion equationsinvolving nonlinearities and/or nonlocal operators representing long-range interactionsare the subject matter of the recent research that we want to report in this paper.
Outline of the paper.
The declared intention is to make a fair presentation of themain aspects of Mathematical Diffusion as seen by an expert in PDEs, and then to devotepreference to the work done by the author and collaborators, especially the more recentwork that deals with free boundaries and with fractional operators. The main topicsare therefore the heat equation, the linear parabolic theory and the fractional diffusionin a first block; the nonlinear models, with emphasis on those involving free boundariescome next; finally, the nonlocal and nonlinear models, and here we will focus on the twoequations that have been most studied by the author in the last decade, both combineporous medium nonlinearities and fractional diffusion operators.Lengthy details are not frequent, but we give some for very recent work of ours andour collaborators. On the other hand, we supply many important explicit solutions andcomment on their role. Indeed, many such examples belong to the class stable diffusivepatterns , that combine their surprising occurrence in numerous real-world applicationswith the beauty of pure mathematics.A large number of connections with other topics is given in the text, as well as hintsfor further reading. More detail on the topics is to be found in the articles, monographs,or in our previous survey papers.
Disclaimer.
Let us comment on an important absence. The stationary states of diffu-sion belong to an important world, the elliptic equations . Elliptic equations, linear andnonlinear, appear in a large number of applications: diffusion, fluid mechanics, waves4f all types, quantum mechanics, ... Elliptic equations are mathematically based on theLaplacian operator, ∆ = ∇ , the most important operator for our community. This is ahuge world. We are not going to cover in any detail the many developments in ellipticequations related to diffusion in this paper, we will just indicate some important factsand connections here and there. We begin our presentation with the linear heat equation (HE): u t = ∆ u proposed by J. Fourier as a mathematical model for heat propagation (“Th´eorie Ana-lytique de la Chaleur”, 1822, [124], with a previous attempt in 1807), and the Fourieranalysis that he promoted. For a long time the mathematical study of heat transport anddiffusion was almost exclusively centered on the heat equation. In these two centuries,the mathematical models of heat propagation and diffusion have made great progressboth in theory and application. Actually, they have had a strong influence on no lessthan 6 areas of Mathematics: PDEs, Functional Analysis, Infinite-Dimensional Dynami-cal Systems, Differential Geometry and Probability, as well as Numerics. And the theoryhas been influenced by its motivation from Physics, and in turn the concepts and methodsderived from it have strongly influenced Physics and Engineering. In more recent timesthis influence has spread further away, to Biology, Economics, and the Social Sciences. • The classical analysis of the heat flow is based on two main mathematical techniques: integral representation (convolution with a Gaussian kernel) and
Fourier analysis , basedon mode separation, analysis, and synthesis. Since this topic is well-known to the readers,see for instance [116], we will stress the points that interest us to fix some concepts andtools.
1. The heat equation semigroup and Gauss.
When heat propagates in free space R N , the natural problem is the initial value problem(1.1) u t = ∆ u, u ( x,
0) = f ( x ) , which is solved by convolution with the evolution version of the Gaussian function(1.2) G ( x, t ) = (4 πt ) − N/ exp ( −| x | / t ) . Note that G has all nice analytical properties for t >
0, but for t = 0 we have G ( x,
0) = δ ( x ), a Dirac mass. G works as a kernel , a mathematical idea that goes back to Greenand Gauss.The maps S t : u (cid:55)→ u ( t ) := u ∗ G ( · , t ) form a linear continuous semigroup of contrac-tions in all L p spaces for all 1 ≤ p ≤ ∞ . This is pure Functional Analysis, a product ofthe 20th century. Asymptotic behaviour as t → ∞ , convergence to the Gaussian . If u is anintegrable function and M = (cid:82) u ( x ) dx (cid:54) = 0, the following convergence is proved(1.3) lim t →∞ t N/ ( u ( x, t ) − M G ( x, t )) = 0 , L p less is needed.So u ( · , t ) increasingly resembles (i.e. as t grows to infinity) a multiple of the Gaussianprofile G ( · , t ). This is the famous Central Limit Theorem in its continuous form (famousin Probability with M = 1, but M (cid:54) = 1 makes no difference as long as M is not zero).The Gaussian function is the most famous example of the many diffusive patternsthat we will encounter, and the previous theorem shows that is not only stable butalso asymptotic attractor of the heat flow (for finite-mass data). Note that the sharpconvergence result needs renormalization in the form of the growth factor t N/ . • These are two classical personalities of the diffusion equation.J. Fourier and K. F. Gauss
2. Matter diffusion.
This is an older subject in Physics, already treated by RobertBoyle in the 17th century with the study of diffusion in solids. After the work of Fourierin heat propagation, Adolf Fick proposed his law of diffusion of matter [122], wherethe mass flux is proportional to the gradient of concentration and goes in the directionof lower concentrations. This leads to the heat equation, HE, as mathematical model.He also pointed out the fundamental analogy between diffusion, conduction of heat, andalso electricity. Actually, Fourier’s law for heat conduction (1822), Ohm’s law for electriccurrent (1827), Fick’s law for diffusion in solids (1855), and Darcy’s law for hydraulicflow (1856) have a similar mathematical gradient form.
3. The connection with Probability.
The time iteration of independent randomvariables with the same distribution led to the theory of random walks, at the early timesof the Bernoullis et al. Soon it was realized that this led to a large-time limit described(after renormalization) by the Gaussian distribution, as textbooks in Probability andStochastic Processes show. The connection of this evolution with the heat equation tookplace after the construction of the
Brownian motion as a mathematically rigorousobject in the form of
Wiener process . In the 1930s Kolmogorov investigated the equiv-alence of the two view points, i.e., the PDE approach via the Heat Equation and thestochastic approach via Brownian motion. This topic is covered by many PDE authors,let us mention [118] and [212].Next we show two opposing diffusion graphs. They show the comparison of ordereddissipation in the heat equation view, as the spread of a temperature or concentration,6ersus the underlying chaos of the random walk particle approach, origin of the Brownianmotion favoured by the probabilistic school.Left, the nice HE evolution of a Gaussian Right, a sample of random walkWe will go back to the latter view in a while. The experimental observation of chaoticmovement in Nature due to mechanical effects at the microscopic level is credited toRobert Brown (1827), see [57], hence the label ‘Brownian motion’.
4. The Fourier Analysis approach to heat flows in bounded domains.
Thesecond classical scenario for heat flows occurs when heat propagates inside a boundeddomain of space. The convolution approach does not work and other ideas have to beproposed. The Fourier approach proposes to look for a solution in the series form(1.4) u ( x, t ) = (cid:88) T i ( t ) X i ( x ) , and then the time factors are easily seen to be negative exponentials of t , while the spacecomponents X i ( x ) form the spectral sequence, solutions of the problems(1.5) − ∆ X i = λ i X i with corresponding eigenvalues λ i . Boundary values are needed to identify the spectralsequence ( λ i , X i ), i = 1 , , . . . This is the famous linear eigenvalue problem , the startingpoint of the discipline of
Spectral Theory.
The time-space coupling implies then that T i ( t ) = e − λ i t . This is nowadays one of the most celebrated and useful topics in AppliedMathematics and is covered in all elementary PDE books.The scheme works for many other equations of the form u t = A ( u ) and in this wayFourier Analysis and Spectral Theory developed with great impetus, as well as SemigroupTheory. Fourier analysis also took a direction towards the delicate study of functions, aproper branch of pure mathematics, which is one of the most brilliant developments inthe last two centuries. Through the work of Cantor this also motivated advances in SetTheory, since the sets of points where a Fourier series does not converge can be quitecomplicated. We now consider a big step forward in the mathematical tools of diffusion. A more generalfamily of models was introduced to represent diffusive phenomena under less idealized7ircumstances and this was done both in the framework of PDEs and Probability. Thishappened in several stages.In the framework of PDEs, the heat equation has motivated the study of other linearequations, which now form the Theory of Linear Parabolic Equations. They are writtenin the form(1.6) u t = (cid:80) i,j a ij ∂ i ∂ j u + (cid:80) b i ∂ i u + cu + f with variable coefficients a ij ( x, t ), b i ( x, t ), c ( x, t ), and forcing term f ( x, t ). Belongingto the parabolic family requires some structure conditions on such coefficients that willensure that the solutions keep the basic properties of the heat equation. In practice,the main condition is the fact that the matrix a ij ( x, t ) has to be definite positive: thereexists λ > (cid:88) i,j a ij ( x, t ) ξ i ξ j ≥ λ (cid:88) i ξ i . This must be valid for all vectors ξ = ( ξ , . . . , ξ N ) ∈ R N and all x, t in the space-time domain of the problem. Let us point out that the theory is developed under someadditional conditions of regularity or size on the coefficients, a common feature of all PDEanalysis. Another prominent feature is that all coefficients can be submitted to differentmore or less stringent conditions that allow to obtain more or less regular solutions.A more stringent uniform positivity condition is(1.8) λ (cid:88) i ξ i ≤ (cid:88) i,j a ij ( x, t ) ξ i ξ j ≤ Λ (cid:88) i ξ i with 0 < λ < Λ. Even the uniform condition (1.7) can be relaxed so that λ > x and t . When these conditions are relaxed we talk about degenerate or singularparabolic equations. This flexibility on the structure conditions makes for a big theorythat looks like an ocean of results. It will be important later in the nonlinear models.In fact, the parabolic theory was developed in the sequel of its more famous stationarycounterpart, the theory for the elliptic equations,(1.9) (cid:88) i,j a ij ∂ i ∂ j u + (cid:88) i b i ∂ i u + c u + f = 0 , with variable coefficients a ij ( x ) , b i ( x ) , c ( x ) and forcing term f ( x ). The main structurecondition is again (1.7) or (1.8), which is usually called uniform ellipticity condition. Inthe time dependent case, (1.7) and (1.8) are called uniform parabolicity conditions.Main steps in the Parabolic Theory are:(1) The first step is the classical parabolic theory in which a ij , b i , c, f are assumed tobe continuous or smooth, as needed. Functional spaces are needed as framework ofthe theory, and these turn out to be C α spaces (H¨older) and the derived spaces C ,α and C ,α . This leads to a well-known theory in which existence and uniqueness results,continuous dependence on data are obtained after adding initial and boundary data tothe problem. And the theory provided us with Maximum Principles, Schauder estimates,Harnack inequalities and other very precise estimates.8his is a line of research that we would like to follow in all subsequent chapters whenfurther models of diffusion will be treated, but unfortunately the direct application ofthe scheme will not work, and to be more precise, the functional setting will not beconserved and the techniques will change in a strong way.The results are extended into the disciplines of Potential Theory and Generation ofSemigroups. These are also topics that will be pursued in the subsequent investigations.(2) A first extension of the classical parabolic theory concerns the case where the coefficients are only continuous or bounded . In the theory with bad coefficients thereappears a bifurcation of the theory into Divergence and Non-Divergence Equations, whichare developed with similar goals but quite different technology. The difference concernsthe way of writing the first term with second-order derivatives.The way stated before is called non-divergent form, while the divergent form is(1.10) (cid:88) i,j ∂ i ( a ij ∂ j u ) + (cid:88) b i ∂ i u + cu + f = 0 . This form appears naturally in many of the derivations from physical principles. Aboutthe structure conditions, we assume the a ij to be bounded and satisfy the uniformparabolicity condition. The basic functional spaces are the Lebesgue and Sobolev classes, L p , W ,p . Derivatives are understood as distributions or more often weak derivatives,and this motivates the label of weak theory . Existence, uniqueness and estimates in W ,p or W ,p norms are produced. Maximum Principles, Harnack inequalities work and C α is often proved. A very important feature is the Calder´on-Zygmund theory, basicto establish regularity in Sobolev spaces. Divergent form equations were much studiedbecause of their appearance in problems of Science and Engineering. We refer to thebooks [135, 179] for the elliptic theory, and to [125, 178, 183] for the parabolic theory.(3) There is nowadays a very flourishing theory of elliptic and parabolic equations withbad coefficients in the non-divergence form (1.6), but we will not enter into it for reasonsof space, since it does not affect the rest of our expos´e. The probabilistic way to address the extension of the previous subsection takes the formof the diffusion process , which is a solution to a stochastic differential equation, SDEfor short. A diffusion is then a continuous-time Markov process with almost surelycontinuous sample paths. This is essentially a 20th century theory, originated in thework of Bachelier, Einstein, Smoluchowski, then Kolmogorov, Wiener and Levy, and thelast crucial step was contributed by Itˆo, Skorokhod, ... The stochastic equation reads(1.11) dX = b dt + σ dW where W is the N -dimensional Wiener process, and b and σ are (vector and matrixvalued respectively) coefficients under suitable conditions. Differentials are understoodin the Itˆo sense. Among the extensive literature we mention Bass [26], Friedman [126],Gihman-Skorohod [134], Oksendal [202], and Varadhan [236] for the relation betweenPDEs and Stochastic processes. Thus, Bass discusses the solutions of linear elliptic9nd parabolic problems by means of stochastic processes in Chapter II. The solution ofstochastic differential equations (1.11) gives a formula to solve the Cauchy problem forthe evolution PDE u t = L u , where L u = (cid:88) a ij ∂ ij u + (cid:88) b i ∂ i u, if we take the vector b = ( b , . . . , b N ) and the symmetric positive semi-definite matrix a = ( a ij ) is given by a = σ · σ T . And there is an associated dual parabolic PDE indivergence form that is solved by the same method. See also Section 7.3 of [202]. Wecall the functions σ and a the diffusion coefficients of the process X t and operator L respectively, while b is the drift vector. Note also that processes with different σ maylead to the same PDE as long as σ · σ T is the same. Let us mention as further usefulreferences [118], [212], and [222]. A comment about real-world practice.
If we consider a field of practical applica-tion like quantitative finance, one may ask the question about which of the two knownapproaches - PDEs versus martingales and SDEs - is more important in the real practiceof derivatives pricing. Here is a partial answer: since the Black-Scholes equation is amodified form of the Heat Equation, understanding PDEs is very important as a practi-cal tool, see [258]. And the American options add a free boundary problem, a topic thatwe will find later in the text. But the stochastic approach contains richer informationthat is useful in understanding aspects of the financial problem, and it can be easier toformulate and compute in some complex models. Fractional diffusion
Replacing the Laplacian operator by fractional Laplacians is motivated by the need torepresent processes involving anomalous diffusion. In probabilistic terms, it features long-distance interactions instead of the next-neighbour interaction of random walks and theshort-distance interactions of their limit, the Brownian motion. The main mathematicalmodels used to describe such processes are the fractional Laplacian operators, sincethey have special symmetry and invariance properties that makes for a richer theory.These operators are generators of stable L´evy processes that include jumps and long-distance interactions. They reasonably account for observed anomalous diffusion, withapplications in continuum mechanics (elasticity, crystal dislocation, geostrophic flows,...),phase transition phenomena, population dynamics, optimal control, image processing,game theory, finance, and others. See [10, 29, 90, 136, 191, 192, 255, 259], see also Section1.2 of [246].After a very active period of work on problems involving nonlocal operators, there is nowwell established theory in a number of directions, like semilinear equations and obstacleproblems, mainly of stationary type. We are interested here in evolution problems.Instead of the Heat Equation, the basic evolution equation is now(2.1) u t + ( − ∆) s u = 0 There has been intense work in Stochastic Processes for some decades on this equation,but not in Analysis of PDEs. My interest in the field dates from the year 2007 in Texas10n collaboration with Prof. Luis Caffarelli, who was one of the initiators, specially inproblems related to nonlinear diffusion and free boundaries.It is known that there is well defined semigroup associated with this equation for every0 < s < s → − ∆ that appears in the classical heat equation and represents Brownianmotion on one side, and the nonlocal family ( − ∆) s , 0 < s <
1, on the other side. Inthe rest of the paper we are going to discuss some of those differences, both for linearand nonlinear evolution equations. We have commented on the origins and applicationsof the fractional Laplacian and other nonlocal diffusive operators in our previous surveypapers [245] and [246].
Before proceeding with the study of equations, let us examine the different approachesand defining formulas for the fractional Laplacian operator. We assume that the spacevariable x ∈ R N , and the fractional exponent is 0 < s < • Fourier approach . First, we may consider the pseudo-differential operator givenby the Fourier transform:(2.2) (cid:92) ( − ∆) s u ( ξ ) = | ξ | s (cid:98) u ( ξ ) . This allows to use the very rich theory of Fourier transforms, but is not very convenientfor nonlinear analysis which is our final goal. Due to its symbol | ξ | s , the fractionalLaplacian can be viewed as a symmetric differentiation operator of fractional order2 s . Even when 2 s = 1, it is not the standard first derivative, just compare the Fouriersymbols. • Hyper-singular integral operator . The formula reads(2.3) ( − ∆) s u ( x ) = C N,s (cid:90) R N u ( x ) − u ( y ) | x − y | N +2 s dy . The kernel is not integrable near x and this motivates the need for the difference inthe numerator of the integrand. The integral is understood as principal value. Withthis definition, the operator is the inverse of the Riesz integral operator ( − ∆) − s u, which has a more regular kernel C | x − y | N − s , though not integrable at infinity. Thefractional Laplacian operator is also called the Riesz derivative . • Numerics and stochastic approach.
Take the random walk for a processes withprobability u nj at the site x j at time t n :(2.4) u n +1 j = (cid:88) k P jk u nk , where { P jk } denotes the transition function which has a fat tail (i.e., a power decaywith the distance | j − k | ), in contrast to the next-neighbour interaction of random11alks. In a suitable limit of the space-time grid you get an operator A as the in-finitesimal generator of a L´evy process: if X t is the isotropic α -stable L´evy process wehave(2.5) Au ( x ) = lim h → h E ( u ( x ) − u ( x + X h )) . The set of functions for which the limit on the right side exists (for all x ) is calledthe domain of the operator. We arrive at the fractional Laplacian with an exponent s ∈ (0 ,
1) that depends on the space decay rate of the interaction | j − k | − ( N +2 s ) ,0 < s < • The Caffarelli-Silvestre extension.
The α -harmonic extension: Find first thesolution of the ( N + 1)-dimensional elliptic problem(2.6) ∇ · ( y − α ∇ U ) = 0 ( x, y ) ∈ R N × R + ; U ( x,
0) = u ( x ) , x ∈ R N . The equation is degenerate elliptic but the weight belongs to the Muckenhoupt A class, for which a theory exists [119]. We may call U the extended field. Then, putting α = 2 s we have(2.7) ( − ∆) s u ( x ) = − C α lim y → y − α ∂U∂y . When s = 1 /
2, i.e. α = 1, the extended function U is harmonic (in N + 1 variables)and the operator is the Dirichlet-to-Neumann map on the base space x ∈ R N . Thegeneral extension was proposed in PDEs by Caffarelli and Silvestre [68], 2007, see also[218]. This construction is generalized to other differential operators, like the harmonicoscillator, by Stinga and Torrea, [230]. • Semigroup approach.
It uses the following formula in terms of the heat flow gen-erated by the Laplacian ∆:(2.8) ( − ∆) s f ( x ) = 1Γ( − s ) (cid:90) ∞ (cid:0) e t ∆ f ( x ) − f ( x ) (cid:1) dtt s . Classical references for analysis background on the fractional Laplacian operator inthe whole space: the books by N. Landkof [180] (1966), E. Stein [229] (1970), and E.Davies [99] (1996). The recent monograph [58] by Bucur and Valdinoci (2016) introducesfractional operators and more generally nonlocal diffusion, and then goes on to study anumber of stationary problems. Numerical methods to calculate the fractional Laplacianare studied e.g. in [205].
Fractional Laplacians on bounded domains
All the previous versions are equivalent when the operator acts in R N . However, inorder to work in a bounded domain Ω ⊂ R N we will have to re-examine all of them. Forinstance, using the Fourier transform makes no sense. Two main efficient alternativesare studied in probability and PDEs, corresponding to different way in which the infor-mation coming from the boundary and the complement of the domain is to be takeninto account. They are called the restricted fractional Laplacian (RFL) and the spectralfractional Laplacian (SFL), and they are carefully defined in Section 8. And there aremore alternatives that we will also discuss there.12 .2 Mathematical theory of the Fractional Heat Equation The basic linear problem is to find a solution u ( x, t ) of(2.9) u t + ( − ∆) s u = 0 , < s < . We will take x ∈ R N , 0 < t < ∞ , with initial data u ( x ) defined for x ∈ R N . Normally, u , u ≥
0, but this is not necessary for the mathematical analysis. We recall that thismodel represents the linear flow generated by the so-called L´evy processes in StochasticPDEs, where the transition from one site x j of the mesh to another site x k has a prob-ability that depends on the distance | x k − x j | in the form of an inverse power for j (cid:54) = k ,more precisely, c | x k − x j | − N − s . The range is 0 < s <
1. The limit from random walkon a discrete grid to the continuous equation can be read e. g. in Valdinoci’s [235].The solution of the linear equation can be obtained in R N by means of convolutionwith the fractional heat kernel(2.10) u ( x, t ) = (cid:90) u ( y ) P t ( x − y ) dy, and the probabilists Blumental and Getoor proved in the 1960s [37], that(2.11) P t ( x ) (cid:16) t (cid:0) t /s + | x | (cid:1) ( N +2 s ) / . Here a (cid:16) b means that a/b is uniformly bounded above and below by a constant. Onlyin the case s = 1 / G t ofthe heat equation (case s = 1). The behaviour as x goes to infinity of the function P t is power-like (with a so-called fat tail ) while G t has exponential spatial decay, see (1.2).This difference is expected in a theory of long-distance interactions. See more on thisissue in [164].Rather elementary analysis allows then to show that the convolution formula generatesa contraction semigroup in all L p ( R N ) spaces, 1 ≤ p ≤ ∞ , with regularizing formulas ofthe expected type (cid:107) u ( t ) (cid:107) ∞ ≤ C ( N, s, p ) (cid:107) u (cid:107) p t − N/ sp . When the data and solutions are not assumed to be Lebesgue integrable, interestingquestions appear. Such questions have been solved for the classical heat equation, whereit is well-known that solutions exist for quite large initial data, more precisely data withsquare-quadratic growth as | x | → ∞ , see Widder [256]. The idea is that the convolutionformula (2.10) still makes sense and can be conveniently manipulated.Likewise, we may study the fractional heat equation in classes of (maybe) large functionsand pose the question: given a solution of the initial value problem posed in the wholespace R N , is it representable by the convolution formula? The paper [25] by B. Barrios,I. Peral, F. Soria, and E. Valdinoci, shows that the answer is yes if the solutions aresuitable strong solutions of the initial value problem posed in the whole space R N , theyare nonnegative, and the growth in x is no more that u ( x, t ) ≤ (1 + | x | ) a with a < s .In the recent paper [47] by M. Bonforte, Y. Sire, and the author, we look for optimalcriteria. We pose the problem of existence, uniqueness and regularity of solutions for the13ame initial value problem in full generality. The optimal class of initial data turns outto be the class of locally finite Radon measures µ satisfying the condition(2.12) (cid:90) R N (1 + | x | ) − ( N +2 s ) dµ ( x ) < ∞ . We call this class M s . We construct weak solutions for such data, and we prove unique-ness of nonnegative weak solutions with nonnegative measure data. More precisely, weprove that there is an equivalence between nonnegative measure data in that class andnonnegative weak solutions, which is given in one direction by the representation formula,in the other one by the existence of an initial trace. So the result closes the problemof the Widder theory for the fractional heat equation posed in R N . We then reviewmany of the typical properties of the solutions, in particular we prove optimal pointwiseestimates and new Harnack inequalities. Asymptotic decay estimates are also found forthe optimal class. Here is the general result in that direction. We want to estimate thebehaviour of the constructed solution u = P t ∗ µ for t > x and t with precise estimates. Here is the main result. Theorem 2.1.
Let u = S t µ the very weak solution with initial measure µ ∈ M + s andlet (cid:107) µ (cid:107) Φ := (cid:82) R N Φ dµ . There exists a constant C ( N, s ) such that for every t > and x ∈ R N (2.13) u ( t, x ) ≤ C (cid:107) µ (cid:107) Φ ( t − N/ s + t )(1 + | x | ) N +2 s . Here M + s are the nonnegative measures in the class M s and (cid:107) µ (cid:107) Φ is the associatedweighted norm with weight Φ( x ) = (1+ | x | ) − ( N +2 s ) / . See whole details in [47], Theorem7.1. The dependence on t cannot be improved. Under radial conditions a better growthestimate in x is obtained. Construction of self-similar solutions with growth in spacealso follows. • Equation (2.9) is the most representative example of a wide class of equations that areused to describe diffusive phenomena with nonlocal, possibly long-range interactions. Wecan replace the fractional Laplacian by a L´evy operator L which is the pseudo-differentialoperator with the symbol a = a ( ξ ) corresponding to a certain convolution semigroupof measures, [164]. Popular models that are being investigated are integro-differentialoperators with irregular or rough kernels, as in [213], where the form is(2.14) u t + b ( x, t ) · ∇ u − (cid:90) R N ( u ( x + h, t ) − u ( x, t )) K ( x, t, h ) dh = f ( x, t ) . See also [8, 55, 83, 101, 121, 219, 233], among many other references. • A different approach is taken by Nystr¨om-Sande [201] and Stinga-Torrea [231], whodefine the fractional powers of the whole heat operator and solve(2.15) ( ∂ t − ∆) s u ( t, x ) = f ( t, x ) , for 0 < s < .
14n this equation the random jumps are coupled with random waiting times. The authorsfind the space-time fundamental solution that happens to be explicit, given by(2.16) K s ( t, x ) = 1(4 πt ) N/ | Γ( − s ) | · e −| x | / t t s = 1 | Γ( − s ) | t s G ( x, t ) , for x ∈ R N , t >
0, where G is the Gaussian kernel. The limits s → , Nonlinear Diffusion
The linear diffusion theory has enjoyed much progress, and is now solidly establishedin theory and applications. However, it was soon observed that many of the equationsmodeling physical phenomena without excessive simplification are essentially nonlinear,and its more salient characteristics are not reflected by the linear theories that had beendeveloped, notwithstanding the fact that such linear theories had been and continueto be very efficient for a huge number of applications. Unfortunately, the mathematicaldifficulties of building theories for suitable nonlinear versions of the three classical partialdifferential equations (Laplace’s equation, heat equation and wave equation) made itimpossible to make significant progress in the rigorous treatment of these nonlinearproblems until the 20th century was well advanced. This observation also applies toother important nonlinear PDEs or systems of PDEs, like the Navier-Stokes equationsand nonlinear Schr¨odinger equations.
The main obstacle to the systematic study of the Nonlinear PDE Theory was the per-ceived difficulty and the lack of tools. This is reflected in a passage by John Nash (1958).In his seminal paper [198], he said
The open problems in the area of nonlinear PDE are very relevant to applied math-ematics and science as a whole, perhaps more so that the open problems in any otherarea of mathematics, and the field seems poised for rapid development. It seems clear,however, that fresh methods must be employed... and he continues in a more specific way:
Little is known about the existence, uniqueness and smoothness of solutions of thegeneral equations of flow for a viscous, compressible, and heat conducting fluid...
This is a grand project in pure and applied science and it is still going on. In order tostart the work, and following the mathematical style that cares first about foundations,he set about the presumably humble task of proving the regularity of the weak solutionsof the PDEs he was going to deal with. More precisely, the problem was to provecontinuity (H¨older regularity) of the weak solutions of elliptic and parabolic equationsassuming the coefficients a ij to be uniformly elliptic (positive definite matrices) but onlybounded and measurable as functions of x ∈ R N . In a rare coincidence of minds, thiswas done in parallel by J. Nash [197, 198] and the then very young Italian genius E. De15iorgi [100] . This was a stellar moment in the History of Mathematics, and the ideasturned out to be “a gold mine”, in L. Nirenberg’s words . The results were then takenup and given a new proof by J. Moser, [194], who went on to establish the Harnackinequality, [195], a very useful tool in the sequel.Once the tools were ready to start attacking Nonlinear PDEs in a rigorous way, it wasdiscovered that the resulting mathematics are quite different from the linear counterparts,they are often difficult and complex, they turn out to be more realistic than the linearizedmodels in the applications to real-world phenomena, and finally they give rise to a wholeset of new phenomena unknown in the linear world. Indeed, in the last decades wehave been shown a multiplicity of new qualitative properties and surprising phenomenaencapsulated in the nonlinear models supplied by the applied sciences. Some of them arevery popular nowadays, like free boundaries, solitons and shock waves. This has keptgenerations of scientists in a state of surprise and delight. Nonlinear Science rests nowon a firm basis and Nonlinear PDEs are a fundamental part of it. The general formula for the nonlinear diffusion models in divergence form is(3.1) u t = (cid:80) ∂ i A i ( u, ∇ u ) + (cid:80) B ( x, u, ∇ u ) , where A = ( A i ) and B must satisfy some so-called structure conditions, the main oneis again the ellipticity condition on the function A ( x, u, z ) as a function of the vectorvariable z = ( z i ). This general form was already posed as a basic research project in the1960s, cf. [216, 17]. Against the initial expectations, the mathematical theory turned outto be too vast to admit a simple description encompassing the stated generality. Thereare reference books worth consulting, like those by Ladyzhenskaya et al. [178, 179],Friedman [125], Lieberman [183], Lions-Magenes [185], and Smoller [221] are quite usefulintroductions. But they are only basic references.Many specific examples, now considered the classical nonlinear diffusion models, havebeen investigated separately to understand in detail the qualitative features and to in-troduce the quantitative techniques, that happen to be many and from very differentorigins and types.My personal experience with nonlinear models of diffusive type lies in two areas calledrespectively ‘Nonlinear Diffusion with Free Boundaries’ and ‘Reaction-Diffusion PDEs’. The work on nonlinear parabolic equations in the mathematical research community towhich I belonged focussed attention on the analysis of a number of paradigmatic modelsinvolving the occurrence of free boundaries, for which new tools were developed andtested. A rich theory originated that has nowadays multiple applications. Strictly speaking, priority goes to the latter, but the methods were different. Nash and Nirenberg shared the Abel Prize for 2015. The Obstacle Problem.
This is the most famous free boundary problem and thereis a huge literature for it, cf. [64, 67, 127, 169] and their references. It belongs to theclass of stationary problems, connected with elliptic equations, hence further away fromour interests in this paper. Let us only say at this point that a free boundary problemis a mathematical problem in which we want to find the solution of a certain equation(normally, a PDE) as well as the domain of definition of the solution, which is also anunkown of the problem. Typically, there exists a fixed ‘physical’ domain D and thesolution domain Ω that we seek is a subset of D , well-determined if we know the freeboundary Γ = ∂ Ω ∩ D .Parabolic free boundaries may move in time. They appear in the ‘four classical sisters’that we will introduce next: • The Stefan Problem (Lam´e and Clapeyron, 1833; Stefan 1880) The problem typ-ically describes the temperature distribution in a homogeneous medium undergoing aphase change (like ice and water). The heat equation must be solved in both separatemedia filling together a certain space D ⊂ R N , and the separation surface is allowed tomove with time according to some transfer law. The mathematical formulation is thus SE : (cid:26) u t = k ∆ u for u > ,u t = k ∆ u for u < . T C : (cid:26) u = 0 , v = L ( k ∇ u − k ∇ u ) . (SE) means state equations, valid in the separate domains Ω = { ( x, t ) : u ( x, t ) > } and Ω = { ( x, t ) : u ( x, t ) < } , which are occupied by two immiscible material phases(typically water for u > u < D is not the whole space then usual boundary conditions have to be given on the fixedboundary ∂D . The main mathematical feature is the existence of a free boundary or moving boundary Γ ⊂ R N × R that separates ice from water and there u = 0, see themonographs[190, 211]. This free boundary Γ moves in time and has to be calculatedalong with the PDE solution u , so that suitable extra information must be given todetermine it: (TC) means transmission condition that applies at the free boundary Γ,and v is the normal advance speed of Γ. Physically, this formula is due to the existenceof latent heat at the phase transition. We not only want to determine the location of Γbut we want to hopefully prove that it is a nice hypersurface in space-time.Summing up, the combination of analysis of PDEs and variable geometry is whatmakes this problem difficult. The correct mathematical solution came only via the weakformulation [159] that allows to eliminate the geometry in a first step and concentratein finding the so-called weak solution. The free boundary comes later as the zero levelset of the weak solution, and finding it needs some regularity theory.A simpler version is the One-phase Stefan problem where ice is assumed to be atzero degrees, roughly u = 0 in Ω . The free boundary is still there but the mathematicaltheory is much easier, hence better known. • The Hele-Shaw cell problem. (Hele-Shaw, 1898; Saffman-Taylor, 1958) Theproblem is posed in a fixed spatial domain D ⊂ R N , and consists of finding Ω( t ) ⊂ D and u ( x, t ) such that u > , ∆ u = 0 in Ω( t ); u = 0 , v = L∂ n u on ∂ f Ω( t ) . also called interface in the literature. t ) ⊂ D , and ∂ n u denotesnormal derivative on the free boundary Γ( t ) = ∂ f Ω( t ), the part of the boundary of theset { x ∈ D : u ( x, t ) > } that lies inside D . Additional conditions are to be given onthe part of fixed boundary ∂D bounding Ω( t ). Once Ω( t ) is known, solving the Laplaceequation for u is standard; notice that it is nontrivial because of the boundary conditions(sometimes there is a forcing term).Mathematically, this is a simplified version of the previous model where there is onlyone phase, and besides the time derivative term disappears from the state equation. Thisincreased simplicity comes together with beautiful analytical properties, some of themrelated to the theory of conformal transformations and complex variables when workingin 2D, see [150, 208]. The Hele-Shaw flow appears in fluid mechanics as the limit of theStokes flow between two parallel flat plates separated by an infinitesimally small gap,and is used to describe various applied problems. The weak formulation is studied in[113]. There are many examples of moving boundaries with interesting dynamics; thus,a peculiar complex variable pattern exhibiting a free boundary with a persistent pointedangle is constructed in [172] in 2D. In that example, the free boundary does not moveuntil the pointed angle is broken, which happens in finite time. On the other hand, widerangles move immediately and the free boundary is then smooth. • The Porous Medium Equation . This is an equation in the nonlinear degenerateparabolic category, u t = ∆ u m , m > . The equation appears in models for gases in porous media, underground infiltration, high-energy physics, population dynamics and many others. We will devote a whole sectionto review the free boundary and other nonlinear aspects of this equation, called PMEfor short, since it has served so much as a paradigm for the mathematics of nonlineardegenerate diffusion, see [12, 243, 244]. Actually, we see that the free boundary does notappear in the formulation, but it will certainly appear in the theory. It is a hidden freeboundary .The equation can be also considered for exponents m <
1, called fast diffusion range ,and further generalized into the class of so-called filtration equations u t = ∆Φ( u ) , whereΦ is a monotone increasing real function. This generality also allows to include the Stefanproblem that can be written as a filtration equation with very degenerate Φ:Φ( u ) = ( u − + for u ≥ , Φ( u ) = u for u < , p > . • The p -Laplacian Equation . This is another model of nonlinear degenerate diffusion(3.2) u t = div ( |∇ u | p − ∇ u ) . Such a model appears in non-Newtonian fluids, turbulent flows in porous media, glaciol-ogy and other contexts. The mathematics of this equation turn out to be closely relatedto the PME: existence, regularity, free boundaries, and so on, but there are subtle differ-ences. Here p > p = 1 (total variation flow, used in image analysis), or p = ∞ (appearingin geometry and transport), [117]. On the other hand, the equation can be generalizedinto the class of equations with gradient-dependent diffusivity of the general form u t − ∇ · ( a ( | Du | ) Du ) = 0 , a is a nonnegative real function with suitable growth assumptions to ensure de-generate parabolicity. Another extension is the doubly non-linear diffusion equation ofthe form u t = ∇ · ( | D ( u m ) | p − D ( u m )) . Here the diffusivity takes the form a ( u, | Du | ) = cu ( p − m − | Du | p − . We use the nota-tions ∇ u = Du for the spatial gradient. This is another important direction taken by Nonlinear Diffusion, in which the nonlinearfeatures originate from a lower-order term with super-linear growth. This may createa mathematical difficulty in the form of blow-up , whereby a solution exists for a timeinterval 0 < t < T and then some norm of the solution goes to infinity as t → T (the blow-up time). In other cases the singular phenomenon is extinction (the solutionbecomes zero every where), or some other kind of singularity formation. • The Standard Blow-Up model:
It is also called the Fujita model (Kaplan, 1963;Fujita, 1966) u t = ∆ u + u p p > . Main feature: If p > (cid:107) u ( · , t ) (cid:107) ∞ of the solutions may go to infinity in finitetime. This depends on the domain and the initial data. For instance, if the space domainis R N and the initial function is constant, then blow-up in finite time always happens.Hint: Integrate the ODE u t = u p . However, when the data are distributed in space thendiffusion and reaction compete and the result is a priori uncertain. This is how a largeliterature arose. Thus, if the initial data are bell-shaped (like the Gaussian function),the domain is bounded and boundary conditions are zero Dirichlet, then small data willnot blow-up and large data will. For other configurations things depend on the exponent p : there exists a critical exponent p F called the Fujita exponent, such that all positivesolutions blow up if p ∈ (1 , p F ). See [129, 131, 163, 181].A number of beautiful blow-up patterns emerge in such evolutions. Galaktionov and theauthor have constructed in [130] a particular one, called the peaking solution , that blowsup in finite time T at a single point x and then continues for later time as a boundedsmooth solution, a clear example of the curious phenomenon called continuation afterblow-up. However, the most common situation in reaction-diffusion systems of thisdiffusive type is complete blow-up at time T with no possible continuation (for instance,the numerical approximation goes to infinity everywhere for t > T ). The intricatephenomenon of bubbling is studied by M. del Pino in another course of this volume[107].As an extension of this elementary reaction-diffusion blow-up model there have beenstudies for many equations of the general form u t = A ( u ) + f ( u, Du )where A is a linear or nonlinear diffusion operator, maybe of porous medium or p -Laplacian type. The studies also include systems. Some of them are systems of mixedtype, one of the most popular ones is the chemotaxis system, where blow-up has a veryinteresting form that is still partially understood, [149].19 The Fisher-KPP model and traveling waves:
The problem goes back to Kol-mogorov, Petrovskii and Piskunov, see [177], that present the most simple reaction-diffusion equation concerning the concentration u of a single substance in one spatialdimension,(3.3) ∂ t u = Du xx + f ( u ) , with an f that is positive between two zero levels f (0) = f (1) = 0. We assume that D > f ( u ) = u (1 − u ) yields Fisher’s equation [123] that wasoriginally used to describe the spreading of biological populations. The celebrated resultsays that the long-time behavior of any solution of (3.3), with suitable data 0 ≤ u ( x ) ≤ N ≥
1, the problem becomes(3.4) u t − ∆ u = f ( u ) in (0 , + ∞ ) × R N , This case has been studied by Aronson and Weinberger in [19, 20], where they prove thefollowing result.
Theorem.
Let u be a solution of (3.4) with u (cid:54) = 0 compactly supported in R N andsatisfying ≤ u ( x ) ≤ . Let c ∗ = 2 (cid:112) f (cid:48) (0) . Then,1. if c > c ∗ , then u ( x, t ) → uniformly in {| x | ≥ ct } as t → ∞ .2. if c < c ∗ , then u ( x, t ) → uniformly in {| x | ≤ ct } as t → ∞ . In addition, problem (3.4) admits planar traveling wave solutions connecting 0 and 1,that is, solutions of the form u ( x, t ) = φ ( x · e + ct ) with − φ (cid:48)(cid:48) + cφ (cid:48) = f ( φ ) in R , φ ( −∞ ) = 0 , φ (+ ∞ ) = 1 . This asymptotic traveling-wave behavior has been generalized in many interesting ways,in particular in nonlinear diffusion of PME or p -Laplacian type, [106], [105], [22]. De-parting from these results, King and McCabe examined in [173] a case of fast diffusion,namely u t = ∆ u m + u (1 − u ) , x ∈ R N , t > , where ( N − + /N < m <
1, and showed that the problem does not admit travelingwave solutions and the long time behaviour is quite different. We will return to thisquestion when dealing with fractional nonlinear diffusion in the work [223], in Section 7. • In the last decades many other models and variants of diffusive systems have beenproposed, in particular in the form of systems, like the various cross-diffusion systems [158]. Cross-diffusion gives rise to instabilities that attract much attention in populationdynamics, since they allow to predict important features in the study of the spatial dis-tribution of species. The seminal work in this field is due to Alan Turing [234]. In orderto understand the appearance of certain patterns in nature with mathematical regulari-ties like the Fibonacci numbers and the golden ratio, he proposed a model consisting ofa system of reaction-diffusion equations. 20
Blow-up problems have appeared in related disciplines and some of them have attractedin recent times the attention of researchers for their difficulty and relevance. We presenttwo cases, a case still requiring more work, a case enjoying big success. The combinationof diffusion with nonlinear reaction is in both cases very intricate and leading to thedeepest mathematics.
The fluid flow models:
The
Navier-Stokes or Euler equation systems for incom-pressible flow. The nonlinearity is quadratic and affects first order terms. Progress isstill partial. There is also much work on the related topic of geostrophic flows. We willnot enter into more details of such a relevant topic that has a different flavor.
The geometrical models:
The
Ricci flow describes the motion of the metric tensorof a Riemannian manifold by means of the Ricci matrix: ∂ t g ij = − R ij . This is anonlinear reaction-diffusion system, even if this information is not clear in the succintformula. Posed in the form of PDEs by R. Hamilton, 1982, it has become a ClayMillenium Problem. Its solution by G. Perelman in 2003 was one of biggest successstories of Mathematics in the 21st century, see [88, 193]. One of the main points inthe proof is the study of the modes of blow-up of this system. In order to see that theevolution system of the Ricci flow is a type of nonlinear diffusion, it is convenient torecall the much simpler case of two-manifolds, since in that case it reduces to a type offast diffusion called logarithmic diffusion, see below.Let us finally mention the equations of movement by curvature to the list of geometricalmodels. Enormous progress has been made in that topic. Basic reading for this chapter:
On Nonlinear Diffusion: [109, 96, 244]. On freeboundaries [108, 127]. Moreover, [94, 93]. Fully nonlinear equations are form a vast topicthat we have not touched, see [59]. PME: degenerate diffusion and free boundaries
A very simple model of nonlinear diffusion in divergence form is obtained by means ofthe equation(4.1) u t = ∇ · ( D ( u ) ∇ u )where D ( u ) is a diffusion coefficient that depends on the ‘concentration variable’ u . Strictparabolicity requires that D ( u ) >
0, and the condition can be relaxed to degenerateparabolicity if we make sure that D ( u ) ≥
0. Now, if we further assume that D ( u ) is apower function, we get the simplest model of nonlinear diffusion equation in the form(4.2) u t = ∇ · ( c | u | γ ∇ u ) = c ∆( | u | m − u ) . with m = 1 + γ and c , c >
0. Exponent m , in principle positive, will play an importantrole in the model, but the constants c i are inessential, we may put for instance c = 1, c = m . The concentration-dependent diffusivity is then D ( u ) = m | u | m − .
21n many of the applications u is a density or concentration, hence essentially nonnegative,and then we may write the equation in the simpler form(4.3) u t = ∆( u m ) , that is usually found in the literature (we have dispensed with useless constants). Butthere are applications in which u is for instance a height that could take negative values,and then version (4.2) is needed, since otherwise D ( u ) would not be positive and theequation would not be parabolic.The equation has enjoyed a certain popularity as a mathematical model for degeneratenonlinear diffusion, combining interesting and varied applications with a rich mathe-matical theory. The theory has many interesting aspects, like functional analysis in theexistence and uniqueness theory, and geometry in the study of the free boundaries, aswell as deep novelties in the long time asymptotics. Our monograph [244] gathers a largepart of the existing theory up to the time of publication (2007). We will devote this sec-tion to review some of the main topics that affect the theory of fractional porous mediummodels of later nonlocal sections, and we will also present the very recent sharp resultson the regularity and asymptotic behaviour of free boundaries, obtained in collaborationwith Kienzler and Koch in [167]. As we have already said, the value of exponent m is an important part of the model.Clearly, if m = 1 we have D ( u ) = 1, and we recover the classical heat equation, u t =∆ u , with its well-known properties, like the maximum principle, the C ∞ regularity ofsolutions, and the infinite speed of propagation of positive disturbances into the wholespace, as well as the asymptotic convergence to a Gaussian profile for suitable classes ofinitial data. • The first interesting nonlinear case is m > D ( u ) degenerates at the level u = 0. This brings as a consequence the existence of weak solutions that have compactsupport in the space variable for all times, though that support expands. We refer tothat situation as Slow Diffusion . As a consequence, free boundaries arise and a wholegeometric theory is needed. All this is in sharp contrast with the heat equation.The differences with the heat equation can be seen by means of an easy calculationfor m = 2. In that case, and under the assumption that u ≥
0, the equation can bere-written as 12 u t = u ∆ u + |∇ u | , and we can immediately see that for values u (cid:29) u ∼ u t = |∇ u | . This last equation is not parabolic , but hyperbolic , with propagation along characteris-tics. The PME equation is therefore of mixed type near the critical value u = 0 whereit degenerates, and it has therefore mixed properties.22he calculation may look very particular, for a specific value of m . But to the initialsurprise of researchers, it extends to all value m >
1, of the slow diffusion range. The pressure transformation v = cu m − allows us to get an equivalent equation for v :(4.4) v t = ( m − v ∆ v + |∇ v | , where we have used the standard normalization c = m/ ( m − m >
1, see [244]. Indeed, it can be provedthat in some weak sense the eikonal equation holds on the free boundary { u = 0 } , andthis implies that the support of the solution spreads with time, another property thatcan be rigorously proved. • The pressure transformation is even more general, and can be applied to the filtrationequation u t = ∆Φ( u ) . If we put v = (cid:82) u (Φ (cid:48) ( s ) /s ) ds , then we can get the pressureequation(4.5) v t = σ ( v )∆ v + |∇ v | , where the function σ ( v ) = Φ (cid:48) ( u ) ≥
0, cf. [56]. • These pressure considerations apply under the assumption that u ≥
0, which isphysically natural for most applications. It must be pointed out the existence anduniqueness theory has been done for signed solutions, according to the generality that issuitable in Functional Analysis. However, many of the estimates on which the qualitativetheory is based do not apply for general signed solutions, and we will forsake them andassume u ≥ • When m < u = 0 in the sense that D ( u ) → ∞ .This range is called Fast Diffusion . We will return to that case in the next Section 5since its properties show a remarkable difference with the PME range m > This application is maybe the best known and has played a role in developing the theoryfor the PME, a clear example of the influence of physics on the mathematics. Accordingto Leibenzon (1930) and Muskat (1933), the flow of gas in a porous medium (they werethinking of the petroleum industry) obeys the laws (cid:40) ρ t + div ( ρ V ) = 0 , V = − kµ ∇ p, p = p ( ρ ) , where ρ is density, p is the averaged pressure and V is the seepage velocity. The firstline is the usual continuity equation from fluid mechanics, and the second line left isthe Darcy law for flows in porous media (Darcy, 1856). Therefore, these porous mediaflows are potential flows due to averaging of Navier-Stokes on the pore scales. We needa precise closure relation which is given by a gas law of the form p = p o ρ γ , with value of23he exponent γ = 1 (isothermal gas) or γ > ρ t = div ( kµ ρ ∇ p ) = div ( kµ ρ ∇ ( p o ρ γ )) = c ∆ ρ γ +1 . In order to get the PME we put u = ρ , m = 1 + γ (which happens to be equal or largerthan 2) and we eliminate useless constants. We point out that the pressure is then p = p o u m − , just the variable that we called v in formula (4.4). No wonder that this equation isimportant. As for the local flow velocity we have V = − c ∇ v in our mathematicalnotation.- There are many other applications, as described in the book [244]: underground waterinfiltration (Boussinesq, 1903) with m = 2, plasma radiation with m ≥
4, (Zeldovich-Raizer, around 1950), spreading of populations (self-avoiding diffusion) m ∼ m = 4, and so on. The way the nonlinear theory of the PME has developed is quite different from the waythe linear heat equation is studied. Indeed, in the early years there were attempts toconstruct a perturbation theory putting m = 1+ ε in (4.3) and then perturbing the linearmodel, but the singular perturbation analysis was not successful. Fortunately, around1958 when the theory started the serious development in Moscow [204], the tools ofnonlinear functional analysis were ready, and in particular the concept of weak solutionand the role of a priori estimates.Here are the main topics of mathematical analysis (1958-2016):- The precise meaning of solution. Since it was realized that classical solutions do notexist if there are free boundaries.- The nonlinear approach: estimates; functional spaces.- Existence of suitable solutions (like weak solutions). Uniqueness. Further in thetheory, variant of the equation showed cases of non-existence or non-uniqueness.- Regularity of solutions: Are weak solutions indeed continuous functions? are they C k for some k ? which is the optimal k ?- Existence, regularity and movement of interfaces: are they C k for some k ?- Asymptotic behaviour: is there something comparable to the Gaussian profile as auniversal attractor? This is a question of emerging patterns . If there is convergence to apattern we want to know that rate of convergence. We also want to know how universalthat convergence is, in other words the basin of attraction of the asymptotic pattern.- Comparison with other approaches like the probabilistic approach. Interesting newtools appear, like Wasserstein metrics and estimates.24he beauty of this plan is that it can be used mutatis mutandis on a huge numberof related models: fast diffusion models, inhomogeneous media, anisotropic media, p -Laplacian models, applications to geometry or image processing; equations involvingeffects, like the chemotaxis models,... These profiles are the alternative to the Gaussian profiles of the linear diffusion case.They are source solutions.
Source means that u ( x, t ) → M δ ( x ) as t →
0. Explicitformulas exist for them (1950):(4.6) B ( x, t ; M ) = t − α F ( x/t β ) , F ( ξ ) = (cid:0) C − kξ (cid:1) / ( m − α = n N ( m − β = N ( m − < / u = Ct − α Free boundary is located at distance | x | = ct β where C > k = k ( m, N ). Since Fourier analysis is not a way forfind them, new ideas are needed. We observe that the solution obeys a scaling symmetry,it is self-similar. In other words, it is invariant under suitable scaling in space and time.This fact is the key to finding the expression, see [244], page 63. An important propertyfor the applications is that (4.6) breaks with the Brownian space-time law: | x | = ct / ,so that it can be classified as anomalous diffusion .If you look for the mathematical properties, we find a surprise with regularity. Put m = 2 for simplicity. B ( x, t ; M ) does not satisfy the equation in a classical sense since u is not even C continuous in space or time. The validity of this physical solution wasa hot problem when it was discovered around 1950. Hence, there is a problem with the concept of solution that will satisfy the mathematicalrequirements (existence and uniqueness for a reasonable class of data, plus stabilityestimates) as well as the physical requirements (to reflect the behaviour that is expectedfrom the evidence obtained in the applications). This problem did not exist for the mainexample of diffusion, the Heat Equation, since classical solutions could be found.Many concepts of generalized solution have been used in developing the mathematicaltheory of the PME, and also in many related equations, not only in the parabolic theory: • Classical solution.
This is the most desirable option, and indeed it happens for non-degenerate situations, u >
0. But it cannot be expected if the Barenblatt solutions areto be included. 25
Limit solution.
This is the practical or computational remedy. To replace the equationby approximated problems with good physical or computational properties and then topass to the limit. The catch is that the approximation may not converge, or we couldbe unable to prove it; even if the approximations do converge, the limit may dependon the approximation. Spurious solutions may appear when the approximation is notefficient, a quality difficult to tell a priori. On the positive side, limit solutions have beensuccessfully used in the diffusive literature with the names of minimal solutions, maximalsolutions, SOLAs (solutions obtained as limits of approximations), proper solutions, ... • Weak solution.
This was a very good solution to the problem of building a theoryfor the PME. The idea is to test the equation against a full set of smooth functions andto eliminate all or most of the derivatives prescribed by the equation on the unknownfunction. It was first implemented on the PME by O. Oleinik and collaborators [204](1958). The simplest weak version reads (cid:90) (cid:90) ( u η t − ∇ u m · ∇ η ) dxdt + (cid:90) u ( x ) η ( x, dx = 0 , while there is a second version, the very weak solution , (cid:90) (cid:90) ( u η t + u m ∆ η ) dxdt + (cid:90) u ( x ) η ( x, dx = 0 . This version is more relaxed than the first. In both cases functional spaces have to bechosen for the solutions to belong to so that the integrals in the formulation make senseand existence and uniqueness can be proved.Once existence and uniqueness of a weak solution was proved for suitable initial data;that it was verified that all classical solutions are weak; and that the Barenblatt solutionsare indeed solutions for t > • Better regularity. Strong solution.
The previous paragraph solves the problem ofthe correct setting in principle. But researchers want to have solutions that have goodproperties so that we can do calculus with them. Fortunately, weak solutions of the PMEare better than weak, they are strong. In this context it means that all weak derivativesentering the original equation are L p functions for some p . • The search for an abstract method to solve a large number of evolution problems ofdiffusive type has led to a functional approach called mild or semigroup solution , thatwe discuss below. • Solutions of more complicated diffusion-convection equations have motivated newconcepts that can be translated to the PME:-
Viscosity solution . Two different ideas: (1) add artificial viscosity and pass to the limit;(2) viscosity concept of Crandall-Evans-Lions (1984); adapted to PME by Caffarelli-V´azquez (1999).-
Entropy solution (Kruzhkov, 1968). Invented for conservation laws; it identifies uniquephysical solution from spurious weak solutions. It is useful for general models withdegenerate diffusion plus convection. 26
Renormalized solutions (by Di Perna - P. L. Lions),
BV solutions (by Volpert-Hudjaev),
Kinetic solutions (by Perthame,...).
Functional Analysis is a power tool for the expert in PDEs, and when used wisely itproduces amazing result. Thus, when faced with the task of solving evolution equationsof the type(4.7) u t + Au = 0 , where A is a certain operator between function spaces, we may think about discretizingthe evolution in time by using a mesh t = 0 < t , ...t K = T and posing the implicitproblems u ( t k ) − u ( t k − ) h k + A ( u ( t k )) = 0 , h k = t k − t k − . In other words, we want to find a discrete approximate solution u = { u k } k such that(4.8) hA ( u k ) + u k = u k − , where we have used equal time spacing h k = h > h , so that we should write u ( h ) = { u ( h ) k } k .This step is called Implicit Time Discretization,
ITD. We start the iteration by assigningthe initial value u ( h )0 = u h , where u h is the given initial data or an approximationthereof. Parabolic to Elliptic . The success of ITD depends on solving the iterated equations(4.8) in an iterative way. In fact, the iteration has always the same format(4.9) hA ( u ) + u = f , since f = u k − is the value calculated in the previous step. When this is used for thefiltration equation u t − ∆Φ( u ) = 0 , we get the stationary equation(4.10) − h ∆Φ( u ) + u = f , and the question reduces to solve for u if f is known. An easy change of variables v = Φ( u ), u = β ( v ) leads to(4.11) − h ∆ v + β ( v ) = f. This is the semilinear elliptic problem that we must solve. We have reduced the theoryof a (possibly nonlinear) parabolic problem to an elliptic problem with a specific form.
Accretive operators. Semigroup generation.
The rest of the story depends onthe theory of accretive operators. If A is an m -accretive map in a Banach space X withdensely defined domain, then the famous Crandall-Liggett Theorem [92] ensures not onlyexistence of the solution of the iterated problems, but also that as h → u ( h ) converge to a function u ( t ) ∈ C ([0 , ∞ ) : X ) that solves the evolution27roblem in a sense called mild sense . The solution is often termed the semigroup solution .Moreover, the set of solutions forms a semigroup of contractions in X .But is this mild solution a solution in some more usual sense? In the case of the PMEit is proved that the operator given by A ( u ) = − ∆( u m ) is m -accretive in the space X = L ( R N ) when properly defined, [27], and also that the mild solution is a weaksolution. We have explained the method in some detail in Chapter 10 of [244]. The next step in the theory is proving that under mild conditions on the data weaksolutions of the PME are indeed continuous, and the free boundary is quite often aregular hypersurface in space-time, or in space for every fixed time. We recall that weare working with nonnegative solutions. We we also dealing with the Cauchy problemin the whole space to save effort and concentrate on the basics, but many results holdfor locally defined solutions. • The regularity theory for solutions relies on the existence of a rather miraculous apriori estimate, called the Aronson-B´enilan estimate [13], that reads:∆ v ≥ − C/t, where v = cu m − is the pressure and C = ( n ( m −
1) + 2) − . Nonnegative solutions withdata in any L p space are then proved to be bounded for positive times. A major stepwas then done by Caffarelli and Friedman (1982) when they proved C α regularity: thereis an α ∈ (0 ,
1) such that a bounded solution defined in a cube is C α continuous. Thisholds in all space dimensions. • What happens to the free boundary? It was soon proved that free boundaries maybe stationary for a while but eventually they must move to fill the whole domain as timepasses. The movement is expansive, the positivity set keeps expanding in time and neverrecedes. Caffarelli and Friedman proved subsequently that if there is an interface Γ, itis also a C α continuous set in space time (properly defined). • How far can you go? The situation is understood in 1D. On the one hand, freeboundaries can be stationary for a time (metastable) if the initial profile is quadraticnear ∂ Ω: v ( x ) = O ( d ), d being distance to the zero set. This time with lack ofmovement is called a waiting time. It was precisely characterized by the author in 1983;it is visually interesting in the experiments with thin films spreading on a table. Inpaper [14] we proved that metastable interfaces in 1D may start to move abruptly afterthe waiting time. This was called a corner point . It implies that the conjecture of C regularity for free boundaries in 1D was false. But in 1D the problems with regularitystop here: 1D free boundaries are strictly moving and C ∞ smooth after the possiblecorner points. See [244] for full details. The situation is more difficult for free boundary behaviour in several space dimensions,and the investigation is still going on. 28 regular free boundary in N-D • Caffarelli, Wolanski, and the author proved in 1987 that if u has compact support,then after some time T > C ,α , and the pressure is also C ,α in alateral sense [73], [74]. Note the lateral regularity is the only option since the Barenblattsolutions are an example of solutions that exhibit a smooth profile that is broken at theFB. The general idea, taken from 1D, is that when the FB moves, the adjoining profileis always a broken profile, since the support of the solution moves forward only if thegradient of the pressure is nonzero (Darcy’s law). • In his excellent doctoral thesis (1997), Koch proved that if u is compactly supportedand transversal then the free boundary is C ∞ after some finite time and the pressureis “laterally” C ∞ . This solved the problem of optimal regularity in many cases, thoughnot all. • The free boundary for a solution with a hole in 2D, 3D is the physical situation inwhich optimal regularity can be tested. Indeed, as the flow proceeds and the hole shrinks,it is observed that the part of the motion of the free boundary surrounding the holeaccelerates, so that at the point and time where the hole disappears (this phenomenonis called focusing), the advance speed becomes infinite. The applied setup is a viscousfluid on a table occupying an annulus of radii r and r . As time passes r ( t ) grows outwhile r ( t ) goes to the origin. In a finite time T the hole disappears. The flow can beregular for t < T but it was suspected from the numerical evidence that it was not atthe focusing time t = T .To prove this fact, a self-similar solution was constructed displaying the focusing be-haviour. It has the form U f ( x, t ) = ( T − t ) α F ( x/ ( T − t ) β ) . with ( m − α = 2 β + 1. The profile is such that F ( ξ ) ∼ | ξ | γ near ξ = 0, with γ = α/β .There is one free parameter, let us say β , that is not known a priori. We find a veryinteresting mathematical novelty, an anomalous exponent , or similarity of the secondkind in the terminology popularized by Barenblatt [23]. The problem was solved by ODEanalysis in 1993 by Aronson and Graveleau [16], and then further investigated by Aronsonand collaborators, like [9, 18]. It is proved that γ < v ∼ ∇ u m − blows-up at the focusing time t = T at x = 0. Moreover, the limit profile U f ( x, T ) is notLipschitz continuous at x = 0, it is only C γ continuous. This is a counterexample to thehypotheses of higher regularity of heat equation and similar diffusive flows: a degenerateequation like PME has limited regularity for nonnegative solutions with moving freeboundaries. Such a phenomenon is known to happen in the other typical evolution freeboundary problems. Stefan, Hele-Shaw and p -Laplacian equation. For the latter see [15].Summing up, higher regularity for PME flows has an obstacle. We may hope to prove29igher regularity if we avoid it, like the situation of compactly supported solutions forlarge times. The question remained for many years to know if we can prove regularity of the solutionsand their free boundaries under some certain geometrical condition on the solution orthe data less stringent than the conditions of compact support and initial transversalityof papers [73], [74], [176]. • Much progress was done recently in paper with H. Koch and C. Kienzler [167], precededby [166]. Here is the main theorem proved in [167] about regularity of solutions that arelocally small perturbations from a flat profile.
Theorem 4.1.
There exists δ > such that the following holds:If u is a nonnegative δ -flat solution of the PME at (0 , on scale with δ -approximatedirection e n and δ -approximate speed , and δ ≤ δ , then for all derivatives we haveuniform estimates (4.12) | ∂ tk ∂ xα ∇ x ( u m − − ( x n + t )) | ≤ Cδ at all points ( t, x ) ∈ ([ − / , × B / (0)) ∩ P ( u ) with C = C ( N, m, k, α ) > . Inparticular, ρ m − is smooth up to the boundary of the support in ( − , × B / , and (4.13) |∇ x u m − − e n | , | ∂ t u m − − | ≤ Cδ .
Moreover, the level sets for positive values of u and the free boundary are uniformlysmooth hypersurfaces inside ( − , × B (0) . The technical assumption is being δ -flat, which means being very close to a flat travellingwave (the special solution that serves as model) in a certain space-time neighbourhood.See Definition 1 in the paper. The size of this neighbourhood is taken to be unit, butthis is not a restriction by the scale invariance of the equation. The very detailed formof the estimates allows us then to derive very strong results for large times, that we willexplain in next subsection. • Theorem 4.1 implies the eventual C ∞ -regularity result for global solutions that wewere looking for. The following result is Theorem 2 of [167]. We use the notation R B ( t ) = c ( N, m ) M ( m − λ t λ , with λ = 1 / ( N ( m −
1) + 2), for the Barenblatt radius forthe solution with mass M located at the origin. Theorem 4.2.
Let u ≥ be a solution of the PME posed for all x ∈ R N , N ≥ , and t > , and let the initial data u be nonnegative, bounded and compactly supported withmass M = (cid:82) u dx > . Then, there exists a time T r depending on u such that for all t > T r we have:(i) Regularity.
The pressure of the solution u m − is a C ∞ function inside the supportand is also smooth up to the free boundary, with ∇ u m − (cid:54) = 0 at the free boundary.Moreover, the free boundary function t = h ( x ) is C ∞ in the complement of the ball ofradius R ( T r ) . ii) Asymptotic approximation.
There exists c > such that t − Nλ (cid:16) a M m − λ − ct − λ − λ | x − x | t λ (cid:17) m − + ≤ u ( t, x ) ≤ t − Nλ (cid:16) a M m − λ + ct − λ − λ | x − x | t λ (cid:17) m − + (4.14) where x = M − (cid:82) xρ ( x ) dx is the conserved center of mass, and a is a certain constant.Moreover, (4.15) B R B ( t ) − ct − λ ( x ) ⊂ supp( u ( · , t )) ⊂ B R B ( t )+ ct − λ ( x )In this way we are able to solve the problem posed in 1987, and improved by Kochin 1997. We use delicate flatness conditions, scalings, heat semigroups and harmonicanalysis. We have eliminated the non-degeneracy condition on the initial data. Theestimates are uniform. The result cannot be improved in a number of directions. Besides,some more information is available: if the initial function is supported in the ball B R (0),then we can write the upper estimate of the regularization time as(4.16) T r = T ( N, m ) M − m R λ . By scaling and space displacement we can reduce the proof to the case M = 1 and x = 0. The fine asymptotic analysis uses also the results of Seis [214]. • Nonlinear Central Limit Theorem revisited
The last part of Theorem 4.2 refers to the way a general solution with compact supportapproaches the Barenblatt solution having the same mass. This kind of result is whatwe have called the PME version of the Central Limit Theorem.It was proved in due time that the standard porous medium flow has an asymptoticstabilization property that parallels the stabilization to the Gaussian profile embodied inthe classical Central Limit Theorem if we take as domain R N and data u ( x ) ∈ L ( R n ).The convergence result is(4.17) (cid:107) u ( t ) − B ( t ) (cid:107) → t → ∞ , as well as(4.18) t Nλ | u ( x, t ) − B ( x, t ) | → , uniformly in x ∈ R N . Here, B ( x, t ; M ) be the Barenblatt with the asymptotic mass M .Note that the factor t Nλ is just the normalization needed to work with relative errorssince B ( x, t ) decays like O ( t − Nλ ). Proofs are due to Kamin and Friedman [128] forcompactly supported solutions, and the author (2001) in full generality. The result isreported with full detail in [244] and explained in [243, 246].An improvement of the result to indicate a definite rate of convergence is due to Carrilloand Toscani (2000). It works for solutions with a finite second moment, (cid:82) u ( x ) | x | dx ,[79] and uses the powerful machinery of entropy methods, that become subsequentlyvery popular in studies of nonlinear diffusion.31he result that we obtain above points out to a finer error rate for compactly supportedsolutions, that can be written as(4.19) t N ( m − λ | u m − ( x, t ) − B m − ( x − x , t ; M ) | = O ( t − λ ) . Seis’ analysis and our paper show optimality of this rate. Note that 2 λ < Other problems.
There are numerous studies of the PME in other settings, likebounded domains with Dirichlet or Neumann conditions, PME with forcing term: u t = ∆ u m + f, PME with variable coefficients or weights, generalized filtration equation, PME withconvection and/or reaction, ...Many of the above results have counterparts for the p -Laplacian flow. Thus, stabiliza-tion to the p -Laplacian version of the Barenblatt solution is proved by Kamin and theauthor in [162]. There are many studies but no comparable fine analysis of the FB hasbeen done. Further reading for this chapter:
On the PME: [244, 243]. On asymptotic behaviour:[240] and [241]. About estimates and scaling: [242]. For entropy methods [11, 79, 78]. The Fast Diffusion Equation
We will consider now that range m < u = 0 in the sense that D ( u ) → ∞ . This range is called FastDiffusion Equation , FDE, and we also talk about singular diffusion . The new range wasfirst motivated by a number of applications to diffusive processes with fast propagation:plasma Physics (Okuda-Dawson law) [28], material diffusion (dopants in silicon) [170],geometrical flows (Ricci flow on surfaces and the Yamabe flow), diffusive limit of kineticequations, information theory, and others, see [242]. Once the mathematics started,it was seen that the FDE offers many interesting mathematical challenges, and someunexpected connections with other disciplines like Calculus of Variations.The common denominations slow and fast for the parameter ranges m > m < u ≈
0. But when large values of u are involved, thenames are confusing since the situation is reversed:- D ( u ) → ∞ as u → ∞ if m > D ( u ) → u → ∞ if m < m − v = cu m − with c = m/ (1 − m ) we geta new pressure equation of the form(5.1) v t = (1 − m ) v ∆ v − |∇ v | , so that the eikonal term is now an absorption term. Note that u → v → ∞ inthe FDE. 32 .1 Barenblatt solutions in the good range We have well-known explicit formulas for source-type self-similar solutions called Baren-blatt profiles, valid for with exponents m less than 1, but only if 1 > m > m ∗ = ( N − /N if N ≥ B ( x, t ; M ) = t − α F ( x/t β ) , F ( ξ ) = ( C + kξ ) − / (1 − m ) . The decay rate and spreading rate exponents are α = N − N (1 − m ) , β = 12 − N (1 − m ) > / . Both exponents α, β → ∞ as m goes down to m ∗ . So the question is what happens for m < m ∗ ? It is a long and complicated story, see a brief account further below.The decay of the Barenblatt FDE profile for fixed time is B = O ( | x | − / (1 − m ) ), a power-like decay that we have termed a fat tail in terms of probability distributions. Theexponent ranges from N to ∞ in the range m ∗ < m <
1. Note that for m ∗ < m < ( N − / ( N + 1) the distribution B ( · , t ) does not even have a first moment.The exponent range m c < m < < of the FDE where the Barenblatt solutions exist iscalled the “good fast diffusion range”, since it has quite nice properties; though differentfrom the linear heat equation, they nevertheless quite satisfying from many points ofview, in particular from the point of view of existence, functional analysis, regularityand asymptotic behaviour. Thus, existence of a classical C ∞ smooth solution is guaran-teed for every nonnegative, locally bounded Radon measure as initial data (no growthconditions like the HE or the PME), the solution is unique, it is also positive everywhereeven if the data are not (they must be nontrivial), decay in time of the solutions dependsin a predictable way from suitable norms of the data. Even unbounded Borel measurescan be taken, see [84].In particular, when u ∈ L ( R N ), a semigroup of contractions if generated, conservationof mass holds, and the solution converges for large time to the Barenblatt solution givenabove, and with the same expression of the Central Limit Theorem that we saw for thePME. The impressive Aronson-B´enilan estimate is now a two-sided universal estimate: − C u/t ≤ u t ( x, t ) ≤ C u/t , which implies better and easier estimates for the rest ofthe theory. Of course, the absence of free boundaries makes it lose part of its power ofattraction. We will not enter into the proofs of these results, than can be found in theliterature, [96, 109], [242],... 33 .2 Comparison of anomalous diffusions The type of diffusion described by the Barenblatt solutions is called anomalous diffusion since it breaks the Brownian spread rate | x | ∼ t / and space decay with an exponentialrate. We have already seen that anomalous behaviour in the linear setting, as thefundamental solution P t ( x ) of the Fractional Heat Equation (FHE) u t + ( − ∆) s u = 0, cf.(2.11). In that case the spreading rate is | x | ∼ t / s . This leads to a formal (and partial)equivalence between anomalous diffusion of FHE and FDE types based on the spreadstrength. It reads 2 s ∼ − N (1 − m ), hence N (1 − m ) ∼ − s ) , which agrees for the classical heat equation: m = 1, s = 1. For the best known case s = 1 / N = ( N − /N , a well-known exponent.For the fundamental solution of the fractional Laplacian diffusion, the decay rate forfixed time is P t ( x ) = O ( | x | − ( N +2 s ) ). This gives the formal equivalence between spatialdecay rates: N + 2 s ∼ / (1 − m ). It does not agree with the heat equation value in thelimit: m = 1, s = 1.One may wonder if we can get complete agreement of exponents and profile functions.This happens in dimension N = 1 for s = 1 / m = 0, a very exceptional case. Theprofile function is the Cauchy distribution(5.2) P ( x ) = 1 π tt + | x | . The situation becomes much more involved once we cross the value m c = ( N − /N for N ≥
3. In the range m c > m > u ∈ L Ploc ( R N ) then we need p > N (1 − m ) / u ∈ L ( R N ),conservation of mass never holds, and in fact such solutions disappear in finite time, aphenomenon called extinction, that is discussed at length in our [242]. Obtaining validversions of the Harnack inequality was challenging in this range [48, 49], see also themonograph [110]. • Logarithmic diffusion.
The limit m → u t = ∇ · ( u m − ∇ u ) = (1 /m )∆ u m . It is proved that the solutions u m ( x, t ) with fixed initial data, say bounded, converge as m → u t = ∇ · ( ∇ u/u ) = ∆ log( u ) , famous in 2D as a model for the evolution of the conformal matric by Ricci flow, asproposed by Hamilton [147] in 1988, where u is the conformal factor. A detailed study34f the surprising mathematical theory is done in [242], where references are given. Thefollowing facts are remarkable: finite mass solutions are not uniquely determined by theinitial data and moreover, they all lose at least 4 π units of mass (which here meanssurface) per unit time. A very beautiful solution happens when we choose surface lossequal to 8 π and the formula is(5.5) U ( x, t ) = 8 a ( T − t )( a + | x | ) , with a > . In the geometrical interpretation it describes the shrinking of a perfect 2D ball to a pointin time
T >
0. The ball is represented on the plane by stereographical projection.Thesolution qualifies as another beautiful diffusive pattern, this time it portrays extinction byRicci flow. We ask the reader to note the difference with the Barenblatt FDE solutions,or with the Cauchy distribution (5.2).Note that there is another natural limit as m →
0, namely the equation u t = ∆ sign( u ).Though it means no flow for positive data, it has an interesting interpretation in termsof total variation flow for signed data, see [40] in 1D and compare with [209]. Totalvariation flow is an very important subject in itself, related to the p -Laplacian, cf. themonograph [7]. • Super-fast diffusion.
Once we cover m = 0 the natural question is, can we cover m <
0. Actually, formula (5.3) makes perfect sense and a theory can be developed thatextends much of what we have seen in the subcritical case 0 < m < m c . Some surprisesarise in the form of nonexistence for integrable data (which seem in principle the mostnatural), see [239]. The range is called very singular diffusion of super-fast diffusion. • Subcritical asymptotic stabilization.
The absence of Barenblatt solutions makesone wonder what happens for large times in the subcritical, logarithmic and very singularcases. This is a complicated topic, that needs lots of mathematics. We refer to [171] and[242] for the earlier extinction analysis for so-called small solutions, and to [39], [36] forstabilization of solution with certain fat tails to the self-similar solutions called pseudo-Barenblatt solutions. The proofs are based on entropy-entropy dissipation methods.References to abundant related work are found.
The subject FDE in the lower m ranges is quite rich. Part of the very interesting resultsconcern problems posed in bounded domains, where the discussion is quite different.However, lack of space leads us not to continue the study of the Fast Diffusion range,and we refer to reader to monographs like [96, 242]. But let us just point out that thereis no unity in the mathematics of the Fast Diffusion comparable to the Porous Mediumrange, and a number of critical exponents m < p -Laplacian equation (PLE), though some remarkabledifferences exist. We refer to the book [242] for an account of our ideas. There is evena transformation that maps all radial solutions of the PME to the corresponding class35f the PLE, see [153]. Of course, p = m + 1 by dimensional considerations, but thetransformation changes also the space dimension. If N >
2, then the corresponding PLEdimension is N (cid:48) = ( N − m + 1) / m .Some studies deal with the Doubly Nonlinear Equation u t = ∇ · ( | D ( u m ) | p − D ( u m )).See [228] for a recent work. Nonlinear fractional diffusion. Potential model
The combination of fractional diffusion and porous medium nonlinearities gives rise tointeresting mathematical models that have been studied in the last decade both becauseof a number of scientific applications and for their mathematical properties. Two mainmodels will be discussed below; a mechanical model has been developed in collaborationwith Luis Caffarelli in Texas, and can be called porous medium flow with fractional po-tential pressure (or more generally, with nonlocal pressure); it has surprising properties.The other one has been developed later but it has better analytical properties. For con-venience we will call them here
PMFP and
FPME . We will also examine models thatinterpolate between both.
We devote this section to introduce model
PMFP . It arises from the consideration ofa continuum, say, a fluid, represented by a density distribution u ( x, t ) ≥ velocity field v ( x , t ), according to the continuity equation u t + ∇ · ( u v ) = 0 . We assume next that v derives from a potential, v = −∇ p , as happens in fluids inporous media according to Darcy’s law, and in that case p is the pressure. But potentialvelocity fields are found in many other instances, like Hele-Shaw cells, and other recentexamples.We still need a closure relation to relate p to u . In the case of gases in porous media,as modeled by Leibenzon and Muskat, the closure relation takes the form of a state law p = f ( u ), where f is a nondecreasing scalar function, which is linear when the flow isisothermal, and a power of u if it is adiabatic. The PME follows. The linear relationshiphappens also in the simplified description of water infiltration in an almost horizontalsoil layer according to Boussinesq’s modelling. In that case we get the standard porousmedium equation, u t = c ∆( u ). See Section 4 on the PME or [244] for these and manyother applications.The diffusion model with nonlocal effects proposed in 2007 with Luis Caffarelli uses thefirst steps of the derivation of the PME, but it differs by using a closure relation of theform p = K ( u ) , where K is a linear integral operator, which we assume in practice tobe the inverse of a fractional Laplacian. Hence, p es related to u through a fractionalpotential operator, K = ( − ∆) − s , 0 < s < , with kernel k ( x, y ) = c | x − y | − ( n − s ) In [246] they were called Type I and Type II in reverse order. − ∆) s p = u . This introduces long-distance effects inthe model through the pressure, and we end up with a nonlocal model, given by thesystem(6.1) u t = ∇ · ( u ∇ p ) , p = K ( u ) where u is a function of the variables ( x, t ) to be thought of as a density or concentration,and therefore nonnegative, while p is the nonlocal pressure, which is related to u via alinear operator K . We can write: u t = ∇ · ( u ∇ ( − ∆) − s u ). A technical observation: thereare problems in defining ( − ∆) − s u in 1D since the kernel may be too singular, but then ∇ ( − ∆) − s u is always well defined, which is enough to perform the calculations that willbe commented upon below.The problem is posed for x ∈ R N , n ≥
1, and t >
0, and we give initial conditions(6.2) u ( x,
0) = u ( x ) , x ∈ R N , where u is a nonnegative, bounded and integrable function in R N . • Particle systems with long range interactions . Equations of the more generalform u t = ∇ · ( σ ( u ) ∇L u )have appeared in a number of applications to the macroscopic evolution of particle sys-tems. Thus, Giacomin and Lebowitz [132], 1997, consider a lattice gas with generalshort-range interactions and a Kac potential, and passing to the limit, the macroscopicdensity profile ρ ( r, t ) satisfies the equation(6.3) ∂ρ∂t = ∇ · (cid:20) σ s ( ρ ) ∇ δF ( ρ ) δρ (cid:21) where σ s ( ρ ) may be degenerate. See also [133]. • Modeling dislocation dynamics as a continuum.
Following old modeling byA. K. Head in [148], Biler-Karch-Monneau [33] considered the one-dimensional case ofmodel (6.1). By integration in x they introduced viscosity solutions `a la Crandall-Evans-Lions. They prove that uniqueness holds, which is very satisfying property. Butthe corresponding mathematical model in several space dimensions looks quite differentfrom (6.1). • Hydrodynamic limit for s = 1 . This is a very interesting limit case. Putting s = 1 makes us lose the parabolic character of the flow that becomes hyperbolic. In 1Dthe situation is rather trivial since when we put p = ( − ∆) − u we get p xx = − u x , andthen u t = ( u p x ) x = u x p x − u Moreover, if v = − p x = (cid:82) u dx , we have v t = up x + c ( t ) = − v x v + c ( t ) , c = 0 this is the Burgers equation v t + vv x = 0, which generates shocks in finite timebut only if we allow for u to have two signs.In several dimensions the issue becomes much more interesting because it does notreduce to a simple Burgers equation. We have(6.4) u t = ∇ · ( u ∇ p ) = ∇ u · ∇ p − u , p = ( − ∆) − u , A very close version to this model has appeared in superconductivity (the Chapman-Rubinstein-Schatzman-E model) see [184, 114], and Ambrosio-Serfaty [6]. In that ap-plication u describes the vortex density. Gradient flow structure for this example isestablished in [5]. • The PME limit.
If we take s = 0, K = the identity operator, we get the standardporous medium equation, whose behaviour is well-known. Therefore we can see thePMFP equation as a nonlinear interpolation between the PME and the hydrodynamiclimit, s = 1. • More generally, it could be assumed that K is an operator of integral type definedby convolution on all of R n , with the assumptions that is positive and symmetric. Thefact the K is a homogeneous operator of degree 2 s , 0 < s <
1, will be important in theproofs. An interesting variant would be the Bessel kernel K = ( − ∆ + cI ) − s . We are notexploring such extensions. Early results on the PMFP have been reported in Proceedings from the Abel Symposium[245], and then in [246], so we will concentrate on general facts and only develop in moredetail some of the new material. For applications of nonlinear nonlocal diffusion see also[65]. • In paper [71] Luis Caffarelli and the author established the existence of weak energysolutions, the basic properties of the solutions, like conservation of mass(6.5) ddt (cid:90) u ( x, t ) dx = 0 , the two energy estimates(6.6) ddt (cid:90) u ( x, t ) log u ( x, t ) dx = − (cid:90) |∇ Hu | dx , where H = ( − ∆) − s/ , and(6.7) ddt (cid:90) | Hu ( x, t ) | dx = − (cid:90) u |∇ Ku | dx, K = ( − ∆) − s . A number of usual properties in diffusive processes do hold here like conservation ofpositivity, as well as L p decay. But we also found lack of a general comparison principle,a major difficulty in developing the theory (such a drawback will not be shared by thesecond model, FPME). And we could not prove uniqueness for general solutions in severalspace dimensions. 38 main goal in the study of this model was to determine whether or not the property offinite propagation holds. The answer turned out to be yes. This is not clear in principledue to the competition between the slow propagation of the PME part with the infinitepropagation of the fractional operator (amounting to long distance effects). The lack ofplain comparison made the proof difficult, and the difficulty was surmounted by a noveluse of the methods of viscosity solutions. Summing up, the degenerate character of thePME wins. On the contrary, infinite propagation was later proved to be true for FPME. • In a second contribution [72] we explored the long-time behaviour in two steps. Wefirst established the existence of self-similar profiles, so-called Fractional Barenblatt so-lutions U ( x, t ) = t − α F ( x t − β ) , β = 1 N + 2 − s , α = N β,
The profile F is compactly supported, a clue to the finite propagation property, and isthe solution of a certain fractional obstacle problem. A different proof in dimension 1follows from paper [33]. The authors of [31] found the self-similar Barenblatt profiles inall dimensions with explicit formulas: F ( x ) = ( A − B | x | ) − s + .Then we introduced the renormalized Fokker-Planck equation and used a suitable en-tropy functional and proved stabilization of general solutions to the previous profilesthat we called fractional Barenblatt profiles. All this is carefully explained in [245]. • The next issue in the programme was the regularity of the solutions. It was studiedin a paper with L. Caffarelli and F. Soria, [70]. Proving boundedness for solutions withintegrable data in L p , 1 ≤ p ≤ ∞ was an important step in the this theory. We candispense with the extension method for fractional Laplacians by using energy estimatesbased on the properties of the quadratic and bilinear forms associated to the fractionaloperator, and then the iteration technique. Theorem
Let u be a weak solution the initial-value problem for the PMFP with data u ∈ L ( R n ) ∩ L ∞ ( R n ) , as constructed before. Then, there exists a positive constant C such that for every t > x ∈ R n | u ( x, t ) | ≤ C t − α (cid:107) u (cid:107) γL ( R N ) with α = N/ ( N + 2 − s ) , γ = (2 − s ) / ( N + 2 − s ) . The constant C depends only on N and s . The major step is then proving C α regularity. The proof uses the DeGiorgi methodwith careful truncations together with very sophisticated energy methods that have toovercome the difficulties of both nonlinearity and nonlocality. A number of ideas comefrom Caffarelli-Vasseur [69] and [66] with difficult modifications due to the degeneratenonlinearity. The theory can be extended to data u ∈ L ( R N ), u ≥
0, giving globalexistence of bounded weak solutions.
The previous results are obtained in the framework of weak energy solutions: The basis ofthe boundedness analysis is a property that goes beyond the definition of weak solution.The general energy property is as follows: for any real smooth function F and such that39 = F (cid:48) is bounded and nonnegative, we have for every 0 ≤ t ≤ t ≤ T , (cid:82) F ( u ( t )) dx − (cid:82) F ( u ( t )) dx = − (cid:82) t t (cid:82) ∇ [ f ( u )] u ∇ p dx dt = − (cid:82) t t (cid:82) ∇ h ( u ) ∇ ( − ∆) − s u dx dt , where h is a function satisfying h (cid:48) ( u ) = u f (cid:48) ( u ). We can write the last integral in termsof a bilinear form (cid:90) ∇ h ( u ) ∇ ( − ∆) − s u dx = B s ( h ( u ) , u )This bilinear form B s is defined as B s ( v, w ) = C (cid:90) (cid:90) ∇ v ( x ) 1 | x − y | N − s ∇ w ( y ) dx dy = (cid:90) (cid:90) N − s ( x, y ) ∇ v ( x ) ∇ w ( y ) dx dy where N − s ( x, y ) = C | x − y | − ( N − s ) is the kernel of operator ( − ∆) − s . After some inte-grations by parts we also have(6.9) B s ( v, w ) = C n, − s (cid:90) (cid:90) ( v ( x ) − v ( y )) 1 | x − y | n +2(1 − s ) ( w ( x ) − w ( y )) dx dy since − ∆ N − s = N − s . It is well known that B s ( u, u ) is an equivalent norm for thefractional Sobolev space W − s, ( R N ). This is the way the fractional Sobolev spacesappear, as dissipated energies that will guarantee compactness in the arguments, see[70]. Fractional Sobolev spaces with a view to their use in PDEs are discussed in [111]. • The particular value s = 1 / distorted geometry . • The Hydrodynamic Limit s → U ( x, t ) = t − F ( x/t /N ) , and F is the characteristic function of a ball. Therefore, even continuity is lost in theregularity of the solutions. This is not a contradiction since the limit equation is nolonger parabolic. Our work is related to work on aggregation models by Bertozzi et al.[30]. • We posed the question of possible rates in the asymptotic convergence to selfsimilarsolutions of Barenblatt type of papers [72, 33]. This question was partially solved in a40aper [77] with J. A. Carrillo, Y. Huang and M. C. Santos, where we showed exponentialconvergence towards stationary states for the Porous Medium Equation with FractionalPressure in 1D. The many-dimensional case seems to be a difficult open problem, it istied to some functional inequalities that are not known. Our analytical approach doesnot seem to apply either. • The questions of uniqueness and comparison are solved in dimension N = 1 thanksto the trick of integration in space used by Biler, Karch, and Monneau [33]. New toolsare needed to make progress in several dimensions.Recent uniqueness results are due to Zhou, Xiao, and Chen, [261]. They obtain local intime strong solutions in Besov spaces. Thus, for initial data in B α , ∞ if 1 / ≤ s < α > N + 1 and N ≥
2. Therefore, Besov regularity implies uniqueness for small times. • The fractional Burgers connection was explored in [80] for N = 1, s = 1 / ∂ x ( − ∆) − / = − H , the Hilbert transform. • The study of the free boundary is in progress, but regularity is still open for small s > • The gradient flow structure of the
PMFM flow in Wasserstein metrics has beenrecently established by S. Lisini, E. Mainini and A. Segatti in [186]. For the generalapproach see the monograph [4]. Previous work in 1D was due to by J. A. Carrillo et al. • The problem in a bounded domain with Dirichlet or Neumann data has not beenstudied, to our knowledge. • Good numerical algorithms and studies are needed.
The interest in using fractional Laplacians in modeling diffusive processes has a wideliterature, especially when one wants to model long-range diffusive interactions, and thisinterest has been activated by the recent progress in the mathematical theory, in theform of a large number works on elliptic equations, mainly of the linear or semilineartype, as well as free boundary problems, like obstacle problems. There are so manyworks on the subject that we cannot refer them here. Let us mention the survey paper[65] by L. Caffarelli, that contains a discussion of the properties of solutions to severalnon-linear elliptic equations involving diffusive processes of non-local nature, includingreference to drifts and game theory. The FPME model and the mixed models
Another natural model for the combination of fractional diffusion and porous mediumnonlinearities is the equation that we will call fractional porous medium equation: ∂ t u +( − ∆) s ( u m ) = 0. In order to be mathematically precise we write the equation as(7.1) u t + ( − ∆) s ( | u | m − u ) = 041ith 0 < m < ∞ and 0 < s <
1. We will take initial data in u ∈ L ( R N ) unlessmention to the contrary. Normally, u , u ≥
0. We will refer to this model as
FPME foreasy reference in this paper. Mathematically, it looks a more direct generalization of thelinear fractional heat equation than the potential model
PMFP studied in the previoussection.This model represents another type of nonlinear interpolation with parameter s ∈ (0 , u t − ∆( | u | m − u ) = 0 for s = 1 and the plain absorptionODE u t + | u | m − u = 0 for s = 0.We have written a detailed description of this model in the survey paper [246], where wegive references to the physical motivations, among them [21, 155, 154, 156], the literature,and the mathematical developments until 2013 approximately. See also Appendix B of[49]. Therefore, we will mention the main items of the research, the references and generalideas, and then proceed to give notice of recent work, that covers different directions. A complete analysis of the Cauchy problem posed for x ∈ R N , t >
0, with initialdata in L ( R N ) was done in two very complete papers coauthored with A. de Pablo, F.Quir´os, and A. Rodr´ıguez: [102] in (2011) and [103] (2012). Using the Caffarelli-Silvestreextension method and the B´enilan-Brezis-Crandall functional semigroup approach, aweak energy solution is constructed, and u ∈ C ([0 , ∞ ) : L ( R N )). Moreover, the set ofsolutions forms a semigroup of ordered contractions in L ( R N ). This is the first instanceof a ‘better behaviour’ than model PMFP . • The second big difference is that
Nonnegative solutions have infinite speed of propa-gation for all m and s , so that there is no nonnegative solution with compact support (wemean, in the space variable). Actually, a very important property of Model PMFP withCaffarelli is that solutions with compactly supported initial data do have the compactsupport property (i.e., they stay compactly supported for all times).Even is propagation is always infinite in this model, we still use the name ‘fractionalFD’ for the range m < • On the other hand, some properties are similar in both models: Conservation ofmass holds for all m ≥
1, and even for some m < L − L ∞ smoothingeffect works; and the C α regularity holds also (unless m is near 0 and solutions are notbounded). Comparison of the models PMFP and
FPME is quite interesting and hasbeen pursued at all levels. • The question of existence of classical solutions and higher regularity for the FPMEand the more general model ∂ t u + ( − ∆) s Φ( u ) = 0(where Φ is a monotone real function with Φ (cid:48) ≥
0) has been studied in two paperswith the same authors (A.deP., F.Q., A.R., J.L.V.). The first paper, [104], treats themodel case Φ( u ) = log(1 + u ), which is interesting as a case of log-diffusion. The secondtreats general nonlinearities Φ and proves higher regularity for nonnegative solutions ofthis fractional porous medium equation, [252]. This is a very delicate result. There is42n extension of this result to prove C ∞ regularity to solutions in bounded domains byBonforte, Figalli and Ros-Ot´on, [41]. • Our paper [247] deals with the construction of what we call the fractional Barenblattsolution of the FPME, which has the also self-similar form:(7.2) U ( x, t ) = t − α F ( xt − β )The construction works for m > m c = ( N − s ) /N , a range that is optimal that remindsus of the Fast Diffusion Equation, Section 5. The difficulty is to find F as the solutionof an elliptic nonlinear equation of fractional type. Such profile is not explicit as in the PMFP model (it is only for some very special exponents [151]). In any case, F hasbehaviour like a power tail F ( r ) ∼ r − ( N +2 s ) . This is important for the applications and it the same as the one predicted by Blumentalfor the linear fractional kernel. This asymptotic spatial behaviour holds for all m ≥ m <
1, but not for some fast diffusion exponents m c < m < t → ∞ follows, and this Barenblatt pattern is proved to bean attractor, as we were expecting from what has been seen along this whole text. Theresult holds for m > m c . Open problem: Rates of convergence have not been found, andthis is an interesting open problem.Extinction in finite time is proved for exponents 0 < m < m c . The correspondingstabilization process must be studied. • Another direction concerns regularity at the local or global level. In collaborationwith M. Bonforte we have obtained a priori upper and lower estimates of intrinsic, localtype for this problem posed in R N , [50]. Quantitative positivity and Harnack Inequalitiesfollow. Against some prejudice due to the nonlocal character of the diffusion, we are ableto obtain them here for fractional PME/FDE using a technique of weighted integrals tocontrol the tails of the integrals in a uniform way. The novelty are the weighted functionalinequalities. This also leads to existence of solutions in weighted L -space for the fastdiffusion version FPME, a restriction that does not appear in the standard FDE.More recent, very interesting work on bounded domains is reported in Section 8. • Symmetrization (Schwarz and Steiner types). This is a project with B. Volzone[253, 254]. Applying usual symmetrization techniques is not easy and we have founda number of open problems. It turns out that Steiner symetrization works and it doesmuch better for fractional FDE than for the fractional PME range. This work wasfollowed by recent collaboration with Y. Sire and B. Volzone to apply the techniques tothe fractional Faber-Krahn inequality, [220]. • We have also investigated very degenerate nonlinearities, like the
Mesa Problem . Thisis the limit of FPME with m → ∞ . We have studied this limit in [248], and the limitflow characterized by the solution of a fractional obstacle problem, that is related to theobstacle problem for the PMFP that was described in [72]. • Numerics for the nonlinear nonlocal diffusion models is being done by a number ofauthors at this moment by : Nochetto et al. [199, 200], Teso [232].43
Fast diffusion and extinction. Very singular fast diffusion. Paper with Bonforte andSegatti [45], on non-existence due to instantaneous extinction, which is the common rulein very singular fractional fast diffusion as shown for standard diffusion in [239]. Paper[251] shows the existence of maximal solutions for some very singular nonlinear fractionaldiffusion equations in 1D in some borderline cases, this is an exception. • We have looked at the phenomenon of KPP propagation in linear and nonlinear frac-tional diffusion with the particular reaction proposed by Kolmogorov-Piskunov-Petrovskiiand Fisher (1938). In the case of standard linear diffusion travelling waves appear andserve as asymptotic attractors. Cabr´e and Roquejoffre [60, 62] studied the diffusionequation with linear fractional diffusion and KPP reaction and showed that there is notraveling wave propagation, and in fact the level sets move out at an exponential ratefor large times. The results are extended to nonlinear fractional diffusion of the FPMEtype for all values of the exponents in work with Stan, [223]. • The potential model
PMFP given in (6.1) is generalized into
PMFP’ (7.3) u t = ∇ · ( u m − ∇ ( − ∆) − s u )with m >
1. This is an extension that accepts a general exponent m , so that thecomparison of both models may take place on more equal terms.The most interesting question seems to be deciding if there is finite and infinite prop-agation for PMFP’ . Recent works with D. Stan and F. del Teso [224] and [226] showthat finite propagation is true for m ≥ m <
2. This isquite different from the standard porous medium case s = 0, where m = 1 is the dividingvalue of the exponent as regards propagation. The problem with existence is delicate forlarge m and is treated in a further paper [227].An interesting and unexpected aspect of the theory is the existence of a transformationthat maps self-similar solutions of the FPME with m ≥ PMFP’ with exponent 1 < m <
2. This applies in particular to theBarenblatt solutions constructed in [247]. The transformation is established in [225]and is quite useful in showing that
PMFP’ has infinite propagation in that range ofparameters. • Work by Biler-Imbert-Karch [32] deals with the variant(7.4) u t = ∇ · ( u ∇ ( − ∆) − s u m − ) . They construct a family of nonnegative explicit compactly supported self-similar solu-tions which are a generalization of the well-known Barenblatt profiles for the classicalporous-medium equation. They also establish the existence of sign-changing weak solu-tions to the initial-value problem, which satisfy sharp hypercontractivity L - L p estimates. • Reference [225] also treats on the double exponent model(7.5) ∂ t u + ∇ ( u m − ∇ ( − ∆) − s u n − ) = 044hat generalizes all the previous models. The paper discusses self-similar transformationsand finite propagation. The transformation of self-similar solutions indicates that finitepropagation holds for m ≥
2, while n >
There is work on equations with other nonlocal linear operators, and also on equationswith lower order terms, leading to reaction-diffusion and blow-up. Nonlinear diffusionand convection is treated in [89, 1]. The chemotaxis systems have been studied withnonlocal and/or nonlinear diffusion, like [115, 34, 53, 182]. We also have geometricalflows, like the fractional Yamabe problem (to be mentioned below). And there are anumber of other options. Operators and Equations in Bounded Domains
We have presented different definitions of the fractional Laplacian operator acting in R N in Section 2, and we have mentioned that all these versions are equivalent. However,when we want to pose a similar operator in a bounded domain Ω ⊂ R N we have to re-examine the issue, and several non-equivalent options appear. This enlarges the theoryof evolution equations of fractional type on bounded domains, and the recent literaturehas taken it into account. Actually, there is much recent progress in this topic and thenext second subsection will describe our recent contributions. A large class of relatednonlocal diffusive operators can be considered in the same framework. There are a number of definitions that have been suggested for the fractional Laplacianoperator (FLO) acting on a bounded domain Ω. The ones we consider here are naturallymotivated, and they give rise to different operators. We will mention three basic options,two of them are mostly used.
The Restricted Fractional Laplacian operator (RFL) . It is the simplest op-tion. It acts on functions g ( x ) defined in Ω and extended by zero to the complement,and then the whole hypersingular integral of the Euclidean case is used. Therefore, it isjust the fractional Laplacian in the whole space “restricted” to functions that are zerooutside Ω.(8.1) ( − ∆ | Ω ) s g ( x ) = c N,s
P.V. (cid:90) R N g ( x ) − g ( z ) | x − z | N +2 s d z , with supp( g ) ⊂ Ω . Here, s ∈ (0 ,
1) and c N,s > − ∆ | Ω ) s is a self-adjoint operator on L (Ω) with a discrete spectrum, with eigenvalues0 < λ ≤ λ ≤ . . . ≤ λ j ≤ λ j +1 ≤ . . . , satisfying λ j (cid:16) j s/N , for j (cid:29)
1. The corresponding eigenfunctions φ j are only H¨oldercontinuous up to the boundary, namely φ j ∈ C s (Ω) , [210].45n important issue is the way in which the additional conditions (formerly boundaryconditions) are implemented for the RFL. It usually takes the form of exterior conditions:(8.2) u ( t, x ) = 0 , in (0 , ∞ ) × (cid:0) R N \ Ω (cid:1) . The behavior of the Green function G plays an important role in the correspondingPDE theory. It satisfies a strong behaviour condition, that we call (K4) condition:(K4) G ( x, y ) (cid:16) | x − y | N − s (cid:18) δ γ ( x ) | x − y | γ ∧ (cid:19) (cid:18) δ γ ( y ) | x − y | γ ∧ (cid:19) , where δ ( x ) is the distance from x ∈ Ω to the boundary. The exponent γ will play a rolein the results derived from the kernel. In the RFL we have γ = s . References.
There is an extensive literature on the RFL operator and the corresponding α -stable process in the probability literature. The interested reader is referred [46]wherewe have commented on relevant works in that direction. The Spectral Fractional Laplacian operator (SFL).
It is defined by the twoequivalent expressions(8.3) ( − ∆ Ω ) s g ( x ) = ∞ (cid:88) j =1 λ sj ˆ g j φ j ( x ) = 1Γ( − s ) (cid:90) ∞ (cid:0) e t ∆ Ω g ( x ) − g ( x ) (cid:1) dtt s , where ∆ Ω is the classical Dirichlet Laplacian on the domain Ω, and ˆ g j are the Fouriercoefficients of f ˆ g j = (cid:90) Ω g ( x ) φ j ( x ) d x , with (cid:107) φ j (cid:107) L (Ω) = 1 . In this case the eigenfunctions φ j are the same as in the Dirichlet Laplacian, smooth asthe boundary of Ω allows. Namely, when ∂ Ω is C k , then φ j ∈ C ∞ (Ω) ∩ C k (Ω) for all k ∈ N . The eigenvalues are powers λ sj of the standard eigenvalues 0 < λ ≤ λ ≤ . . . ≤ λ j ≤ λ j +1 ≤ . . . and λ j (cid:16) j /N . It is proved that the eigenvalues of the RFL are smallerthan the ones of SFL: λ j ≤ λ sj for all j ≥
1, [87].Lateral boundary conditions for the SFL are different from previous case. They can beread from the boundary conditions of the Dirichlet Laplacian by the semigroup formula.They are often defined by means of the equivalent formulation that uses the Caffarelli-Silvestre extension defined in a cylinder adapted to the bounded domain, as done in[54, 63, 230]. If U is the extended function, then we impose U = 0 on the lateralboundary x ∈ ∂ Ω, y > γ = 1. Remarks.
Both SFL and RFL admit another possible definition using the so-calledCaffarelli-Silvestre extension. They are the two best known options for a FLO. Thedifference between RFL and SFL seems to have been well-known to probabilists, it wasdiscussed later in PDEs, see Servadei-Valdinoci [217], Bonforte and the author [51], andMusina-Nazarov [196]. In this last work the denomination
Navier fractional Laplacian is used. The debate about the proper names to be used is not settled.46 he Censored Fractional Laplacians (CFL) . This is another option appearingin the probabilistic literature, it has been introduced in 2003 by Bogdan, Burdzy andChen, [38]. The definition is(8.4) L g ( x ) = P . V . (cid:90) Ω ( g ( x ) − g ( y )) a ( x, y ) | x − y | N +2 s d y , with 12 < s < , where a ( x, y ) is a measurable, symmetric function bounded between two positive con-stants, satisfying some further assumptions; for instance a ∈ C (Ω × Ω). In the simplestcase we put a ( x, y ) = constant. On the other hand, [38] point out that in the excludedrange s ∈ (0 , /
2] the censored 2sstable process is conservative and will never approachthe boundary. The CFL is also called regional fractional Laplacian .The Green function G ( x, y ) satisfies condition (K4) with γ = s − , as proven by Chen,Kim and Song [86]. See also [145]. Note.
We have presented 3 models of Dirichlet fractional Laplacian. The estimates(K4) show that they are of course not equivalent . Our work described in the nextsubsection applies to those operators and a number of other variants, that are listed in[52] and [42]. For instance, sums of operators of the above types and powers of saidoperators are included.
We report here on very recent work done in collaboration with M. Bonforte, and alsoY. Sire and A. Figalli, on nonlinear evolution equations of porous medium type posedin bounded domains and involving fractional Laplacians and other nonlocal operators.The papers are [46], [51], [52], [41], and [42].We develop a new programme for nonlocal porous medium equations on bounded do-mains aiming at establishing existence, uniqueness, positivity, a priori bounds, regularity,and asymptotic behaviour for a large class of equations of that type in a unified way. Weinclude the set of suitable versions of FLO in a bounded domain. The main equation iswritten in abstract form as(8.5) ∂ t u + L Φ( u ) = 0 , where Φ a continuous and nondecreasing real function, most often a power function. • A problem to be settled first is the suitable concept of solution. We use the “dual”formulation of the problem and the concept of weak dual solution , introduced in [51],Definition 3.4, which extends the concept of weighted very weak solution used before.In brief, we use the linearity of the operator L to lift the problem to a problem for thepotential function U ( x, t ) = (cid:90) Ω u ( y, t ) G ( x, y ) dy where G is the elliptic Green function for L . Then ∂ t U = − Φ( u ) . • Class of operators.
In our recent work we have extended the evolution theoryto cover a wide class of linear operators L that satisfy the following conditions. L :47om( L ) ⊆ L (Ω) → L (Ω) is assumed to be densely defined and sub-Markovian, moreprecisely, it satisfies (A1) and (A2):(A1) L is m -accretive on L (Ω);(A2) If 0 ≤ f ≤ ≤ e − t L f ≤ L − can be written as L − [ f ]( x ) = (cid:90) Ω K ( x, y ) f ( y ) d y , The kernel K is called the Green function and we assume that there exist constants γ ∈ (0 ,
1] and c , Ω , c , Ω > x, y ∈ Ω :(K2) c δ γ ( x ) δ γ ( y ) ≤ K ( x, y ) ≤ c | x − y | N − s (cid:18) δ γ ( x ) | x − y | γ ∧ (cid:19) (cid:18) δ γ ( y ) | x − y | γ ∧ (cid:19) , where we adopt the notation δ ( x ) := dist( x, ∂ Ω). We will also use φ , the first eigen-function of L , and we know that φ (cid:16) dist( · , ∂ Ω) γ . Further assumptions will be madein each statement, depending on the desired result we want, in particular (K4) that wehave already mentioned. • Sharp bounds.
Under these assumptions, we obtain existence and uniqueness ofsolutions with various properties, like time decay in L p spaces. We will not delve inthis basic theory that is covered in the papers [46, 52]. We will stress here one ofour main contributions in [42]: we prove sharp upper and lower pointwise bounds fornonnegative solutions, both at the interior and close to the boundary. Indeed, we mustpay close attention to the boundary behaviour, that turns out to be different for differentoperators in this class. However, only some options appear, as we describe next. Let usintroduce first an important exponent σ = 1 ∧ smγ ( m − . Notice that σ = 1 for the RFL and the CFL, but not always for the SFL unless m = 1.The results that follow are taken from the last work, [42]. In the next results (CDP)means the Cauchy-Dirichlet problem with zero lateral data, and solutions means dualweak solutions. We make the default assumptions (A1), (A2), and (K2) on L . Case 1. Nonlocal operators with nondegenerate kernels.
We assume here more-over that the kernel of L is non degenerate at the boundary, namely(8.6) L f ( x ) = (cid:90) R N (cid:0) f ( x ) − f ( y ) (cid:1) K ( x, y ) d y , with inf x,y ∈ Ω K ( x, y ) ≥ κ Ω > . Under these assumptions we can prove the following first version of the Global HarnackPrinciple.
Theorem 8.1.
Let (A1), (A2), (K2), and (8.6) hold. Also, when σ < , assume that K ( x, y ) ≤ c | x − y | − ( N +2 s ) for a.e. x, y ∈ R N and that φ ∈ C γ (Ω) . Let u ≥ be a weak ual solution to the (CDP) corresponding to u ∈ L φ (Ω) . Then, there exist constants κ, κ , so that the following inequality holds: (8.7) κ (cid:18) tt + t ∗ (cid:19) mm − φ ( x ) σ/m t m − ≤ u ( t, x ) ≤ κ φ ( x ) σ/m t m − for all t > and all x ∈ Ω . Here, t ∗ = κ ∗ (cid:107) u (cid:107) − ( m − φ (Ω) , and this time will appear in the other theorems. For largetimes both lower and upper bounds are similar. We point out that the result holds inparticular for the Restricted and Censored Fractional Laplacians, but not for the SpectralFractional Laplacian. The lower bound is false for s = 1 (in view of the finite speed ofpropagation of the standard PME). Case 2. Matching behaviour for large times.
We can prove that previous GlobalHarnack Principle for large times without using the non-degeneracy of the kernel, underthe following conditions on σ : either(i) σ = 1 (i.e., 2 s > γ ( m − /m ), or(ii) σ <
1, and we have an improved version of (K2)(K4) K ( x, y ) (cid:16) c | x − y | N − s (cid:18) δ γ ( x ) | x − y | γ ∧ (cid:19) (cid:18) δ γ ( y ) | x − y | γ ∧ (cid:19) , and the initial data are not small: u ≥ cφ σ/m for some c > Theorem 8.2 (Global Harnack Principle II) . Let (A1), (A2), and (K2) hold, and let u ≥ be a weak dual solution to the (CDP) corresponding to u ∈ L φ (Ω) . Assumethat either (i) or (ii) above hold true. Then there exist constants κ, κ > such that thefollowing inequality holds: (8.8) κ φ ( x ) σ/m t m − ≤ u ( t, x ) ≤ κ φ ( x ) σ/m t m − for all t ≥ t ∗ and all x ∈ Ω . The constants κ, κ depend only on
N, s, γ, m, κ , κ Ω , and Ω . The conditions on σ are sharp. Actually, the proof in the case σ = 1 includes theclassical PME (i.e., the non fractional equation, for which finite propagation holds, sothat there can be no positive a priori lower bound for short times). The case of a really degenerate kernel.
We assume moreover that we assumemoreover that the kernel of L exists and can be degenerate at the boundary (actually,excluding the local case, this is the most general assumption) in the form(8.9) L f ( x ) = P.V. (cid:90) R N (cid:0) f ( x ) − f ( y ) (cid:1) K ( x, y ) d y , with K ( x, y ) ≥ c φ ( x ) φ ( y ) ∀ x, y ∈ Ω . This is an assumption that holds for the Spectral Fractional Laplacian operator. Toour knowledge, precise information about the kernel of the SFL was not known beforeLemma 3.1 of [42]. 49ote that, for small times, we cannot find matching powers for a global Harnack in-equality (except for some special initial data), and such result is actually false for s = 1(in view of the finite speed of propagation of the PME). Hence, in the remaining cases,we have only the following general result. Theorem 8.3 (Global Harnack Principle III) . Let (A1), (A2), (K2), and (8.9) hold.Let u ≥ be a weak dual solution to the (CDP) corresponding to u ∈ L φ (Ω) . Then,there exist constants κ, κ > , so that the following inequality holds: (8.10) κ (cid:18) tt + t ∗ (cid:19) mm − φ ( x ) t m − ≤ u ( t, x ) ≤ κ φ ( x ) σ/m t m − for all t > and all x ∈ Ω . This is what we call non-matching powers for the spatial profile at all times. Thepaper gives analytical and numerical evidence that such non matching behaviour doesnot happen in the associated elliptic problems, and came as a surprise to the authors.For some class of initial data, namely u ≤ ε φ we can prove that for small times κ (cid:18) tT (cid:19) mm − φ ( x ) t m − ≤ u ( t, x ) ≤ κ T m − φ ( x ) t m − for all 0 ≤ t ≤ T and all x ∈ Ω . Numerics.
This work has been improved in January 2017 with numerics done at BCAMInstitute by my former student F. del Teso and collaborators, [95], 2017, that validatesthe different behaviour types. • Asymptotic Behaviour.
An important application of the Global Harnack inequal-ities of the previous section concerns the sharp asymptotic behavior of solutions. Moreprecisely, we first show that for large times all solutions behave like the separate-variablessolution U ( t, x ) = S ( x ) t − m − . The profile S is the solution of an elliptic nonlocal prob-lem. Then, whenever the Global Harnack Principle (GHP) holds, we can improve thisresult to an estimate in relative error. Theorem 8.4 (Asymptotic behavior) . Assume that L satisfies (A1), (A2), and (K2),and let S be as above. Let u be any weak dual solution to the (CDP). Then, unless u ≡ , (8.11) (cid:13)(cid:13)(cid:13) t m − u ( t, · ) − S (cid:13)(cid:13)(cid:13) L ∞ (Ω) t →∞ −−−→ . We can exploit the (GHP) to get a stronger result, using the techniques of paper [46].
Theorem 8.5 (Sharp asymptotic behavior) . Under the assumptions of Theorem 8.4,assume that u (cid:54)≡ . Furthermore, suppose that either the assumptions of Theorem 8.1 orof Theorem 8.2 hold. Set U ( t, x ) := t − m − S ( x ) . Then there exists c > such that, forall t ≥ t := c (cid:107) u (cid:107) − ( m − φ (Ω) , we have (8.12) (cid:13)(cid:13)(cid:13)(cid:13) u ( t, · ) U ( t, · ) − (cid:13)(cid:13)(cid:13)(cid:13) L ∞ (Ω) ≤ m − t t + t . We remark that the constant c > only depends on N, s, γ, m, κ , κ Ω , and Ω . omments on related work. Construction of the solutions of the FPME on boundeddomains with the SFL was already used in [102, 103] as an approximation to the problemin the whole space, but the regularity or asymptotic properties were not studied. Kimand Lee in [168] study the Fast Diffusion range m < p -Laplacian type We report here about our work [250]. It deals with a model of fractional diffusioninvolving a nonlocal version of the p -Laplacian operator, and the equation is(8.13) ∂ t u + L s,p u = 0 , L s,p ( u ) := (cid:90) R N Φ( u ( y, t ) − u ( x, t )) | x − y | N + sp dy = 0where x ∈ Ω ⊂ R N , N ≥ , Φ( z ) = c | z | p − z, p ∈ (1 , ∞ ) and s ∈ (0 , L s,p is theEuler-Lagrange operator corresponding to a power-like functional with nonlocal kernelof the s -Laplacian type. The study of the equation is motivated by the recent increasinginterest in nonlocal generalizations of the porous-medium equation. In the paper wecover the range 2 < p < ∞ . Note that for p = 2 we obtain the standard s -Laplacianheat equation, u t + ( − ∆) s u = 0, which was discussed before; on the other hand, it isproved that in the limit s → p (cid:54) = 2, we get the well-known p -Laplacian evolutionequation ∂ t u = ∆ p ( u ), after inserting a normalizing constant.We consider the equation in a bounded domain Ω ⊂ R N with initial data(8.14) u ( x,
0) = u ( x ) , x ∈ Ω , where u is a nonnegative and integrable function. Moreover, we impose the homoge-neous Dirichlet boundary condition that in the fractional Laplacian setting takes theform(8.15) u ( x, t ) = 0 for all x ∈ R N , x (cid:54)∈ Ω , and all t > . When then apply the integral operator on the set of functions that vanish outside of ΩThe first result of this paper concerns the existence and uniqueness of a strong non-negative solution to an initial-boundary value problem for (8.13) in bounded domainΩ ⊂ R N , with zero Dirichlet data outside Ω × (0 , ∞ ). The boundedness of the solutionis established after proving the existence of a special separating variable solution of theform U ( x, t ) = t / ( p − F ( x ) , called the friendly giant. The profile function F ( x ) of the friendly giant solves theinteresting nonlocal elliptic problem (cid:90) R N Φ( F ( y ) − F ( x )) | x − y | N + sp dy = c F ( x ) . The friendly giant solution provides a universal upper bound and also gives the large-time behaviour for all the nonnegative solutions of initial-boundary value problems withhomogeneous Dirichlet boundary conditions.51he fractional p -Laplacian has recently attracted the attention of many researchersfor its mathematical interest. See among other related works the papers by Caffarelliet al. [66], Maz´on et al. [189], Puhst [206]. Another approach in the form of non-local gradient dependent operators is taken by Bjorland-Caffarelli-Figalli in [35]. Thecorresponding stationary equation is also studied in the literature, see previous refer-ences. Finally, the work [152] by Hynd and Lindgren deals with the doubly nonlinearmodel | u t | p − u t + L s,p u = 0, that has a special homogeneous structure. Regularity andasymptotic behaviour follow. Further work on related topics
Equations of the form D αt u = L u + f are another form of taking into account nonlocal effects. Here L represents the diffusionprocess with long-distance effects in the family of the fractional Laplacian operators.The symbol D αt denotes the fractional time derivative. There are a number of variantsof this concept, the most popular being maybe the Caputo fractional derivative , whichwas introduced by M. Caputo in 1967 [76] and reads(9.1) Ca D αt f ( t ) = 1Γ( n − α ) (cid:90) ta f ( n ) ( τ ) dτ ( t − τ ) α +1 − n , n − < α ≤ n Indeed, fractional time derivatives are the most elementary objects of Fractional Calcu-lus, a branch of mathematical analysis that studies the possibility of taking real numberpowers (real number fractional powers or complex number powers) of the differentiationoperator D = d/dx, and the integration operator.The foundations of the theory of fractional derivatives were laid down by Liouville in apaper from 1832. Different definitions use different kernels, but all them make weightedaverages in time.Some recent work: Dipierro and Valdinoci derive the linear time-fractional heat equa-tion in 1D in a problem of neuronal transmission in cells, [112]; Allen, Caffarelli andVasseur study porous medium flow with both a fractional potential pressure and a frac-tional time derivative,[2]. • The study of the heat equation posed on a Riemannian manifold comes from long timeago, the diffusive operator being the Laplace-Beltrami operator(9.2) ∆ g ( u ) = 1 (cid:112) | g | ∂ i ( g ij (cid:112) | g | ∂ j u ) , • Generalization of the Caffarelli-Silvestre extension method allows to define extensionsand boundary operators of the fractional Laplacian type when M is the boundary of aconformally compact Einstein manifold. Combining geometrical and PDE approaches,Chang and Gonz´alez in [82] related the original definition of the conformal fractionalLaplacian coming from scattering theory to a Dirichlet-to-Neumann operator for a relatedelliptic extension problem for M , see also [137]. It is possible then to formulate fractionalYamabe-type problems for the conformally covariant operators P γ , [138]. For more recentwork in this problem see [157, 97]. • Work on the Porous Medium Equation on manifolds was done in the 2000s, like [43],[244]; fast diffusion was treated in [44]; general Aronson-B´enilan estimates and entropyformulae for porous medium and fast diffusion equations on manifolds were obtained in[187]. • Recently the author studied the PME on the hyperbolic space [249], and constructedthe fundamental solution and proved the asymptotic convergence and free boundarypropagation rates. The fact that the fundamental is not explicit or self-similar is notpleasant, and looking for some higher symmetry properties a remarkable object appearedto play an asymptotic role. Namely, there exists nonnegative weak solution of the PMEdefined on the whole of H N for all t >
0, that has a strong algebraic structure. In thePoincar´e upper half-space representation it is given by the formula(9.3) U ( x, y, t ) m − = a (log( ct γ y )) + t with m, N >
1, 1 /a = m ( N −
1) and 1 /γ = ( N − m − x ∈ R N − , y >
0. Notethat U has zero initial trace at t = 0 on the half-space, but it has a singularity as trace at y = + ∞ , which corresponds to a singularity at the North Pole in the standard Poincar´eball representation. Therefore, we can say that the special solution (or geometricalsoliton) U comes from the infinite horizon and expands to gradually to fill the wholespace; for any t > { ( x, y ) : y =(1 /c ) t − γ } . Recall that geodesic distance is given by the formula ds = dy /y . A detailfor analysts: U represents an example of non-uniqueness of nonnegative solutions for theCauchy Problem in hyperbolic space.This family of special solutions is the pattern to which all solutions with compactlysupported initial data are proved to converge as t → ∞ . Accordingly, we get the followingsharp estimates for all solutions u ≥ (cid:107) u ( · , t ) (cid:107) ∞ ∼ ct − / ( m − log( t ) / ( m − , S ( t ) ∼ γ log( t )where S ( t ) is the location of the free boundary measured in geodesic distance. • The asymptotic analysis of PME flows is extended to more general manifolds in [144].The work on the asymptotic behaviour on hyperbolic space is extended to fast diffusionby Grillo et al. in [141]. In another direction, Amal and Elliott [3] study fractionalporous medium equations on evolving surfaces, a very novel subject.53 .3 Diffusion in inhomogeneous media
We have already mentioned the inhomogeneity of the medium as a reason for the intro-duction of coefficients in the passage from the heat equation to the parabolic class. Inview of the important practical consequences, there is no surprise in finding coefficientsappear in most of the models we have considered above, both linear and nonlinear, localand nonlocal. Let us just mention some well-known references like [161, 207, 160], whereit appears as diffusion with weights, see also the more recent [142] with two weights.There is an interesting connection between weighted diffusion in Euclidean space andLaplace-Beltrami diffusion on manifolds, that has been studied in [249] and is beingfurther investigated.
The main equation in this case is an equation for a scalar unknown θ driven by theequation(9.4) ∂ t θ + v · ∇ x θ = L ( θ ) , where v · ∇ x T is the convective term with velocity vector v , and L ( θ ) is the diffusionoperator; that diffusion can be linear of nonlinear, local or nonlocal. A very importantaspect is the relation of v to the rest of the variables. Thus, when v = v ( x, t ) is a givenfunction, no essential new problems arise if v is smooth. But serious difficulties happenfor nonsmooth v . All this is reflected for instance in the seminal paper by Caffarelli andVasseur [69] where the equation is ∂ t θ + v · ∇ x θ + ( − ∆) / ( θ ) = 0and v is a divergence-free vector field. In the popular quasi-geostrophic model v is givenin terms of θ that makes the problem more involved, but the results stated in [69] donot depend upon such dependence. The proof of regularity needs to establish delicatelocal energy estimates, despite the fact that the diffusion operator ( − ∆) / is non-local.It also uses DeGiorgi’s methods in an essential way. There is much work in geostrophicflows, like [188, 174, 175].There is a vast literature on this important issue. In some cases v is given by Darcy’slaw in an incompressible fluid, and the papers refer to the problem as “flow in porousmedia”, like [81]. Let us point out that this use is quite different from our use of “porousmedia” in the present paper, the difference being often stressed by calling their use“incompressible flow in porous media”, [120]. -Minimal surfaces are an important subject which uses many methods of the nonlinearelliptic and parabolic theory, Recently, it has developed a new branch, nonlocal minimalsurfaces. Work on both aspects is reported in detail in another contribution to thisvolume by Cozzi and Figalli [91]. Related items are fractional perimeters and fractionalphase transition interfaces. We will not enter into that area.54The study of the combined effects of diffusion and aggregation is a very active fieldwhere the methods of diffusion in its different forms must be combined with the countermechanism of attraction. We refer to the contribution by Calvez, Carrillo and Hoffmannto this volume, [75].- In the study of nonlinear diffusion we have chosen to present almost exclusively equa-tions with diffusion terms in divergence form. There is a large body of work involvingFully Nonlinear Parabolic Equations (they are non-divergence equations), both ellipticand parabolic. We will not touch such theories here. Addendum and final comment
Here is the complete Wikipedia list of diffusion topics:Anisotropic diffusion, also known as the Perona-Malik equation, enhances high gra-dients; Anomalous diffusion, in porous medium; Atomic diffusion, in solids; Brownianmotion, for example of a single particle in a solvent; Collective diffusion, the diffusionof a large number of (possibly interacting) particles; Eddy diffusion, in coarse-graineddescription of turbulent flow; Effusion of a gas through small holes; Electronic diffusion,resulting in electric current; Facilitated diffusion, present in some organisms; Gaseous dif-fusion, used for isotope separation; Heat flow, diffusion of thermal energy; It¯o-diffusion,continuous stochastic processes; Knudsen diffusion of gas in long pores with frequent wallcollisions; Momentum diffusion, ex. the diffusion of the hydrodynamic velocity field; Os-mosis is the diffusion of water through a cell membrane; Photon diffusion; Random walkmodel for diffusion; Reverse diffusion, against the concentration gradient, in phase sep-aration; Self-diffusion; Surface diffusion, diffusion of adparticles on a surface; Turbulentdiffusion, transport of mass, heat, or momentum within a turbulent fluid. • The reader may wonder whether mathematical diffusion is a branch of applied math-ematics? In principle it would seem that the answer is an obvious yes, and yet it is notso clear. As we have seen, the mathematical theories of diffusion have developed into acore knowledge in pure mathematics, that encompasses several branches, from analysisand PDEs to probability, geometry and beyond. We hope that the preceding pages willhave convinced the reader of this trend, and will also motivate him/her to pursue someof many avenues open towards the future.
Acknowledgments.
This work was partially supported by Spanish Project MTM2014-52240-P. The text is based on series of lectures given at the CIME Summer School heldin Cetraro, Italy, in July 2016. The author is grateful to the CIME foundation for theexcellent organization. The author is also very grateful to his collaborators mentioned inthe text for an effort of many years. Special thanks are due to F. del Teso, N. Simonovand D. Stan for a careful reading and comments on the text.55 eferences [1]
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Address : Juan Luis V´azquezDepartment de Matem´aticasUniversitas Aut´onoma de Madrid28049 Madrid, Spain e-mail: [email protected]
Mathematics Subject Classification.
Keywords and phrases.
Diffusion, nonlinear diffusion, nonlocal diffusion, fractionaloperators.