Entropy methods for diffusive PDEs

Transcription

1 Entropy methods for diffusive PDEs Dr. Nicola Zamponi Vienna University of Technology Summer term 17 Lecture notes from a lecture series at Vienna University of Technology Contents 1 Introduction A few applications of the concept of entropy Some ideas involving entropy The Bakry-Emery approach 1.1 The linear Fokker-Planck equation Convex Sobolev inequalities Logarithmic Sobolev inequality Weighted Poincaré inequality Beckner inequality The heat equation: convergence to the self-similar solution Linear Fokker-Planck equation: generalizations Fokker-Planck equation with variable diffusion Non-symmetric Fokker-Planck equation Degenerate Fokker-Planck equation Nonlinear Fokker-Planck equations Cross-Diffusion PDEs Examples of cross-diffusion PDEs Population dynamics: the SKT model Ion transport Tumor-growth models Multicomponent fluid mixtures Derivation of some cross-diffusion models Derivation from random-walk lattice models Derivation from fluid models Entropy structure

2 3.3.1 Relation to thermodynamics Relation to hyperbolic conservation laws About the symmetry of B and the eigenvalues of A The Boundedness-by-Entropy Method Proof of the general existence theorem for cross-diffusion systems in the volume-filling case A few examples Population Models Ion-Transport Models About uniqueness of weak solutions Further examples of cross-diffusion PDEs Energy-transport models A cross-diffusion system derived from a Fokker-Planck equation... 68

3 1 Introduction The term entropy, which will be as the Reader may expect the leading concept, the key idea of the course, was born in 1865 by the mind of Rudolf Clausius, who used this word to denote the amount of energy which is no longer usable for physical work e.g. heat produced by friction. In 1877, Ludwig Boltzmann suggested a statistical interpretation of the concept of entropy, stating the the entropy S of an ideal gas is proportional to the logarithm of the number of microstates W corresponding to the gas macrostate, i.e. S = k B log W carved on his gravestone in Vienna s Zentralfriedhof. Here k B m kg s K 1 is the well-known Boltzmann constant, a recurring object in statistical mechanics. Willard Gibbs gave a similar definition of entropy: S = k B i p i log p i, where p i is the probability of the microstate i, and the summation is on all the microstates associated to the macrostate. One of the most celebrated results by Boltzmann is the H-theorem, which states that the H-function: H[f] = fx, v, t log fx, v, tdxdv. 1 R d R d is nonincreasing in time along the solutions f os the Boltzmann equation. However, since the entropy S of the system is proportional to H[f], this implies that S is nondecreasing in time, which can be seen as a formulation of the second law of thermodynamics the physical entropy of a closed system cannot decrease in time. We will see that the H- theorem is a kind of prototype for the entropy methods for PDEs. Before we go on, let us introduce the mathematical convention of putting a minus in front of the entropy : the mathematical entropy equals minus the physical entropy. For us, an entropy will be, tipically, a convex Lyapounov functional for some PDE, that is, a functional which will be nonincreasing along the solutions of some partial differential equation or system of them. The results hereby presented can be found in [3]. 1.1 A few applications of the concept of entropy There are many examples which display the usefulness of the concept of entropy in mathematics. We discuss here some of them. Hyperbolic conservation laws. When dealing with systems of hyperbolic conservation laws, i.e. PDEs like t u + div fu =, x R d, t >, where f : R n R n d is the flux, the fact that the equation has no strong solution, while weak solutions are not unique, can be really annoying. Luckily, the entropy lends us a hand in such a difficult predicament. An entropy solution of is a weak solution u : R d, R n of such that for all convex functions h : R n R a function q : R n R d exists such that t hu + div qu 3

4 in some distributional sense. The function h is called entropy density, while q is the entropy flux. Moreover, the entropy Hu = R d hudx is nonincreasing in time. It turns out that the entropy solutions are unique. Kinetic theory. Take the Boltzmann equation t f + v x f = Qf, f, x, v R d R d, t >, 3 where f = fx, v, t is the system distribution function and Qf, f is the binary collision operator. Recall the definition 1 of H, which we will call simply entropy. The entropy production P [f] is defined as P [f] := d H[f]. The properties of Q ensure that P [f], dt and that P [f] = if and only if f = M, where d/ m Mv = ρ e m v u /k B T πk B T is the so-called Maxwellian, ρ is the particle density, u is the mean velocity, and T is the temperature. As a consequence d H[f], i.e. H[f] is nonincreasing in time. dt You, curious Reader, may ask: is it true what they say, that ft M as t? If so, how fast? The usual idea is to consider the relative entropy H[f M] := H[f] H[M] and see if it can be dominated by the entropy production, that is ΦH[f M] P [f] for some increasing Φ : R R such that Φ =. 4 Replacing P [f] inside 4 with its definition leads to d H[f] + ΦH[f] H[M] t >. dt Let us assume that H[ft] > H[M] otherwise f = M, since M can be shown to be a strict minimum point. Therefore H[f ] H[M] H[ft] H[M] ds Φs t t >. The integral on the left-hand side of the above inequality must explode as t. Since the only singularity of 1/Φs is at s =, we conclude that H[f] H[M], and therefore f M since H[M] = min f H[f]. If Φs = λs, then we deduce that H[ft] H[M] e λt H[f ] H[M], t >. Bakry-Emery technique. An idea developed first by Barky and Emery in 1985 consists in estimating the second time derivative of the entropy by means of the first time derivative. Assume we can prove d dt H[f] κ d H[f] t >, dt 4

5 for some constant κ >. Assume moreover that both d H[f], H[f] tend to zero as t. dt Integrating the above inequality in the time interval t, leads to d H[f] κh[f], t >. dt Again, Gronwall s Lemma implies that H[ft] e κt H[f ]. Cross-diffusion systems. A system of PDEs of the form t u = div Au u x R d, t >, 5 where u :, R n is the vector of the unknowns and A : R n R n n is the diffusion matrix, is called a cross-diffusion system. In applications u models chemical concentrations or population densities, and the matrix A is often not symmetric nor positive semidefinite, making any analytical study of 5 tricky. For example, tools like maximum principles or parabolic regularity theory cannot be applied. However, not everything is lost, as sometimes cross-diffusion systems have an entropy, i.e. a Lyapounov functional. This fact provides us with useful a-priori estimates, while it also allows us to make a change of variables which yields a often symmetric positive definite diffusion matrix and makes it possible to prove nonnegativity and even uniform boundedness of the original variables u without exploiting any maximum principle. These new variables, which we will call entropy variables, are simply what is thermodynamics is termed chemical potentials. 1. Some ideas involving entropy We sketch here a few mathematical uses of the entropy. Long-time behaviour. Let us consider, as a toy problem, the heat equation on the d dimensional torus T d =, 1 d : t u = u x T d, t >, ux, = u x x T d, 6 where u L 1 T d, u in T d is the initial datum. It is well known that 6 has a unique smooth solution u = ux, t having the same mass as u, i.e. ux, tdx = u T d T d xdx, t >. Let u = u T d xdx be this mass, which coincides with the average of u since T d has measure 1. We call u the steady state of the system. We want to show that ut u as t in some sense. How do we proceed? We can, for example, define the convex, nonnegative functional H [u] = u u dx. 7 T d Clearly H [u] is just the square L norm of the difference between u and the steady state u. If we can find an upper bound for H [ut], where ut is the solution of 7, and such upper bound tends to as t, then we have achieved our goal. 5

6 In order to find this upper bound for H [ut], we ask ourselves: How does H vary along the solutions of 6? The answer is easily found by computing the time derivative of H [ut]: d dt H [ut] = ut u utdx = ut dx, 8 T d T d where, in order to obtain the second equality, we integrated by parts and exploited the periodic boundary conditions. Now we should find an upper bound for the right-hand side of 8. Poincaré-Wirtingen inequality provides us with the following result: ut u dx C P T d ut dx. T d 9 Now we just have to put 8, 9 together to obtain The Gronwall Lemma implies d dt H [ut] H [ut] t >. 1 C P ut u L T d = H [ut] H [u ]e t/c P t >. 11 Therefore, we proved that ut u in L T d as t exponentially with a rate equal to 1/C P. What about other metrics? Can we prove, for example, a convergence result in L 1 T d with a different possibly bigger rate? Well, of course we can. Let us consider the Boltzmann entropy: H 1 [u] = T d u log u u dx = T d u log u u + u dx. 1 u Since u log u u u + u = hu hu h u u u with hu = u log u u and h is convex in,, we deduce that u log u u u + u and therefore H 1 [u]. We take the time derivative of H 1 [ut] and get d dt H 1[ut] = log ut ut dx = 4 ut dx. T u d T d Now we need to bound the left-hand side of the above inequality by something proportional to H 1 [ut]. This time we exploit the logarithmic Sobolev inequality: ut log ut dx C L ut dx. 13 T u d T d So we deduce d dt H 1[ut] 4 H 1 [ut] t >, C L 6

7 which, thannk to Gronwall s Lemma, implies H 1 [ut] e 4t/C L H 1 [u ] t >. To get a convergence rate in L 1 T d we can apply Csiszár-Kullback inequality: and conclude ut u L 1 u H1 [ut], ut u L 1 Ce t/c L t >, where C > is a suitable constant, depending on H 1 [u ] and u. Therefore, we proved that ut u in L 1 T d as t exponentially with a rate equal to /C L. What about nonlinear equations? The technique works in this context, too. Consider for example the DLSS Derrida-Lebowitz-Speer-Spohn equation, modeling quantum electron transport in a semiconductor under suitable assumptions: t u = div u u t >, u = u in T d. 14 u Differentiate H 1 [u] in time along a nonnegative solution ut of 14: d dt H 1[ut] = log u div u u dx T u d = u u dx = u u dx. T u d T u d It is possible to prove that u u dx κ u dx, T u d T d where κ = 4d 1. At this point we employ the higher-order log-sobolev inequality dd+ Let us put everything together to get T d u log u u dx 1 8π 4 T d u dx. d dt H 1[ut] 4π 4 κh 1 [ut] t >, which, by Gronwall s lemma, implies that H 1 [ut] exponentially as t with rate 4π 4 κ. At this point we can summarize the above ideas into a general strategy to prove convergence of solutions to PDEs towards a steady state. Let us imagine we have an evolution equation with the form t u + Aut = t >, u = u, 7

8 where u :, B, B is some Banach space with dual B, and A : B B is some nonlinear mapping. Furthermore imagine that we are given some relative entropy functional H = H[u] and a steady state u i.e. a solution of Au =. What we have to do is to compute the entropy production, i.e. minus the time derivative of H[u] along the solutions on the evolution equation, and then find a possibly linear relation between the entropy production and the entropy itself. Finally, Gronwall s lemma will allow us to conclude that H[ut] with some rate exponential, algebraic... as t. Global existence and boundedness of weak solutions. Entropy methods can also be employed to prove existence of positive, uniformely bounded weak solutions to systems of reaction-diffusion and cross-diffusion PDEs. For example, consider 5 with n = Au given by u1 u Au = u 1 + u u + u The considered equations are a special case of a Maxwell-Stefan system and describe a fluid mixture of 3 components with equal molar masses under isobaric, isothermal conditions; u i is the mass fraction of the component i, for i = 1,, 3. The mass fractions must be nonnegative, i.e. the constraints u 1, u, u 1 + u 1 must be satisfied. One can prove the nonnegativity of u 1, u with a minimum principle, but no maximum principle is available which allows for the proof of the uniform boundedness of u 1 + u. But we need no maximum principle. First we have to derive an entropy balance inequality for the entropy H[u] = hudx, R d hu = 3 u i log u i u i. One can show the main obstacle is to find a uniform lower bound for some u dependent quadratic form that d H[u] u 1 + u dx. dt R d So we have a nice a-priori estimate which would prove itself to be quite useful in an analytical study of the system. However, having an entropy also means that we can define new variables, called entropy variables: i=1 w i = h u i = log u i u 3 i = 1,. What s the use of these new variables? First, 5 can be rewritten as t u = div Bw w, Bw Auwh uw 1. It turns out that B is symmetric and positive definite. This, of course, would prove invaluable in the analysis. 8

9 Second, let us have a look at uw: u i w = e w i 1 + e w i + e w, i = 1,. Clearly u i for i = 1,, 3, that is, we would have positivity and uniform boundedness for the physical variables without using any maximum or minimum principle. Therefore, if we can turn these a-priori estimates and smart ideas into a rigorous proof, we can have global existence of nonnegative, uniformely bounded weak solutions for a system of nonlinear PDEs with a diffusion matrix which is neither symmetric nor positive semidefinite. That doesn t sound too bad, does it? Uniqueness of weak solutions. Let us spend a few words about how entropy can help in showing uniqueness of weak solutions to PDEs. We can e.g. consider the drift-diffusion model in the d dimensional torus t u = div u + u V x T d, t >, ux, = u x x T d. 15 The model describes the semiclassical transport of electrons in a semiconductor under certain simplifying assumptions. Here u is the electron density and V is a given electric potential. We want to prove that 15 has at most one solution. The most natural thing to do would be to test 15 against u 1 u the difference of two solutions with the same initial datum and try to control the drift term by means of the diffusion term plus some Sobolev embedding. However, this strategy works only as long as the potential is smooth enough, e,g. V L p T d with p > big enough. But what if this is not true? Actually, we can achieve our goal by means of an entropy-based idea. Let us define [ ] u1 + u F [u 1, u ] = H[u 1 ] + H[u ] H, H[u] = u log u u dx. T d Taking the time derivative of F [u 1, u ] leads to d dt F [u 1, u ] = 4 u1 + u u 1 + u dx. 16 T d However, it is possible to prove that the so-called Fisher information F[u] T d u dx is a convex functional, ands therefore the right-hand side of 16 is nonpositive. This means that, for t >, F [u 1 t, u t] F [u 1, u ] = since u 1 = u at initial time. However, the strict convexity of H also implies that F [u 1, u ] and the equality holds if and only if u 1 = u. Therefore u 1 t = u t for t >. 9

10 The Bakry-Emery approach The method that we are going to present in this section was first developed by D. Bakry and M. Emery in the 198s [5] and consists in computing the second derivative of the entropy with respect to time and estimating it by means of the first derivative of the entropy, that is, the entropy production..1 The linear Fokker-Planck equation The Barky-Emery method is usually explained by applying it to the linear Fokker-Planck equation: u t = div u + u V, t >, x R d, 17 ux, = u x x R d, 18 which arises from many applications, e.g. semiconductor transport, plasma physics, stellar dynamics. The function V depends only on x and represents a potential, while the unknown function u = ux, t is a density. Existence results for 17, 18 are available in literature; see e.g. [45]. For the purpose of these lecture notes, we will assume that the solution exists and is sufficiently smooth to justify the computations that will be carried out. We assume that the initial datum is nonnegative and has mass equal to 1: u R d dx = 1. Since 17 is in divergence form the mass is conserved: ux, tdx = u xdx = 1 for t >. R d R d Concerning the potential V, we assume that V is smooth enough and 1 e V L 1 R d, λ > : d xi x j V xw i w j λ w x R d, w R d. 19 i,j=1 The steady state u of 17 is defined as the only positive constant-in-time solution u = u of u+u V = in R d such that R d u dx = R d u dx. Since u+u V = u log u+v and we are looking for positive solutions, this implies that log u + V must be constant in R d, and therefore u x = V x e R d e V y dy Let now φ :, [, a smooth function such that x R d. φ1 = φ 1 =, φ 1 = 1, φ >, 1/φ in [,. 1 1 Constraint 19 on the Hessian of V is called Bakry-Emery condition. 1

11 Examples of functions φ satisfying 1 are φs = s log s s + 1, φs = sα 1 αα 1 α 1, ]. The function φ generates the following relative entropy u H φ [u] = u dx. We are going to prove the following R d φ Theorem.1. Assume 19 1 hold, and let H[u ] <. Then any smooth solution u : R d, [, to 17, 18 converges exponentially to the steady state u, in the sense that ut u L 1 R d e λt H φ [u ] t >. Proof. We begin by differentiating H φ [u] with respect to t. Let ρ = u/u, so that 17 takes the form u t = div u ρ = u ρ + u ρ. 3 The entropy production can be computed through an integration by parts as follows: P φ [u] = d dt H φ = φ ρu t dx = φ ρ ρ u dx. 4 R d R d The nonnegativity of P results from the convexity of φ. It is now time to compute the second time derivative of the entropy: d dt H φ = φ ρ ρ t u + φ ρ ρ t ρ u dx = I 1 + I, 5 R d I 1 φ ρ ρ t u dx, I φ ρ ρ t ρ u dx. 6 R d R d In the following we will denote with f the Hessian with respect to x of a scalar function f = fx,.... Moreover, for any matrix A R d d, we define A tra A = d i,j=1 A ij. Let us compute I 1 by using 3 and integrating by parts: I 1 = φ ρ ρ ρ u dx R d = φ ρ ρ 4 + φ ρ ρ ρ ρu dx. R d Before computing I, let us first deal with the term ρ t ρ: u t ρ = 1 u div u ρ = ρ + ρ log u = ρ ρ V, 11

12 ρ t ρ = ρ ρ ρ ρ V ρ V ρ, and by writing ρ ρ = div ρ ρ ρ we get ρ t ρ = div ρ ρ ρ ρ ρ V ρ V ρ. Therefore I becomes I = φ ρdiv ρ ρu dx + φ ρ ρ + ρ ρ V u dx 7 R d R d + φ ρ ρ V ρ u dx. R d We integrate by parts the first integral on the right-hand side of 7, while using 19 to estimate the third integral: I φ ρ ρ ρ ρu dx R d + φ ρu ρ + ρ ρ u + u V dx R d + λ φ ρ ρ u dx. R d Since u is the steady state, then u + u V = ; moreover the last integral on the right-hand side of the above equation equals the entropy production. Therefore I φ ρ ρ ρ ρu dx + R d φ ρ ρ u dx + λp φ [u]. R d Summing I 1, I and applying 5 leads to d dt P φ[u] φ ρ ρ 4 + 4φ ρ ρ ρ ρ + φ ρ ρ u dx. + λp φ [u] R d By adding and subtracting φ ρ R d φ ρ ρ 4 u dx to the right-hand side of the above inequality we deduce d dt P φ[u] + R d φ ρ ρ + φ ρ ρ ρ R φ d ρ φ ρ φ ρ φ ρ u dx ρ 4 u dx + λp φ [u]. However, thanks to 1, φ ρ φ ρ 1 φ ρ = φ ρ, φ ρ 1

13 so d dt P φ[u] λp φ [u]. 8 Gronwall s lemma allows us to deduce that lim t P [ut] =. We would like to deduce that lim t H[ut] =, too, but the proof of this claim is quite technical; see [4, Sect. ] for details. Therefore let us just assume that lim t H[ut] = and go on with the proof... Let us integrate 8 in the time interval [t, and use the definition of P φ [u] and well as the relations lim t P [ut] = lim t H[ut] = : Applying Gronwall s Lemma leads to P φ [ut] = d dt H φ[ut] λh φ [ut]. 9 H φ [ut] H φ [u ]e λt t >. The above estimate, together with Csiszár-Kullback-Pinsker inequality [47] allows us to conclude the proof. u u L 1 R d H φ[u],. Convex Sobolev inequalities The Reader, lost in the computational details of the proof of Theorem.1, may have not noticed that in the previous section we actually proved something more than the exponential decay of the solution to the linear Fokker-Planck equation towards the steady state. As a matter of fact, we showed also the following Corollary.1 Convex Sobolev inequalities. Let φ, V satisfy 19 1, and let u be given by. Then u u dx 1 u φ u u λ R u d u u dx, 3 R d φ for all functions u : R d [, such that the above integrals are convergent. Proof. The left-hand side of 3 equals H φ [u], while the right-hand side equals P φ [ut] see eq. 4. Therefore eq. 3 can be obtained from 9 by replacing the solution ut of 17, 18 with a generic function u : R d [,. Ineq. 3 is actually a family of integral inequalities. By choosing φ is suitable ways we can obtain specific inequalities. There is, however, an intuitive argument which allows us to understand why that limit holds. Eq. 4 provides us with a handy expression for P φ [u]. If we assume that lim t ρt exists in some suitable sense, then it should be equal to some function ρ such that R d φ ρ ρ u dx =. Since φ > in [,, this implies that ρ must be constant, and given the fact that R d ρu dx = 1 this means that ρ = 1, i.e. lim t ut = u. Therefore, we expect that lim t H[ut] =. 13

14 ..1 Logarithmic Sobolev inequality Let φs = s log s s + 1, s >. Let u L 1 R d, u. Assume that udx = 1. Then R d 3 becomes the log-sobolev inequality: u log u dx u R u d λ R u d u dx. 31 Letting f = u/u, dµ = u dx into 31 yields the so-called Gaussian form of the log-sobolev inequality: 3 f f log R f d R dµ dµ f dµ. 3 λ d R d We point out that we added the factor 1/ f dµ inside the logarithm in order to get rid R d of the constraint f dµ = 1. Moreover, the Reader should notice an interesting fact: the constant /λ inside 3 does not depend on the dimension d of the space. However, the measure dµ depends on d through its normalization factor. If V x = x / and therefore λ = 1, then the log-sobolev inequality can be rewritten in another, more explicit form. In fact, in this case u = π d/ e λ x /, u log u dx = u log udx + d Rd R u d R logπ + x udx, d u R u d u dx = u u log u u dx R d u u = u dx u log u dx + u log u dx R d R d R d = u dx + log u + log u u dx, R d R d which implies u log udx + d R logπ + d u dx. R d However, since x we conclude R d u x log u 1 log u log u 1 log u = d, dx u log udx + d R logπ + d u dx. 33 d R d Ineq. 33 constitutes the so-called Euclidean form of the log-sobolev inequality. 3 The reason why it is called Gaussian form probably lies in the fact that the simplest possible choice for potential, i.e. V x = x /, leads to the Gaussian measure dµ = π d/ e x / dx. 14

15 .. Weighted Poincaré inequality What happens if we choose φs = s 1 in 3? We obtain R d u u 1 u dx 1 λ u R u d u dx. However, since f = u/u and R d u dx = 1 = R d fu dx = R d fu dx it follows f u dx fu dx 1 f u dx, 34 R d R λ d R d which is known as the weighted Poincaré inequality...3 Beckner inequality More in general, what if we choose φs = s α 1 in 3 with α 1, ]? With the same procedure as before we find the Beckner inequalities: α 1 f α u dx fu dx α f α f u dx. 35 α 1 R d R λ d R d Clearly by choosing α = we recover 34. On the other hand, by taking the limit α 1 in 35 4 we obtain the log-sobolev inequality The heat equation: convergence to the self-similar solution I bet the Reader has seen this thing before: u t = u t >, u = u in R d. 36 It s the heat equation of course. We take the initial datum u to be nonnegative of course and with unit mass. From the well-known explicit expression for u: it follows immediately that ux, t = 4πt d/ R d e x y / u ydy, t >, ut L 1 R d = 1, ut L R d 4πt d/, and therefore ut log utdx ut log ut L R dx d log4πt as t. d R d R d 4 To take the limit in the right-hand side of 35, one can use e.g. l Hopital theorem. 15

16 So, the Boltzmann entropy of u tends to as t. In particular, the entropy method cannot be applied to determine a convergence rate for the solution towards the steady state this is reasonable, since the steady state is, which does not have unit mass. However, we can use the entropy to study the so-called intermediate asymptotic of the equation, that is, the rate of convergence of the solution towards the self-similar solution Ux, t = πt + 1 d/ exp We are going to show the following x t + 1, x R d, t >. 37 Theorem. Relaxation to self-similarity. Let u : R d [, such that u R d dx = 1, x u R d xdx <, u log udx <. Let u be the solution to 36, U be given by 37, R d H[u] be the Boltzmann entropy. Then H[u ] ut Ut L 1 R d, t >. 38 t + 1 Proof. Let us do the following rescaling: y = x t + 1, s = 1 logt + 1, vy, s = eds ue s y, 1 es 1. As a consequence, v satisfies v s = div y y v + yv s >, v = u in R d. 39 Eq. 39 is a linear Fokker-Planck with quadratic potential V y = y /. The only steady state of 39 is the Gaussian Theorem.1 implies that v y = π d/ e y / = t + 1 /d Ux, t. vs v L 1 R d e s H[u ], s >. It is time to go back to the original variables x, t. Since vs v L 1 R d = ut Ut L 1 R d, e s = t + 1 1/, ineq. 38 follows. This finishes the proof..4 Linear Fokker-Planck equation: generalizations. The results presented in this Section about the long-time behaviour of the solution to the linear Fokker-Planck equation can be generalized in several ways. Here we present three possible generalizations: Fokker-Planck equation with variable diffusion, non-symmetric Fokker-Planck equation, degenerate Fokker-Planck equation. 16

17 .4.1 Fokker-Planck equation with variable diffusion Let us consider an equation of this kind: u t = div Dx u + u V t >, u = u in R d, 4 where D : R d, is a smooth function. We assume u : R d, is an L 1 R d function such that u R d dx = 1, while e V L 1 R d. Moreover, we assume that 1 d 1 4 D D D + 1 D D V I + D V + 1 D V + V D D λi in R d. 41 The steady state of 4 is unique and given by. The entropy H φ is again given by where φ :, R satisfies 1. Theorem.3 Long-time behaviour of 4. If the above-stated assumptions hold, and if H φ [u ] <, then any smooth solution ut to 4 satisfies H φ [ut] H φ [u ]e λt t >. 4 Furthermore, the following convex Sobolev inequality holds: H φ [u] 1 u u Dxdx. 43 λ R d φ u Hints of the proof. Let ρ = u/u, so that we can rewrite 4 as u t = div Du ρ. The first time derivative of the entropy reads as d dt H φ[ut] = φ ρ ρ u Dxdx. R d The second time derivative of H φ [u] can be estimated as d dt H φ[ut] trabu dx + λ φ ρ ρ u Dxdx, R d R d where φ A = ρ φ ρ φ ρ φ ρ and B is a suitable matrix depending on D, ρ and their derivatives up to order. By using Sylvester s criterion it is straightforward to see that 1, 41 imply the positive semidefiniteness of A, B, respectively actually, constraints φ >, 1/φ and 41 are equivalent to the nonnegativity of deta, detb. As a consequence trab and therefore d dt H φ[ut] λ φ ρ ρ u Dxdx = λ d R dt H φ[ut]. d Integrating the above inequality in the time interval t, and exploiting the relations d lim t H φ [ut] =, lim t H dt φ[ut] = again, the second limit follows straightforwardly from Gronwall s lemma, while the first one is more difficult to prove lead to d H dt φ[ut] + λh φ [ut], which imply both 4 and 43. This finishes the proof. 17 u

18 .4. Non-symmetric Fokker-Planck equation What happens when we consider a Fokker-Planck equation with a nonconstant diffusion coefficient AND a non-conservative force? I m talking about something like this: u t = div Dx u + uf t >, u = u in R d, 44 with D a positive smooth function. The idea to deal with an equation like that is to decompose the force F as sum of a gradient term an a perturbation, i.e. F = V + F, div DxF u = in, t >, 45 where u is given again by. To this decomposition there corresponds a splitting of the Fokker-Planck operator L into a symmetric part L s [u] and a skew-symmetric part L ss [u], defined as L s [u] = div Dxu u, L ss [u] = div DxF u. u It s clear that L s [u ] = L ss [u ] =, right? Let us verify that L s, L ss are symmetric and skew-symmetric with respect to L u 1 dx, respectively. We start with L s. For arbitrary u, v it follows L s [u], v L u 1 dx = L s [u]vu 1 dx = u Dx u v dx R d R u d u which is symmetric in u, v. Therefore L s is symmetric in L u 1 dx. Now let us deal with L ss. Eq. 45 implies that L ss [u] = DxF u u/u, thus L ss [u], v L u 1 dx + L ss[v], u L u 1 dx = R d DxF u uv u dx =, where the last equality follows from an integration by parts and 45. Therefore L ss is skew-symmetric in L u 1 dx. The decomposition L = L s + L ss helps a lot in the proof of Theorem.4 Long-time behaviour of 44. Under the above-stated assumptions and the hypothesis that 41 holds with V replaced by V F, any smooth solution ut to 44 having finite initial entropy H φ [u ] < satisfies H φ [ut] H φ [u ]e λt t >. Hints of the proof. Let ρ = u/u. The first time derivative of H φ [ut] reads as d dt H φ[ut] = φ ρ ρ u Dxdx + φ ρdiv DF udx. R d R d The second integral on the right-hand side of the above equation is actually zero. In fact, φ ρdiv DF udx = R d φ ρdxf u ρ dx = R d DxF u φρ dx R d 18

19 which vanishes after an integration by parts since 45 holds. So, it seems that the nonsymmetric perturbation does not change the form of the entropy dissipation. However, F plays a role in the computation of the second time derivative of H φ [ut]. Such a computation is similar to the one carried out in the proof of Thr..1, but more involved, and will not be presented here..4.3 Degenerate Fokker-Planck equation We present here a class of Fokker-Planck equations whose diffusion coefficient D is a possibly singular matrix: u t = div D u + ucx t >, u = u in R d. 46 We assume that D R d d is constant and positive semidefinite, while C R d d is a constant matrix. Since D has not full rank, the entropy production can vanish for functions other than the steady state, and the second time derivative of the entropy might not have a constant sign. The idea to deal with 46 is to employ a modified entropy; see [3] for details. We assume that: {v R d v is an eigenvector of C } kerd =, 47 all eigenvalues of C have positive real part. 48 Assumption 47 is a technical hypothesis which ensures existence of smooth, positive solutions to 46, provided that the initial datum u is positive and L 1. Assumption 48 implies the existence of a confinement potential. The steady state u is given by u x = e x Kx/ R d e y Ky/ dy x R d, where K R d d is the unique symmetric and positive definite solution to the Lyapounov equation CK + KC = D. We decompose the Fokker-Planck operator L as L = L s +L ss with L s [u] = div u D ρ and L ss [u] = div u R ρ, where R = 1CK KC. It is possible to show that L s, L ss are symmetric and skew-symmetric in L u 1 dx, respectively. Furthermore, let us define µ = min{rλ λ is an eigenvalue of C}. Theorem.5 Long-time behaviour for 46. Assume the same hypothesis of Thr..3. Moreover assume that 47, 48 hold. Let ut be the smooth solution to 46. i If for all eigenvalues λ of C such that Rλ = µ it holds geometric multiplicity of λ = algebraic multiplicity of λ 19

20 then there exists a constant κ > such that H φ [ut] κh φ [u ]e µt t >. ii If there exists an eigenvalue λ of C such that Rλ = µ and geometric multiplicity of λ algebraic multiplicity of λ then for every ε > there exists a constant κ ε > such that H φ [ut] κ ε H φ [u ]e µ εt t >. Sketch of the proof. We only show i. It is possible to prove that a symmetric, positive definite matrix D R d d exists such that KC K 1 D + D KC K 1 µd. Let us define the functional P [u] = φ ρ ρ D ρ u dx. R d Since D is positive definite, we can find a constant η > such that D ηd; therefore P [ut] ηp [ut] = d H dt φ[ut]. So, if we can find a suitable upper bound for P [ut], then we are done. It is possible to see that d dt P [u] = φ ρ ρ D RK 1 D + D K 1 D + R ρ u dx R d trabu dx R d where φ A = ρ φ ρ φ ρ φ ρ and B is a suitable matrix depending on D, ρ and their derivatives up to order. Again, both the matrices A, B are positive semidefinite, so trab. Furthermore, it holds D RK 1 D + D K 1 D + R = KC K 1 D + D KC K 1 µd. This leads to d dt P [ut] µp [ut] P [ut] P [uδ]e µt δ t > δ.

21 A suitable convex Sobolec inequality implies H φ [ut] 1 λ P P [ut] 1 λ P P [uδ]e µt δ. We wish to set δ = in the above inequality, but if we do it, we get an estimate for H φ [ut] depending on P [u ], which is not optimal. However, from [3, Thr. 4.8] it follows that for some c 1, c >. It follows P [ut] c 1 t 1+c H φ [u ] t >, H φ [ut] e µt δ λ P c 1 δ 1+c H φ [u ] = cδe µt H φ [u ] t > δ 49 with cδ = eµδ λ P c 1 δ 1+c. Estimate 49 only holds for t > δ; however, since H φ [ut] H φ [u ] and e µt e µδ for t δ, we conclude that H φ [ut] κe µt H φ [u ] t > for some positive constant κ. This finishes the proof..5 Nonlinear Fokker-Planck equations The Bakry-Emery method, applied successfully to the linear Fokker-Planck equation, can be extended to the case of nonlinear Fokker-Planck equations. We consider here equations of the form u t = div fu + u V t >, u = u in, 5 where R d is a bounded and convex domain. We consider no-flux boundary conditions: fu + u V ν = on, t >. 51 Equations similar to 5 are employed e.g. in the study of porous-media flow, charge transport in semiconductors and population dynamics. While 5 has been studied for more general functions fu and V x, we will consider here for the sake of simplicity the case of power functions, that is V x = λ x x, fu = u m u, m 1 1, m 1. 5 d Such an ansatz leads e.g. to the porous medium equation [48]. The cases m < 1 and m > 1 are referred to as slow diffusion and fast diffusion, respectively. The steady state is unique and given by the Barenblatt profile: u x = N 1/m 1 λm 1 m x,

22 where y + max{y, }, y R, while N > is a constant which can be determined by imposing the constraint of mass conservation: u dx = u dx. At this point, we would like to define the relative entropy as in, but we run into trouble: the steady state given by 53 has compact support. Therefore, we define the relative entropy H[u u ] as u m 1 H[u u ] = H[u] H[u ], H[u] = u m 1 + λ x dx. 54 The following result holds: Theorem.6. Let 5, 53 hold, and let u L 1 be nonnegative such that H[u ] <. If ut is a solution of 5, 51, then where C > is a suitable positive constant. ut u L 1 Ce λt t >, 55 Hints of the proof. We are not going to present all the details of the proof, since the ideas are basically the same as in the proof of Thr..1. For the complete picture see [3, pp ]. We define the so-called entropy variable µ = mu m 1 /m 1 + λ x /. We point out that µ is simply the partial derivative of the integrand in 54 with respect to u. As a consequence 5 can be rewritten in gradient-flow form as u t = div u µ in, t >. 56 Moreover, 51 implies ν µ = on, t >. Therefore by testing 56 against µ we obtain d dt H[ut u ] = u µ dx, t >. 57 The next step in the proof is to compute the second time derivative of H[ut u ]. We omit the lengthy computations, which are quite similar philosophically speaking to the ones carried out to show 8, and simply state that the following inequality holds: d dt H[ut u ] + λ d dt H[ut u ] u m µ ν dσ. 58 So, it seems that working with a proper subdomain of R d is not without consequences: now we have to deal with a surface integral. This is actually the main difference between this proof and the proof of Thr..1. Is the right-hand side of 58 nonnegative? Of course it is. To see this, we just need to apply the following lemma, whose proof can be found e.g. in [6, Lemma 5.1]: Lemma.1. Let R d d 1 be a convex domain with C boundary and let µ H 3 satisfy µ ν = on. Then µ ν on.

23 Now we just have to integrate 58 in the time interval t, and exploit the facts d that lim t H[ut u dt ] = and lim t H[ut u ] =. While the second limit follows directly from 58 through a Gronwall argument, the proof of the first limit is more technical and will be skipped in these lecture notes. The curious Reader can find more details in [3, p. 35]. Therefore we are left with which implies, thanks to Gronwall s Lemma d dt H[ut u ] + λh[ut u ], H[ut u ] e λt H[u u ] t >. 59 Now we wish to apply Csiszár-Kullback-Pinsker inequality, but there is a problem: the steady state u might vanish in a positive measure set. Therefore we cannot apply the aforementioned inequality in a straightforward way; however, it can be showed see [38, pp. 3-31] for details that a similar result holds: u u L 1 C H[u u ], for some constant C >. This finishes the proof. It is now natural, given what we have seen in the linear case, to ask the question: is the nonlinear Fokker-Planck equation with f, V given by 5 related to some functional inequality? The answer is yes of course: Proposition.1 Gagliardo-Nirenberg inequality one of many. Let either d =, p > 1 or d 3, 1 < p < d/d. Moreover let q = p + 1/p 1, q1 p θ/ q d C = πd q where Γ is the Euler Gamma function. Then θ/ θ/ Γq, θ = Γq d/ d1 1/p d + d p, v L p R d C v θ L R d v 1 θ L p+1 R d, v H1 R d L p+1 R d. 6 Furthermore, if vx = N + x x 1/1 p + for any N >, x R d, then equality holds in 6. In particular, the constant C is optimal. 3

24 3 Cross-Diffusion PDEs Many physical systems coming from the applied sciences e.g. physics, biology, chemistry can be modeled through a set of reaction-diffusion PDEs with cross-diffusion: u t = div Au u + Du φ + fu, t >, u = u in. 61 In 61 R d d 1 is a bounded domain with smooth boundary, the vector-valued function u :, D R n is the unknown of the system, tipically representing densities or concentrations of a multicomponent physical system, D is the domain of the physical variables, A : D R n n is the diffusion matrix, D : D R n n is the drift matrix, φ : R is a potential, and f : D R n is the reaction term. The notation div Au u is to be understood as d div Au u i = A ij u u j i = 1,..., n. x k x k k=1 j=1 We impose homogeneous Neumann boundary conditions, which describe the conservation of the species: Au u + Du φ ν = t >, on Examples of cross-diffusion PDEs. Let us see a few examples of cross-diffusion PDEs coming from the applied sciences Population dynamics: the SKT model. Shigesada et alii proposed in [4] a famous model for a system of two populations which share the same environment and are subject to intra-specific and inter-specific population pressures. The evolution of the densities u 1, u of the populations species is described by 61, 6 with a1 + a Au = 11 u 1 + a 1 u a 1 u 1 u1, D =, 63 a 1 u a + a 1 u 1 + a u u while the reaction term is of Lotka-Volterra type: b1 b fu = 11 u 1 b 1 u u b b 1 u 1 b u u In 63, 64, the parameters a ij, b i are nonnegative. The potential φ describes inhomogeneities of the environment e.g. if it is favorable to the species or not. Under certain assumptions on the coefficients roughly speaking, a 1, a should be small compared to a 1, a 1, eqs. 61 admit nonconstant steady state, which biologically represent pattern formation. 4

25 We point out that the diffusion matrix A given by 63 is in general not symmetric nor positive semidefinite; it has, however, positive eigenvalues. Due to this fact, the derivation of a-priori estimates is tricky, and the global existence of solutions has been an open problem for decades, until the s [13, 14, 5] Ion transport. If you wish to describe the transport of ions in biological cells or in multicomponent fluid mixtures, you would probably use the Poisson-Nernst-Planck equations for the ion concentrations and the electric potential. The derivation of the equations works fine under the assumption that the concentrations levels are far from the saturation points no volumefilling case; however, the concentrations are allowed to saturate which may happen in reality, different equations are to be employed. An alternative set of equations is constituted by 61 with A ij u = D i u i + u n δ ij, D ij u = u i u n δ ij i, j = 1,..., n, 65 where u n 1 n 1 k=1 u k and D 1,..., D n are positive constants. The functions u 1,..., u n 1 are the ion concentrations, u n is the solvent concentration, while φ is the electric potential, which is either a solution to the Poisson equation or a given function. Again, A is not symmetric nor positive semidefinite. The upper bound n 1 i=1 u i < 1 should hold for consistency with the physics volume-filling case, but in general no maximum principle is available for system of PDEs with cross-diffusion. Derivation of suitable a-priori estimates is also challenging. Global existence of bounded weak solutions to 61, 65 was proved in the two species case n = in [7] and for arbitrary n in absence of potential in [31, 5] Tumor-growth models. There are three stages in the process of tumor growth. The first stage is the avascular growth: the tumor cells proliferate by relying on the body s healthy blood vessels for oxygen and nutritional substances supply. However, as the tumor grows bigger, the amount of available oxygen at its center decreases, which means that the tumor cannot grow in size more than a millimeter or so without its own blood supply. The second phase of the tumor growth is the vascular growth: the tumor starts developing an independent blood supply by stimulating the formation of new blood vessels inside the tumor. The third and final stage of tumor growth is the metastatic phase, during which the tumor cells are able to escape from the tumor via the circulatory system and lead to the formations of other tumors in the body. Avascular tumor growth can be described by fluid-dynamic models. For example, in [8] a continuous model is derived for avascular tumor growth in one space dimension, under the assumption that the tumor-host environment consists of tumor cells, the extracellular matrix ECM, which provides support for the tumor cells, and water. The volume fractions of tumor cells, ECM and water are denoted with u 1, u, u 3 respectively and sum up to 5

26 one: u 1 + u + u 3 = 1 volume-filling case. From the mass and momentum balance equations for the system a set of PDEs is derived, which has the structure 61 with A, f given by u1 1 u Au = 1 βθu 1 u βu 1 u 1 + θu 1, 66 fu = u 1 u + βθ1 u u βu 1 u 1 + θu 1 α1 u 1 1 u 1 u α u 1 α 3 u 1 u 1 u 1 u, 67 where β >, θ >, α 1, α, α 3 are parameters. There is no potential. Again, the diffusion matrix is in general neither symmetric not positive definite. Relation u 1 + u 1 should be fulfilled for consistency with the physics Multicomponent fluid mixtures. The well-known Maxwell-Stefan equations describe the evolution of a multicomponent gaseous mixture under some suitable assumptions: ideal gas, zero baricentric velocity, isobaric and isothermal conditions, same molar mass for all components. They were suggested by J. C. Maxwell in 1866 for dilute gases and by J. Stefan in 1871 for fluids. The Maxwell- Stefan equations are constituted by the mass and reduced force balance equations for the mixture and read as t u i + div J i = f i u, u i = k=1 k i u i J k u k J i D ij, i = 1,..., n. 68 Maxwell-Stefan s model represents a generalization of Fick s law: while in the latter the flux J i depends linearly only on u i, in the former u i depends on all the fluxes J 1,..., J n. The Maxwell-Stefan equations can predict phenomena that are beyond the reach of Fick s law, i.e. osmotic diffusion in multicomponent mixtures. The mathematical difficulties that one has to deal with when solving 68 are three. First, the matrix associated with the linear relations J 1,..., J n u 1,..., u n is singular, and therefore expressing the fluxes J 1,..., J n in terms of the concentrations gradients u 1,..., u n is not straightforward. Second, the diffusion matrix Au that one obtains after carrying out the aforementioned inversion process is in general not symmetric nor positive semidefinite, Third, nonnegativity and boundedness of u 1,..., u n have to be proved for consistency with the physics, and this is far from simple for the lack of general minimum/maximum principles for systems of cross-diffusion PDEs. 3. Derivation of some cross-diffusion models. At this point, the curious Reader might ask: but how to derive the cross-diffusion presented above from other models? What will now follow aims at partially answering this question. In fact, cross-diffusion PDEs can be obtained by performing all kind of nefarious activities 6

27 on other, more basic models, like for example random walk lattices, fluid models, kinetic equations, stochastic PDEs, et cetera. We are going to focus on two methods, which involve taking suitable limits in space-discrete random walk equations and continuous fluid models, repsectively Derivation from random-walk lattice models. Let us consider a one-dimensional lattice, whose cell j Z has a uniform size h > and midpoint x j, so that x j x j 1 = h, for j Z. Moreover, let u 1,..., u n population densities defined on the lattice {x j : j Z}, i.e. u i x j, t represents the proportion of population i in the cell j at time t. The species can move from cell j to one of the neighbouring cell j ± 1 with transition rates T j± i. The densities u 1,..., u n evolve according to the following master equation: t u i x j = T j 1,+ i u i x j 1 + T j+1, i u i x j+1 T j,+ i + T j, i u i x j, 69 for i = 1,..., n, j Z, t >. How to model the rates T j,± i? The basic idea is that, if the departure cell is more crowded than the arrival cell, then the tendency of the species to leave the cell is higher. Therefore, a possible expression for the rate is T j,± i = σ hp i u 1 x i,..., u n x i q i u n+1 x j±1. 7 In the above equality, σ h > is a suitable scaling constant, u n+1 1 n i=1 u i is the volume fraction unoccupied by the species, p i, q i are suitable functions. Expression p i u 1 x i,..., u n x i, q i u n+1 x j±1 measures the tendency of species i to leave cell j, while q i u n+1 x j±1 represents a damping of this tendency due to the crowding of the two neighbouring cells j + 1, j 1. We are going to see that, if σ h is chosen wisely, then 69, 7 converge in the limit h to the cross diffusion system t u = x Au x u x R, t >, 71 A ij u = δ ij p i uq i u n+1 + u i p i uq iu n+1 + u i q i u n+1 p i u u j = q i u n+1 ui p i u i, j = 1,..., n. 7 u j q i u n+1 This argument is the same of [5, Appendix]. It is convenient to introduce the following abbreviations: p j i = p iu 1 x j,..., u n x j, q j i = q iu n+1 x j, k p j i = p i u k u 1 x j,..., u n x j, q j i = q iu n+1 x j. Thus, we can rewrite the master equation as σ 1 t u j i = qj i pj 1 i u j 1 i + p j+1 i u j+1 i 7 p j i uj i qj+1 i + q j 1 i. 73

28 Set D = x. We compute the Taylor expansions of p i and q i i = 1,..., n and replace u j±1 k u j k by the Taylor expansion ±hduj k + 1 h D u j i + Oh3. Then, collecting all terms up to order Oh, we arrive at p j±1 i = p j i + h k p j i Duj k + h k p j i D u j k + klp j i Duj k Duj l + Oh 3, q j±1 i k=1 k=1 k,l=1 = q j i ± h pj i Duj n+1 + h q j i D u j n+1 + q j i Duj n+1 + Oh 3 = q j i h qj i Du j k + h q j i D u j k + q j i Du j k Duj l + Oh 3. k=1 k=1 In the last step, we have used u n+1 = 1 n k=1 u k. We insert these expressions into 73 and rearrange the terms. It turns out that the terms of order O1 and Oh cancel, and we end up with σ 1 k,l=1 h t u j i = D u j k qj i pj i δ ik + q j i uj i kp j i + pj i uj i qj i k=1 + Du j k Duj l qj i kp j i δ il + q j i uj i klp j i pj i uj i q j i. k,l=1 We choose σ = h and pass to the limit h : t u i = D p i u k q i p i δ ik + q i u i + p i u i q i u k k=1 p i p i + Du k Du l q i δ il + q i u i p i u i q i. u k u k u l k,l=1 A lenghty but straightforward computation shows that the last sum equals k=1 p i Du k D q i p i δ ik + q i u i + p i u i q i, u k and we end up with t u i = D k=1 p i Du k q i p i δ ik + q i u i + p i u i q i, u k which is identical to 71, 7. Volume-filling and non-volume-filling models. If we do not want to incorporate 8

29 volume-filling effects in the model, then q i 1 for i = 1,..., n is the right choice. Then A ij u = u j u i p i u and therefore 71 becomes t u i = u i p i u i = 1,..., n. Choosing n =, p i u = a i +a i1 u 1 +a i u i = 1, yields the SKT model 61, 63 with no potential. So we can state that the basic assumption of the SKT model is the linear dipendence of the transition rates on the species densities. If we want to incorporate volume-filling effects, then q i must be nonconstant and vanish at zero. A relatively simple model of this kind can be obtained by setting p i 1, q i s = D i s for i = 1,..., n, s >, where D 1,..., D n are positive constants. The equations we obtain constitute the ion transport model 61, 65 without potential. 3.. Derivation from fluid models. Let us consider a fluid of n components. The mass and momentum balance equations for the fluid read as t u i + div u i v i = r i, 74 t u i v i + div u i v i v i S i = p u i + u i b i + f i 75 for i = 1,..., n. Here u i, v i are the mass density and drift velocity of species i, respectively, r i is the mass production rate e.g. die to chemical reactions, S i is the stress tensor, p is the phase pressure, f i the momentum production rate. The sum p u i + u i b i represents the force acting on species i: p u i is the interphase force coming from the phase pressure, while b i u i is the body force. We are going to derive a cross-diffusion model from 74, 75. We impose the following hypothesis: 1. the total mass density is constant: n i=1 u i = 1;. the baricentric velocity is zero: n i=1 u iv i = ; 3. all species have the same molar masses; 4. the total body force is zero: n i=1 b iu i = ; 5. the stress S i is made up by the contributions of the phase pressure p and the partial pressures P i = P i u: S i = u i p + P i ui; 6. the partial pressure of the n th components vanishes: P n u ; 7. the momentum production is proportional to the velocity differences: f i = k ij u i u j v j v i i = 1,..., n, j=1 with symmetric positive coefficients k ij = k ji. 9