Semidiscretization and long-time asymptotics of nonlinear diffusion equations

SEMIDISCRETIZATION AND LONG-TIME ASYMPTOTICS OF NONLINEAR DIFFUSION EQUATIONS JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI Abstract. We review several results concerning the long time asymptotics of nonlinear diffusion models based on entropy and mass transport methods. Semidiscretization of these nonlinear diffusion models are proposed and their numerical properties analysed. We demonstrate the long time asymptotic results by numerical simulation and we discuss several open problems based on these numerical results. We show that for general nonlinear diffusion equations the long-time asymptotics can be characterized in terms of fixed points of certain maps which are contractions for the euclidean Wasserstein distance. In fact, we propose a new scaling for which we can prove that this family of fixed points converges to the Barenblatt solution for perturbations of homogeneous nonlinearities for values close to zero. 1. Introduction The fine description of long-time asymptotics for nonlinear diffusion equations has attracted the attention of many researchers in the last few years. This revival has been devoted to new ideas brought up in the subject from different communities: the entropy approach from kinetic theory having its roots in the famous H-theorem for the Boltzmann equation [15, 22], the optimal mass transport theory giving a geometric point of view of these equations based on suitable metrics in the space of probability measures [24, 17, 1, 2] and variational techniques related to new Gagliardo-Nirenberg inequalities [18]. This review is intended to show some recent topics of research concerning long-time asymptotics of equations of the form ∂u ∂t = div(u∇V+∇f(u)), x ∈RN, t > 0,(1.1) in which V(x)∈W1,1 loc (RN) is positive and f(u) verifies (F1) fbelongs to C([0,+∞)) ∩C1((0,+∞)), (F2) fstrictly increasing such that f(0) = 0 and f0(u)>0 for all u > 0. 1 2 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI This family of equations includes nonlinear Fokker-Planck equations where V(x) is assumed to be confining (see next section for details) and general nonlinear diffusion equations where V(x) = 0. The Cauchy problem for general nonlinear diffusion equations is well-posed by classical results [8, 7] and for the nonlinear Fokker-Planck equations we refer to [14] and the references therein. In fact, nonlinear diffusion equations without confinement V(x) = 0 are expected to diffuse as t→ ∞, and thus, solutions should vanish as t→ ∞ with an expansion of their support or their tails depending whether the diffusion is slow or fast. On the contrary, nonlinear Fokker-Planck equations are expected to stabilize towards an steady state defined by setting the flux to zero in (1.1). The rigorous proof of this stabilization was done in [14] in L1 by using an entropy-entropy production approach. The stationary state was characterized as the unique minimizer in the space of integrable functions with given mass of a suitable functional that we call entropy. This entropy functional was then proved to be a Lyapunov functional for the equation and thus, the study of its evolution gave the desired convergence rate. We refer to [15, 14, 22] for details. Moreover, generalized Log-Sobolev inequalities were obtained in [14] relating the entropy to the entropy production. Both equations share remarkable properties with respect to Wasserstein distances for probability measures. These distances are well-known in the probability community since probability measures spaces with suitable bounded moments endowed with these distances become complete metric spaces. The remarkable property of the family of equations (1.1) is that assuming that V(x) is convex, their flow map is a global contraction for the 2-Wasserstein distance [24, 17, 1, 2]. Moreover, in the 1D case equations (1.1) under the convexity assumption on V(x), are contractions for all Wasserstein distances [16, 12]. Recently, J. L. V´azquez [34] has shown a very nice counterexample proving that this result is not true for large index in the Wasserstein distance in any other dimension. However, it is an open problem to show if it is asymptotically, for large times, true. Nonlinear Fokker-Planck equations with confining potential V(x) = 1 2|x|2 and homogeneous nonlinearity f(u) = umare equivalent through a explicit change of variables to the nonlinear diffusion equation with V(x) = 0 and f(u) = umand therefore, the study of their long-time behaviour is equivalent too. In fact, the stabilization towards equilibrium of the nonlinear Fokker-Planck equation translates into a self-similar behavior as t→ ∞ for the nonlinear diffusion equation, in the sense that all solutions resemble a self-similar profile as t→ ∞. This self-similar profile is the well-known Barenblatt profile for homogenous nonlinear diffusions. NONLINEAR DIFFUSION EQUATIONS 3 We will show in section 2 that the semidiscretization of equations (1.1) based on the implicit Euler scheme preserves the non-increasing behavior of the entropy functional. Moreover, in section 2.2 we will analyse how to discretize nonlinear diffusion Fokker-Planck equations and the stability problems arising due to the stabilization to equilibria in which the diffusion is degenerate. This numerical scheme enjoying entropy decay does not preserve theoretically the contractivity of Wasserstein metrics at a discrete level. Let us mention that 1D schemes have recently been proposed in [19] preserving the contractivity of metrics. In section 3, the contractivity of Wasserstein distances in one space dimension for general nonlinear diffusion equations is used in the case of asymptotically homogeneous nonlinearities to obtain a bound on the expansion rate of the support of solutions. They will behave basically like the Barenblatt profile corresponding to the exponent to which the nonlinearity resembles for small values of u. This result is already known since the work of J.L. V´azquez [31] but here we will give a original proof related to the recent result in [12]. Although several qualitative properties of the solutions for general nonlinear diffusion equations have been obtained [25], there is no result concerning asymptotic profiles of general diffusion equations except in the case of power-like behavior for small values of u. One of the objectives of this review is to summarize the results recently obtained in [11], in which asymptotic profiles for general nonlinear diffusion equations are obtained characterized by being fixed points of suitable contractions in Wasserstein distances. This result will be explained in detailed in section 4. Moreover, we will try to elucidate numerically the behavior of these asymptotic profiles as t→ ∞ using the entropy preserving scheme introduced before. Finally, we propose in section 5 an alternative scaling to the one introduced in [11] to characterize again asymptotic profiles by means of a sequence of fixed points whose convergence for large times can be studied at least in the case of asymptotic homogeneity. In order to do that continuity arguments with respect to the nonlinearity developed in [7] become relevant. 2. Semidiscretization of nonlinear diffusion equations We consider the general non-linear Fokker-Planck equation ∂u ∂t = div(u∇V+∇f(u)), x ∈RN, t > 0,(2.1) u(x, 0) = uI(x), x ∈RN,(2.2) 4 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI where 0 ≤uI∈L1(RN)∩L∞(RN). A potential V≥0 will be called confining if it satisfies (V1) Vis uniformly convex function and V(x)−→ +∞as |x| −→ +∞. Much is already known for the problem (2.1)-(2.2) about existence and uniqueness (see [14] and references therein). We recall that Definition 1. Assume (V1), (F1) and (F2). A generalized solution of (2.1) is a function u:RN×R→Rsuch that (i) u∈L∞(RN×(0, T)) for all T > 0, (ii) ∇f(u)∈L1 loc(RN×R:RN), (iii) for all ψ∈C∞ o(RN×R)compactly supported it holds −ZRN uI(x)ψ(x, 0) dx −ZRN×Rµu∂ψ ∂t ¶(x, τ)dxdτ +ZRN×R (u∇V+∇f(u))(x, τ)∇ψ(x, τ)dxdτ = 0. Problem (2.1)-(2.2) has the mass conservation property, i.e., ku(·, t)kL1(RN)=kuI(·)kL1(RN) for all t > 0 and it’s also known that its solution uconverges exponentially fast to the stationary state u∞, solution of the problem u∞∇V+∇f(u∞) = 0,ku∞(·)kL1(RN)=kuI(·)kL1(RN).(2.3) The proof of this result is based on the so-called entropy dissipation method, in which the convergence towards equilibrium is concluded using the time monotonicity of the physical entropy E(u(t)) := ZRN [u(x, t)V(x) + φ(u(x, t))] dx, (2.4) where φis a strictly convex function solving the problem φ00(u) = f0(u) u, φ0(1) = 0, φ(0) = 0 (2.5) see [14, 15, 28] for details. As already mentioned in the introduction, in the special case V(x) = 1 2|x|2and f(u) = um, if N(m−1) + 2 >0, there exists a time-depending scaling which transforms equation (2.1) into the diffusion equation vt= ∆vm, x ∈RN, t > 0,(2.6) v(x, 0) = uI(x). This result allows us to translate any result on the asymptotic behaviour of equation (2.1), if V(x) = 1 2|x|2and f(u) = um, into results of asymptotic NONLINEAR DIFFUSION EQUATIONS 5 behaviour in time of equation (2.6). In fact it has been shown (see [15]) that the similarity solution of (2.6) B(|x|, t) = t−N λµC−(m−1) 2mλ |x|2t−2 λ¶1 m−1 + ,(2.7) where λ:= N(m−1) + 2, evaluated at the time t=1 λ, coincides with the unique compactly supported equilibrium state u∞of (2.1). In this work we consider a fully implicit Euler semidiscretization for equation (2.1) uk(x)−uk−1(x) ∆t= div(uk(x)∇V(x) + ∇f(uk(x))),(2.8) where uk(x) denotes the approximation u(x, k∆t), k∈N. The main formal properties of the above semidiscretization are worked out in the forthcoming paper [13] and summarized in the following proposition. Proposition 2. Let 0≤uI∈L1(RN)∩L∞(RN)and f,Vbe satisfy (V1), (F1) and (F2). If uIV∈L∞(RN), there exists a nonnegative weak solution uk∈L2(RN)∩L∞(RN)of (2.8) fulfilling ZRN ∇ψ· ∇f(uk)dx = =−RRNuk∇ψ· ∇V dx −1 ∆tRRNψ(uk−uk−1)dx, (2.9) for all ψ∈H1(RN)compactly supported. Moreover an appropriate interpolation of {uk}k∈Nconverges to a generalized solution of (2.1) when ∆t→0. The basic property of the above numerical scheme is the decay of the relative entropy E(u|u∞) := ZRN [φ(u)−φ(u∞)−φ0(u∞)(u−u∞)] dx. (2.10) Let us also remember that for general nonlinear Fokker-Planck equations the entropy and the relative entropy satisfy E(u)−E(u∞)≥E(u|u∞) being equal if and only if u∞is positive everywhere (see [14, Proposition 5]). Moreover the difference can be explicitly written as E(u)−E(u∞)−E(u|u∞) = ZRN [V(x) + φ0(u∞(x))](u−u∞)) dx. (2.11) 6 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI Lemma 3. Let uk,k∈N, weak solution of problem (2.9). Assuming that ZRN uI|∇V+φ00(uI)∇uI|2dx < +∞,(2.12) and that the potential V(x)is uniformly convex with convexity constant α, i.e., D2V(x)≥αIN, then E(uk)−E(u∞)≤(E(uI)−E(u∞))(1 + 2α∆t)−k, k ∈N.(2.13) Proof. Here we give a formal proof of this lemma, based on the generalized Log-Sobolev inequality (see [14, Theorem 17]). Let D(uk) := ZRN uk|∇V+φ00(uk)∇uk|2dx, be the entropy production for the functional E(uk|u∞) defined in (2.10). Then the generalized Log-Sobolev inequality asserts that E(uk|u∞)≤E(uk)−E(u∞)≤1 2αD(uk)∀k∈N.(2.14) Let ukthe solution of (2.8), with k∈N. Using the convexity of φ, it holds E(uk|u∞)≥ ≥ZRN φ0(uk+1)(uk−uk+1) + φ(uk+1)−φ(u∞)−φ0(u∞)(uk−u∞)dx =ZRN φ(uk+1)−φ(u∞)−φ0(u∞)(uk+1 −u∞) + +ZRN φ0(u∞)(uk+1 −uk) + φ0(uk+1)(uk−uk+1)dx =E(uk+1|u∞) + ZRN [φ0(u∞)−φ0(uk+1)](uk+1 −uk)) dx. NONLINEAR DIFFUSION EQUATIONS 7 Now, using (2.11) we get E(uk)−E(u∞)≥ ≥E(uk+1|u∞) + ZRN [φ0(u∞)−φ0(uk+1)](uk+1 −uk)) dx +ZRN [V(x) + φ0(u∞(x))](uk−u∞)) dx =E(uk+1|u∞)−ZRN [V(x) + φ0(uk+1)](uk+1 −uk)) dx +ZRN [V(x) + φ0(u∞(x))](uk−u∞)) dx +ZRN φ0(u∞)(uk+1 −uk)) dx +ZRN V(x)(uk+1 −uk)) dx =E(uk+1|u∞) + ZRN [V(x) + φ0(u∞(x))](uk+1 −u∞)) dx −ZRN [V(x) + φ0(uk+1)](uk+1 −uk)) dx =E(uk+1)−E(u∞)−ZRN [V(x) + φ0(uk+1)](uk+1 −uk)) dx. Therefore, using equation (2.8) and integrating by parts, we deduce ZRN [V(x) + φ0(uk+1)](uk+1 −uk)) dx = =−∆tZRN uk+1|∇V+φ00(uk+1)∇uk+1|2dx, and thus, E(uk)−E(u∞)≥E(uk+1)−E(u∞)+∆t D(uk+1). From inequality (2.14) it holds E(uk)−E(u∞)≥(1 + 2α∆t) (E(uk+1)−E(u∞)), and the theorem follows now recursively. ¤ Let us point out that previous lemma was already observed in the linear case in [3] with a proof simplified by the fact V(x) + φ0(u∞(x)) = Cfor all ∈RNfor linear diffusions. It is the discrete version of the exponential decay with rate 2αin the continuous case obtained in [14, 15]. The rigurous proof of this lemma is done by approximations of the nonlinear function in the same spirit as in [14]. In the case of general nonlinear diffusion equations we have also a decay estimate for the corresponding entropy. 8 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI Corollary 4. Let vkbe solution of (2.8) with V≡0and E(k) := ZRN φ(vk(x)) dx. (2.15) For all k∈Nit holds E(k)≥ E(k+ 1).(2.16) Proof. From equations (2.8) and (2.5) we get E(k)− E(k+ 1) = =ZRN φ(vk)−φ(vk+1)dx =ZRN φ0(vk+1)(vk−vk+1)dx =−∆tZRN φ0(vk+1)div(∇f(vk+1)) dx = ∆tZRN f0(vk+1)2 vk+1 |∇vk+1|2dx. ¤ Let us mention that an alternative semidiscretization derived from a discrete variational scheme based on the Wasserstein distance was introduced in the linear case in [20] and generalized to the nonlinear case in [1] and the references therein. This semidiscretization also verifies a decay of the entropy similar to Corollary 4. Let us show some numerical results related to problem (2.6) and (2.1) in the case f(u) := umfor some m > 1. 2.1. The porous medium equation. We introduce the fully discretization of equation (2.6) in a uniform grid using central finite differences in space to obtain: vk(i)−vk−1(i) ∆t=D+D−(f(vk(i))), k ∈N, i = 1, .., M, (2.17) where D+and D−are the standard forward and backward first order finite difference operators, defined for any discretized function (z(i))i=1,..,M as follows D+z(i) := z(i+ 1) −z(i) ∆x, i = 1, .., M −1, D−z(i) := z(i)−z(i−1) ∆x, i = 2, .., M. The resulting nonlinear system of equations is iteratively solved by Newton’s method at each time step. Time stepping is set to constant. NONLINEAR DIFFUSION EQUATIONS 9 (a) −8 −6 −4 −2 0 2 4 6 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 x v(x,t) 2 t=0 t=5⋅ 10−1 t=5 t=20 (b) 0 5 10 15 20 0.1 0.15 0.2 0.25 0.3 0.35 t Entropy Figure 1. Numerical results and entropy decay for (2.6) in case m= 2 with initial condition (2.18) (a) Time evolution of v(x, t), (b) Entropy evolution E(t). Figure 1 shows numerical results for f(v) := v2with initial data vI(x) =    2[(4−x2)−3.9 cos(π 4x)] k2[(4−x2)−3.9 cos(π 4x)]kL1(−2,2) if −2≤x≤2, 0 otherwise, (2.18) Let us point out that the expected Barenblatt asymptotic profile (2.7) is fixed by mass conservation ZR B(|x|, t)dx =ZR vI(x)dx. The results show the convergence to the selfsimilar profile given by the Barenblatt profile as t→ ∞ and the decreasing character of the entropy. Note that in this case the decay rate of the entropy is not exponential but rather algebraic. In fact, for the porous medium equation the entropy becomes the Lmof the solution that decays like m−1 mλdue to the L1-L∞ effect [4] which asserts kv(·, t)kL∞(RN)≤C t−N N(m−1)+2 kvIkL1(RN). In figure 2 the numerical approximations of the two dimensional porous medium equation with f(v) = v3together with the entropy decay are plotted. 16 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI (see [16, 12]) |inf{supp U1} − inf{supp U2}| ≤ W∞(U1, U2), |sup{supp U1} − sup{supp U2}| ≤ W∞(U1, U2).(3.6) Moreover, whenever U1has compact support, then W∞(U1, U2) is finite if and only if Vhas compact support. Let us consider for the moment the case of a homogeneous nonlinearity f(u) = um, which corresponds to the porous medium equation in (3.1). We are interested in computing the Wasserstein distance between any nonnegative solution with unit mass wof the rescaled equation ws= (yw + (wm)y)y, and the corresponding self–similar Barenblatt profile w∞with mass 1 defined in (2.7) written in similarity variables y, s, i. e. the stationary profile w∞(y) = µC−m−1 2m|y|2¶1 m−1 + ,(3.7) for some constant C. In order to study the evolution of such a quantity, because of all the previous remarks about the one–dimensional case, we set F(y, s) = Zy −∞ w(z, s)dz G(y, s) = Zy −∞ w∞(z)dz. Let F−1, G−1: (0,1) →Rbe the pseudo–inverses of Fand Grespectively. Then, F−1satisfies the following equation (see [16, 12]), ∂F −1 ∂s =−F−1−∂ ∂ρ "µ∂F−1 ∂ρ ¶−m#,(3.8) while G−1satisfies G−1+∂ ∂ρ "µ∂G−1 ∂ρ ¶−m#= 0.(3.9) Equations (3.8) and (3.9) provide a direct computation of the L2p–norms of the difference F−1−G−1, and therefore, an estimate of the 2p–Wasserstein distance between wand w∞, as shown in the paper [12]. As we will see later on, the diffusion term −∂ ∂ρ "µ∂F−1 ∂ρ ¶−m# in equation (3.8) is dissipative, while the term −F−1provides an exponential decay. More precisely, we have W2p(w(s), w∞)≤e−sW2p(wI, w∞). NONLINEAR DIFFUSION EQUATIONS 17 In the original variables, we have the following contraction property W2p(v(t), v∞(t)) ≤W2p(vI, v∞(0)), where v∞(t) = B(x, t+1 λ) is the Barenblatt profile in the original variables. Sending p→ ∞ yields contraction of the W∞and, thanks to (3.6), |inf{supp v(t)} − inf{supp v∞(t)}| ≤ W∞(vI, v∞(0)), |sup{supp v(t)} − sup{supp v∞(t)}| ≤ W∞(vI, v∞(0)).(3.10) In fact, last lines can even be improved to prove the finite speed of propagation without be assumed from the beginning. Since the W∞distance is contractive and v∞(t) has compact support at any time t≥0, taking compactly supported initial data vI, the quantity W∞(v(t), v∞(t)) stays finite and immediately it follows that v(t) has compact support. In [12] the above arguments are performed rigorously, by means of an approximation of the original Cauchy problem by an initial boundary value problem on a closed interval eventually tending to the whole real line. Our purpose is to generalize the previous approach to the perturbed case f(u) = um+ Ψ(u) according to assumption (F3). Before proceeding further, let us remark that these results were already proven in [31] by comparison arguments. The novelty here is the proof by completely different arguments that might be generalized to larger dimensions at least asymptotically. We shall perform the estimate of the Wasserstein distances directly on the solutions of the Cauchy problem. The computations below can be made rigorous in a similar fashion as in [12]. To justify heuristically our procedure, let us turn back to the rescaled equation (3.3). By formally letting stend to +∞, we deduce that the evolution of the solution w(s) for large sis governed by the power term wm. Therefore, we expect that the Wasserstein distance between the rescaled solution w(·, s) of equation (3.3) and the corresponding stationary Barenblatt profile w∞(3.7) tends to zero as sgoes to infinity with the same exponential rate as for the porous medium equation. Given a solution wto (3.3) and given the stationary Barenblatt profile with unit mass w∞corresponding to the power nonlinearity um, we denote once again F(y, s) = Zy −∞ w(z, s)dz G(y, s) = Zy −∞ w∞(z)dz. The two corresponding pseudo–inverses F−1and G−1satisfy ∂F −1 ∂s =−F−1−∂ ∂ρ (·∂F−1 ∂ρ ¸−m +emsΨÃe−sµ∂F−1 ∂ρ ¶−1!),(3.11) G−1+∂ ∂ρ "µ∂G−1 ∂ρ ¶−m#= 0. 18 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI We next state our result concerning Wasserstein distances. Theorem 5. Let f(v)satisfy conditions (F1),(F2) and (F3) stated before. (a) Let w(y, s)be the solution to (3.3) with vI∈L1 +(R)having mass 1 and finite second moment. Then, for any p≥1, W2p(w(s), w∞) = ·Z1 0¯¯F−1(ρ)−G−1(ρ)¯¯ 2pdρ¸ 1 2p ≤Ce−s,(3.12) holds, where C=C0+W2p(vI, w∞)and C0depends only on f. (b) Let w(y, s)be the solution to (3.3) with vI∈L1 +(R)having mass 1 and compact support. Then W∞(w(s), w∞)≤Ce−s,(3.13) where C=C0+W∞(vI, w∞)and C0depends only on f. In the original variables (3.1), part (b) of the previous theorem provides our result concerning with expansion rate of the support of any solution v(x, t) having compactly supported initial datum vI. Indeed, since the support of the Barenblatt profile is a ball of radius C(t+1 λ)1 m+1 for some fixed constant C > 0, we easily obtain the following result. Corollary 6. Let v(x, t)be solution to (3.1) with vI∈L1 +(R)having compact support, let v∞(t)be the Barenblatt profile with same mass. Then, there exist a fixed constant Csuch that |inf{supp v(t)} − inf{supp v∞(t)}| ≤ C |sup{supp v(t)} − sup{supp v∞(t)}| ≤ C(3.14) Remark 7. We found a lot of references in the literature concerning the finite speed of propagation property in slow diffusion equations (see [26, 25] for the general nonlinear case). Most of them are based on heavy analytic tools. Our result is more complete in the general nonlinear case, and covers a wider class of nonlinearities. Moreover, our technique seems to be applied to this problem in a very natural way. Proof of Theorem 5. To perform the proof of theorem 5, we compute the evolution of W2p(w(s), w∞) by means of the one–dimensional representation formula (3.5). The calculations below are formal, in the sense that we should need the pseudo–inverse function F−1to be smooth enough. We observe that this occurs when the initial datum u0is supported on a interval. We could make this argument rigorous by means of standard approximation tools (see [12]). We skip these details and suppose that F−1is smooth. NONLINEAR DIFFUSION EQUATIONS 19 Moreover, we need to know a priori that the speed of propagation of the support of the solution is finite. This property, which actually characterizes slow diffusion equations, was proved by Kalashnikov, Oleinik and Yiu–Lin (see [25] and the references therein). In fact, a refinement of the argument lead to prove that the speed of propagation is finite without assuming it by showing the control on W∞distance by approximations. Using the notations of the previous subsection, thanks to (3.11) and after integration by parts, we have d ds Z1 0£F−1−G−1¤2pdρ = 2pZ1 0£F−1−G−1¤2p−1∂ ∂sF−1(ρ, s)dρ =2p Z1 0£F−1−G−1¤2p−1 "−F−1−∂ ∂ρ õ∂F −1 ∂ρ ¶−m +emsΨ Ãe−sµ∂F −1 ∂ρ ¶−1!!+ +G−1+∂ ∂ρ õ∂G−1 ∂ρ ¶−m!# =−2p Z1 0£F−1−G−1¤2pdρ−2p(2p−1) Z1 0£F−1−G−1¤2p−2 µ∂F −1 ∂ρ −∂G−1 ∂ρ ¶× ×ems "fÃe−sµ∂F −1 ∂ρ ¶−1!−fÃe−sµ∂G−1 ∂ρ ¶−1!#dρ −2pZ1 0£F−1−G−1¤2p−1∂ ∂ρ ÃemsΨÃe−sµ∂G−1 ∂ρ , s¶−1!!.(3.15) We observe that, due to the compact support of the solutions, the boundary term coming from integration by parts disappears (see [16, 12]). In fact, this boundary term is given by X i=0,1 (−1)i2p£F−1(i, s)−G−1(i)¤2p−1"µ∂F −1 ∂ρ ¶−m (i, s)+ emsΨÃe−sµ∂F−1 ∂ρ ¶−1!(i, s)−µ∂G−1 ∂ρ ¶−m (i, s)#. The first bracket is bounded at any sbecause of the finite speed of propagation property of the solutions. The second bracket is a sum of positive powers of the solution wand of the Barenblatt function w∞evaluated at the boundary of their support respectively. Hence, this second bracket equals zero. Now, since the function fis increasing, the second integral at the end of (3.15) is nonnegative. This observation is the key point in this computation 20 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI (see again [12]). In fact, thanks to this we can get rid of the nonlinear term, and we have only to estimate the term depending on the Barenblatt profile, which is known. Indeed, after some calculations in the very last term of (3.15), due to the equation satisfied by G−1in (3.11), we obtain the following inequality d ds Z1 0£F−1−G−1¤2pdρ ≤ −2pZ1 0£F−1−G−1¤2pdρ −2p me(m−1)s Z1 0£F−1−G−1¤2p−1G−1Ψ0Ãe−sµ∂G−1 ∂ρ ¶−1!µ∂G−1 ∂ρ ¶m−1 dρ. We can assume that Ψ(v) = vng(v), with g0(v) = O(vk), k > −1, as v→0. Then, it follows that Ψ0(v) = O(vn−1), as v→0. Hence, by H¨older inequality, the last integral above can be estimated from above by the term C(w∞)pe−(n−m)sµZ1 0¯¯F−1−G−1¯¯ 2p−1dρ¶ 2p−1 2p , where the constants C(w∞) is given by C(w∞) = kw∞kL∞(R)max{| inf{supp w∞}|,|sup{supp w∞}|} (this quantity depends only on the mass and on the exponent m). We now apply the variation of constants formula in order to get the rate of convergence to zero of Wp(w(s), w∞). In order to perform this task, we set for simplicity Xp(s) = Z1 0£F−1(ρ, s)−G−1(ρ)¤2pdρ. Hereafter, Cdenotes a fixed positive constant, independent on pand s. So far we have proved that d dsXp(s)≤ −2pXp(s)+2p Ce−(n−m)sXp(s)2p−1 2p.(3.16) By Young inequality we get d dsXp(s)≤ −2p³1−Ce−(n−m)s´Xp(s) + Ce−(n−m)s, and the variation of constants formula implies that Xp(s)≤(Xp(0) + C)e−min{2p,(n−m)}s.(3.17) In case that n−m < 2p, the exponential rate of convergence in (3.17) can be improved iteratively by substituting the above inequality in the last addend of (3.16), until it reaches the value e−2ps. Obviously, the number of steps depends on p. We have thus proved (3.12). The inequality (3.13), then, easily follows by sending p→ ∞.¤ NONLINEAR DIFFUSION EQUATIONS 21 Figure 6 shows the evolution of the p-Wasserstein distance defined in (3.5) for several values of pbetween v1(x, t) and v2(x, t), solutions of (2.17) with initial conditions (2.18) and (2.20) respectively in case f(v) := v2. The integral in (3.5) for the forthcoming tests is computed by numerical quadrature and we are using the fully discretized implicit Euler scheme described in subsection 2.1. It is interesting to observe that although the distance between the solutions is only known to be a contraction, this distance is in fact decaying quickly as t→ ∞. Let us point out that the two initial data have zero center of mass and therefore are well centered. In fact, it is a conjecture to prove that there is a decay of the distance between the solutions when you fix the center of mass of the initial data. In the case of expansion rate of supports, this was already observed by J. L. V´azquez in [31]. (a) 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 t p=2 (b) 0 5 10 15 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t p=5 (c) 0 5 10 15 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 t p=15 Figure 6. Wasserstein p-metric Wp(v1, v2). 22 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI 4. Intermediate asymptotics for general nonlinearities In this section we summarize a result contained in [11] concerning the long time behavior for a general nonlinear diffusion equation (2.6) vt= ∆f(v), x ∈RN, t > 0,(4.1) where fsatisfies the the following assumptions: (NL1) f∈C[0,+∞)∩C1(0,+∞), f(0) = 0 and f0(v)>0 for all v > 0, (NL2) ∃C > 0 and m > N−2 Nsuch that f0(v)≥Cvm−1for all v > 0, (NL3) f(v) v1−1/N is nondecreasing on v∈(0,∞). Assumption (NL1) ensures that the Cauchy problem (4.1) is well–posed for any initial datum in L1 +(RN) (see [6, 35], see also [8, 7] for more qualitative properties of solutions). Under the additional assumption on the nonlinearity f(NL2), equation (4.1) enjoys an L1–L∞regularizing property. Indeed, it’s proved that the solution to (4.1) with initial datum in L1 satisfies the following temporal decay estimate (see [4] for the power law case, [35] for the general nonlinear case) kv(·, t)kL∞(RN)≤C t−N λkvIkL1(RN).(4.2) Assumption (NL3) implies that the entropy functional associated to equation (4.1) is displacement convex [23] and thus, the flow map of the nonlinear diffusion (4.1) is a non-expansive contraction in time with respect to the euclidean Wasserstein distance W2in probability measures [24, 1, 17]. As previously observed, in the power law case f(v) = vmself–similar solutions of the form (2.7) can be seen as stationary profiles of equation (4.1) written in similarity variables, i.e. the nonlinear Fokker–Planck equation (2.1) with V(x) = |x|2 2and f(u) = um. Moreover, relative entropy tools provide exponential convergence towards such stationary profiles. In case of a general non homogeneous nonlinearity f(v) there are no time dependent scalings which allow to rewrite the equation (4.1) in the form of (2.1); therefore it is an open problem how to detect any special solution as a reasonable candidate to be the universal asymptotic profile for the equation (4.1). Let us then propose our approach. We define the “temperature” of a solution uof (4.1) as its second moment, i.e., θv(t) = ZRN |x|2 2v(x, t)dx. NONLINEAR DIFFUSION EQUATIONS 23 Given a solution v(t) of (4.1) we will study the long time behavior of θv(t)N/2v(θv(t)1/2x, t).(4.3) Similar scalings have been used in the analysis of homogeneous cooling states in granular media equations (see for instance [5, 29] and the references therein). The nonlinear time dependent scaling (4.3) can be also seen as the projection of the solution v(·, t) onto the manifold of probability measures with unit temperature (see also [10]) M=½µ∈ P2(RN),1 2ZRN |x|2dµ(x) = 1¾. In order to have the above scaling (4.3) well defined for all positive times t, in addition we must require the following natural assumption on the nonlinearity f, namely (FT) for any solution v(x, t) to (4.1), the following relation holds, Z|x|2v(x, 0)dx < ∞,⇒Z|x|2v(x, t)dx < ∞for all t > 0. In [11] we prove that the above assumption is satisfied under reasonable requirements for the function f, also including fast diffusion ranges of nonlinearities. For further reference, we state the following lemma, which ensures that θv(t) tends to infinity as t→+∞(see [11] for the proof). Lemma 8. Suppose that fsatisfies assumptions (NL1)–(NL2)–(FT). Then θ(v)(t)≥C0t2 d(m−1)+2 (4.4) and C0depends only on the mass of v. Our main result is the following: Theorem 9 (Asymptotic profile for general nonlinear diffusions). Given fverifying the hypotheses (NL1)–(NL3) and (FT), there exists t∗>0and a one parameter curve of probability densities v∞(t), with unit temperature defined for t≥t∗such that, for any solution of (4.1) with initial data (1 + |x|2)vI∈L1 +(RN)of unit mass and temperature, W2³θv(t)N/2v(θv(t)1/2·, t), v∞(t)´−→ 0as t→ ∞. Moreover, the asymptotic profile v∞(t)is characterized as the unique fixed point of the renormalized flow map S(t) S(t)vI:= θv(t)N/2v(θv(t)1/2·, t), where v(·, t)is the solution to (4.1) with initial datum vI. 24 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI The main ingredient in previous theorem is the proof of a contraction property for the maps S(t), t≥t∗>0, obtained as compositions of the flow map for the nonlinear diffusion equation (4.1) and the projection of the solution onto the unit second moment manifold M. The asymptotic profile v∞(t) is nothing else but the Barenblatt-Prattle solution at the time in which it has unit temperature (therefore is constant in time) in the case of the homogeneous nonlinearities f(v) = vm[30]. Indeed, thanks to our point of view we can generalize the classical notions of self–similarity and source– type solution by means of the idea of invariance of the solution orbit after projection onto the subset M. The proof of contraction of S(t) with respect to the euclidean Wasserstein distance makes use of the L1−L∞regularizing effect, needed to control from below the behavior of the temperature of the solution as t→ ∞. Then, the proof is based on elementary properties of the euclidean Wasserstein distance, which allow to generalize our approach to more situations. Open problems are the eventual convergence of v∞(t) to a unique limit point as t→ ∞, and convergence results in Lpspaces for solutions of (4.1) to such a limit. We can provide an answer only in case f(v) satisfies assumption (F3) of the previous section, i.e. when fis a higher order perturbation of a power law vm. In that case, we can prove convergence towards the corresponding Barenblatt profile. Theorem 10 (Asymptotically homogeneous nonlinearities). Let v(x, t)be the solution to (4.1), with nonlinearity fsatisfying hypotheses (NL1)-(NL3),(F3) and (FT) above and with initial datum vI∈L1 +(RN) such that ZRN£vm I(x) + |x|2vI(x) + φ(vI(x))¤dx < +∞. Let B(|x|, t)be the Barenblatt self–similar function with the same mass as vIcorresponding to the exponent m. Then, the following estimate holds for all t≥0 kv(·, t)−B(|·|, t)kL1(RN)≤C(t+ 1)−δ λα , where λ=N(m−1) + 2, α =(2if m≤2 mif m≥2,δ= min{2, Nn}, nis given by condition (F3) and Cdepends only on the initial datum vI. 4.1. Numerical results. The aim of this section is to compute numerically the asymptotic profiles v∞(t) and try to clarify if that profile has a unique limit as t→ ∞. All the results in this section are obtained by applying a simple fixed point iteration on the maps S(t) for several values of t chosen uniformly over a time interval [T1, T2]. Therefore, one uses the fully NONLINEAR DIFFUSION EQUATIONS 25 discretized Euler implicit scheme introduced in subsection 2.1 to compute the solution of the Cauchy problem for (4.1) for each value of the iteration till eventual convergence of the fixed point iteration upto a fixed tolerance. Let us finally remark that all theoretical results were written by projecting the solution onto the manifold of measures with unit second moment, of course, everything can be generalized by projecting onto the manifold with an arbitrary fixed temperature and we will do so for numerical convenience. Figure (7) is a benchmark for the scheme since we know theoretically that the asymptotic profile of the problem vt= (v2)xx is the Barenblatt (2.7) B(x, τ) with Cfixed by conservation of mass: kvI(·)kL1(R)=kB(·, t)kL1(R) and τfixed by the initial second moment: ZR x2vI(x)dx =ZR x2B(|x|, τ)dx. We take as initial data vIdefined by (2.18) and we have plotted all the v∞(t) fixed points onto a time interval [5,14] that match each other and B(x, τ) as expected. −3 −2 −1 0 1 2 3 0 0.05 0.1 0.15 0.2 0.25 x w∞(x,t) Figure 7. asymptotic profile of (2.6) with m= 2. Figure (8) shows the asymptotic profile for the equation with f(v) := v2+ v3and taking the initial data vI(x) defined by (2.18) to fix the mass and the temperature of the asymptotic profile like before and to start the fixed point iteration. Since this case is a perturbation of the homogeneous equation with f(v) = v2in the sense of hypothesis (F3), then the asymptotic profile should approach the Barenblatt corresponding to f(v) = v2with initial mass and time fixed by the initial second moment. Results shown in this figure support this fact since the computed values of the asymptotic profile 32 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI where Vα(x, t) solves (∂tVα=αmN ∆f¡α−NVα¢ V(x, 0) = αNv0(αx).(5.6) This is easily seen by solving the above problem, by performing the scaling αNv¡αx, αλt¢=Vα(x, t), and by observing that vsolves (5.1). Finally, one immediately realizes that Vα(x, 1) is exactly Rαv0(x). In the homogeneous case f(v) = vm, the equation in the Cauchy problem (5.6) is again the original equation (5.1) (i.e. the porous medium equation). This is due to the well known invariance property of the porous medium equation under its similarity transformation. In such case, one can prove that the family of fixed points is independent on the parameter αand it coincides with the rescaled Barenblatt profile which is a stationary solution to a nonlinear Fokker–Planck equation. As a consequence of what we have proved in this section, for any fixed α we have v∞ α(x) = Vα(x, 1), where Vαis the solution to (5.6) with initial datum αNv∞ α(αx). Hence, one can try to investigate the limiting behavior of the solutions to (5.6) as α→ ∞ in order to detect an eventual limit as α→ ∞ for the family of fixed points v∞ α. In order to perform this task, we use the following strategy. We first analyze the asymptotic behavior with respect to the parameter αof the rescaled solutions Vαunder quite general assumptions on the initial data. In particular, under certain extra assumptions on fwe shall be able to detect a unique limit point V∞(in some sense to be specified afterwards) for the family {Vα}αby means of standard energy techniques. Then, we apply such convergence result to some suitable choice of initial datum v0in order to obtain convergence with respect to the d2distance of Rα(v0) as α→+∞ towards the unique limit point V∞. Finally, we obtain convergence in d2 of v∞ αto V∞by means of (5.5) and by triangulation. We remark that the parameter αappears both in the equation in (5.6) and in the rescaled initial datum αNv0(α·), so that we are dealing both with a problem of continuous dependence on the initial data (with initial data eventually approaching a measure, as we will see later) and with a continuous dependence on the nonlinearity function. In order to pursue our goal, let us first identify the limit of the rescaled initial data Vα(·,0) = αNv0(α·) for some v0∈L1 +∩L∞as α→ ∞. We have the following lemma, the proof of which is straightforward. Lemma 14. Vα(·,0) = αNv0(α·)→δ0as α→ ∞ in the sense of measures. NONLINEAR DIFFUSION EQUATIONS 33 Next, let us characterize those nonlinearities for which αNmf¡α−NV¢ admits a limit as α→+∞for fixed V > 0. Let us then impose αNmf¡α−NV¢→Φ(V),as α→+∞. By a change of variable α−NV=z, we have lim z→0Vmf(z) zm= Φ(V) and therefore f(z)∼Czmas z→0 for some constant C. Thus, Φ(V) = CV m. This heuristic limit procedure suggests us that we may hope to get some result when passing to the limit at least in case fbehaves like a power in zero. Let us then impose the following assumption. (NL5) lim v→0 f(v)−Cvm vm= 0,for some positive constant C. We now aim to prove some strong compactness for the family of functions {Vα(·,·)}αin order to get the limit as α→+∞in a suitable way. To perform this task we generalize the standard energy method for the porous medium equation (see for instance [33]). We shall prove the following lemma. Lemma 15. For any 0< t1< t2and for any α > 2, we have Zt2 t1ZRd"¯¯¯¯ ∂ ∂tαmN f(α−NVα(x, t))¯¯¯¯ 2 +¯¯αmN ∇f(α−NVα(x, t))¯¯ 2#dxdt ≤C (5.7) for some constant Cdepending only on the mass, on the function fand on t1. Proof. In what follows we shall assume that the solution Vαto (5.6) is smooth enough to perform the computations below. The assertion above for a general weak solution can be obtained by standard approximation. We first observe that the family of solutions Vαenjoys a bound in L∞([t0,+∞]× RN) which is uniform with respect to α. This comes from the usual L1– L∞regularizing property for nonlinear diffusion, which is a consequence of assumption (NL2) above. Such bound eventually holds in L∞([0,+∞]× RN) if the initial data are bounded. Let us multiply the equation in (5.6) by α−Nf(α−NVα) and integrate over RN×[t1, t2]. Integration by parts yields α(m−1)NZt2 t1ZRN¯¯∇f(α−NVα(x, t))¯¯ 2dxdt ≤ZF(α−NVα(x, t1))dx, where F(v) = Rv 0f(ζ)dζ. Then, since f(v)≤Cvmas v∈[0, M], we have F(v)≤C(m)vm+1 on the same interval. Thus, due to uniform bound in 34 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI L∞for the family Vα, we deduce α2mN Zt2 t1ZRN¯¯∇f(αmN Vα(x, t))¯¯ 2dxdt ≤C, (5.8) for some Cindependent on αand depending on t1and on the initial mass. In order to estimate the time derivative term in (5.7) we compute 1 2 d dt ZRN¯¯∇αmN f(α−NVα)¯¯ 2=α2mN ZRN ∇f(α−NVα)· ∇∂tf(α−NVα)dx =−α2mN ZRN ∆f(α−NVα)∂tf(α−NVα)dx =−αN(m−1) ZRN |∂tVα|2f0(α−NVα)dx. Let us define G(v) = Rv 0f0(ζ)1/2dζ. We have, then 1 2 d dt ZRN¯¯∇αNmf(α−NVα)¯¯ 2=−ZRN¯¯¯¯ ∂ ∂t ³αN(m+1)/2G(α−NVα)´¯¯¯¯ 2 dx. Therefore, integration with respect to tover (t1/2, t2) yields Zt2 t1/2ZRN (t−t1/2) ¯¯¯¯ ∂ ∂t ³αN(m+1)/2G(α−NVα)´¯¯¯¯ 2 dx =−1 2Zt2 t1/2 (t−t1/2) d dt ·ZRN¯¯αmN ∇f(α−NVα(x, t))¯¯ 2dx¸dt =−1 2µt2−t1 2¶ZRN¯¯αmN ∇f(α−NVα(x, t2))¯¯ 2dx + +1 2Zt2 t1/2ZRN¯¯αNm∇f(α−NVα)¯¯ 2dxdt. Thanks to (5.8), we can find a constant Cdepending on t1such that Zt2 t1ZRN¯¯¯¯ ∂ ∂t ³αN(m+1)/2G(α−NVα)´¯¯¯¯ 2 dx ≤C. Moreover, assumption (NL4) and the uniform bound in L∞of Vαimply ¯¯¯∂tαN(m+1)/2G(α−NVα)¯¯¯ 2=¯¯¯αN(m−1) 2G0(α−NVα)∂tVα¯¯¯ 2 =·αN(1−m) 2G0(α−NVα) f0(α−NVα)¸2 ¯¯∂t¡αNmf(α−NVα)¢¯¯ 2 =αN(1−m)f0(α−NVα)−1¯¯∂t¡αNmf(α−NVα)¢¯¯ 2 ≥c0¯¯∂t¡αNmf(α−NVα)¢¯¯ 2, NONLINEAR DIFFUSION EQUATIONS 35 for some constant c0depending on the function fand on the initial mass. Therefore, we can recover an L2–estimate for the time derivative of αmdf(αNVα(x, t)) and the proof is completed. ¤ As a consequence of the previous lemma, by Sobolev embedding we recover αNmf(α−NVα(x, t)) ⊂⊂ L2 loc,x,t. Now, we prove that assumption (NL5) implies in particular, for all M > 0, lim α→+∞sup 0≤v≤M¯¯αNm[f(α−Nv)−Cα−Nmvm]¯¯→0 as α→+∞.(5.9) To see this, let us fix a positive ε > 0. Then, thanks to (NL5) there exists aα0such that ¯¯αNm[f(α−N)−Cα−Nm]¯¯<ε Mm as α > α0. Let us set α1:= α0M1/N . Then, for all α > α1and for all 0≤v≤Mwe have α v1/N > α0, which implies ¯¯αNmv−m[f(α−Nv)−Cα−Nmvm]¯¯<ε Mm≤ε vm for all 0 ≤v≤M, and this yields sup 0≤v≤M¯¯αNm[f(α−Nv)−Cα−Nmvm]¯¯< ε, for all α > α1, which proves (5.9). Thanks to (5.9) and thanks to the uniform estimate kVα(·, t)kL∞≤M, Mdepending only on the initial mass (which is a consequence of the L1–L∞ regularizing property), in view of the compactness of αNmf(α−NVα(x, t)), we can extract a sequence αn→+∞such that (Vαn(·,·)→V∞ αmd nf(α−NVαn)→CV m ∞ almost everywhere in (x, t), for some V∞∈L2m, as n→+∞. We observe that, in principle the sequence Vαnmay depend on the time interval [t1, t2]. We can construct the desired sequence by standard diagonal procedure. Finally, we prove that all the limit points of the family {Vα(x, t)}α>2 must coincide. Let ψbe a test function on [t1, t2]×RN. By definition of 36 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI distributional solution of the equation in (5.6) we obtain, for all α > 2, −ZT 0ZRN αmN f(α−NVα)∆ψdxdt +ZRN Vα(x, T)ψ(x, T)dx −(5.10) ZRN Vα(x, 0)ψ(x, 0)dx = 0. By taking the limit as α→+∞, by means of the dominated convergence theorem and in view of lemma 14 and by the uniform bound in L∞for Vα, we obtain −ZT 0ZRN CV m ∞∆ψdxdt +ZRN v∞(x, T)ψ(x, T)dx −δ0(ψ) = 0, which means that V∞is the only measure valued solution to the porous medium equation vt=C∆vmwith a Dirac mass as initial datum, i. e. the corresponding Barenblatt solution (see [27] for uniqueness of solutions with initial datum a measure). Then, we can state the following assertion Vα(·,·)→V∞(·,·) almost everywhere in (0,+∞)×RN. Then, there exists a t0such that 0 < t0<1 such that Vα(·, t0)→V∞(·, t0) almost everywhere in RN. Let us then compute the distance between Vα(·, t0) and V∞(·, t0) in L1. For R > 0 we have ZRN |Vα(x, t0)−V∞(x, t0)|dx =Z|x|>R |Vα(x, t0)−V∞(x, t0)|dx +Z|x|≤R |Vα(x, t0)−V∞(x, t0)|dx ≤Z|x|≤R |Vα(x, t0)−V∞(x, t0)|dx +1 R2Z|x|>R |Vα(x, t0)−V∞(x, t0)| |x|2dx. (5.11) The first term in the last line above tends to zero as α→+∞by uniform bound in L∞for the Vαand because of the dominated convergence theorem. The second term is bounded by 2θ0/R2, and therefore it is arbitrarily small for large R. This proves that kVα(·, t0)−V∞(·, t0)kL1(RN)→0 as α→+∞. So far we did not require any extra assumption on the initial datum v0in (5.6) more than v0∈ Mθ0. As announced previously, let us then perform the special choice for the initial datum v0∈L1 +∩Cb(RN), v0strictly positive and such that RRNv0(x)|x|4dx < ∞. Then, the corresponding rescaled solution Vα(v0) will be continuous with respect to t. Moreover, in a similar fashion NONLINEAR DIFFUSION EQUATIONS 37 as in (5.11), we can easily prove that Vα(v0) belongs in C([t0,+∞); L1(RN)) for any α > 2, so that we can apply a continuous dependence result stated in [7], by taking t0as initial time. Such results implies that, Vαconverges to V∞as α→+∞in C([t0,+∞); L1(RN)). In particular, such convergence holds pointwisely at t= 1, that is Rα(v0)→v∞:= V∞(·,1) almost everywhere in RN. Thanks to our choice of v0, we can perform the following estimate for the fourth moment of Vα. Integration by parts and assumption (NL4) yields d dt ZRN Vα(x, t)|x|4dx = =C(N)αNm RRNf(α−NVα(x, t))|x|2dx ≤C(N)RRNVm α(x, t)|x|2dx. Then, after a change of variable we have ZRN Vα(x, 1)|x|4dx ≤1 α4Zv0(x)|x|4dx +C(N)θ0, which implies a uniform bound of the fourth moment of Rα(v0) with respect to α. Thanks to this, we can recover a tightness property of the family Rα(v0). Namely, for R > 0 we have Z|x|>R Rα(v0)(x)|x|2dx ≤1 R2Z|x|>R Rα(v0)(x)|x|4dx, and the left hand side converges to zero uniformly with respect to α > 2 as R→+∞. We recall that such tightness property plus the weak convergence in the sense of measures of the family Rα(v0) to V∞(·,1) (which is a trivial consequence of the convergence almost everywhere and of the uniform bound in L∞of Vλ) are equivalent to the convergence of Rα(v0) to V∞(·,1) in the d2topology (see [36], theorem 7.12). We have thus proven that d2(Rα(v0), v∞)→0 as α→+∞.(5.12) Finally, (5.5) with v0as initial datum together with (5.12) imply d2(v∞ α, v∞)→0 as α→+∞, and therefore, by triangulation lim α→+∞d2(αNv(α·, α1/α), v∞) = 0.(5.13) Therefore, we have proven the following theorem. Theorem 16. Given fsatisfying assumptions (NL1)–(NL5), for any solution v(x, t)to the Cauchy problem (vt= ∆f(v) v(x, 0) = vI(x),(5.14) 38 JOS´ E A. CARRILLO, MARCO DI FRANCESCO AND MARIA P. GUALDANI with v0∈L1 +(RN)having finite second moment, the following limiting relation holds d2(αNv(α·, α1/α), v∞)→0 as α→+∞, where v∞is the Barenblatt solution to the porous medium equation vt=C∆vm, with Cand mas in assumption (NL5), evaluated at time t= 1. Moreover, v∞is characterized as the unique limit with respect to the d2as α→+∞of the family v∞ αof fixed points of the renormalized flow map v07→ Rαv0(x) = αNv³αx, α1/α´, with v(x, t)solution to (5.14) with initial datum v0. Remark 17. We remark that, even though the above theorem holds only in case fbehaves like a power near zero, nevertheless we do not need to assume any extra hypothesis on the perturbation term f(v)−Cvmmore than being faster than vmas vtends to 0. In the existing literature about this topic, as well as in our theorem 10 in the previous section, the perturbation term need to behave like a power vnwith n > m, see for instance [9]. Acknowledgements: The authors acknowledge support from the European IHP network HYKE “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Applications” HPRN-CT-2002-00282. JAC acknowledges partial support from DGI-MCYT/FEDER project BFM2002-01710 and CIRIT 2001SGR/00172. MPG has been supported by the Deutsche Forschungsgemeinschaft, grant JU 359/5 (Priority Program “Multi-scale Problems”) and by the DFG-Project JU 359/3 (Gerhard Hess Award). JAC and MPG have been supported by the DAAD-Acciones Integradas Project. We thank A. Arnold, G. Toscani and A. J¨ungel for fruitful discussions. References [1] M. Agueh, Existence of solutions to degenerate parabolic equations via the mongekantorovich theory, C. R. Acad. Sci. Paris S´er. A-B (2002). [2] L. Ambrosio, N. Gigli, and G. Savar´e, Gradient flows in metric spaces and in the Wasserstein spaces of probability measures, to appear in Lecture Notes in Mathematics. [3] A. Arnold, A. 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