Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem

In this paper we study the asymptotic behavior of a very fast diffusion PDE in 1D with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in \cite{CGI1}. We prove exponential convergence to equilibrium under minimal assumptions on the data, and we also provide sufficient conditions for $W_2$-stability of solutions.


Introduction
During the last years, asymptotic analysis for solutions of nonlinear parabolic equations have attracted a lot of attention, also in connection with gradient flows and entropy methods.
The aim of the present paper is to investigate the dynamics of the PDE where r > 1, and ρ > 0 and f (t, ·) are probability densities on [0, 1] with periodic boundary conditions. When ρ = 1, this equation takes the form which belongs to the general class of fast diffusion equations ∂ t u = div(u m−1 ∇u), m < 1.
We recall that, when the problem is set on the whole space R n , the value of m plays a crucial role: solutions are smooth and exist for all times if m > m c := (n − 2)/2, while they vanish in finite time if m ≤ m c (the existence of such an extinction time motivates the name "very fast diffusion equations"). There is a huge literature on the subject, and we refer the interested reader to the monograph [14] for a comprehensive overview and more references.
Our case corresponds to the range m = −r < −1. It is interesting to point out that (1.2) set on the whole space R or with zero Dirichlet boundary conditions has no solutions, since all the mass instantaneously disappear [12,Theorem 3.1] (see also [6,10,11] for related results). It is therefore crucial that in our setting the equation has periodic boundary conditions, so that the mass is preserved We observe that this kind of equations has the property of diffusing extremely fast. In particular, if f 0 is a non-negative and bounded initial datum, the solution becomes instantaneously positive. As we are only interested in the long time behaviour of solutions, to simplify the presentation we will only consider initial data that are bounded away from zero and infinity.
Our equation (1.1) is motivated by the so-called quantization problem. The term quantization refers to the process of finding the best approximation of a d-dimensional probability distribution by a convex combination of a finite number N of Dirac masses. This problem arises in several contexts and has applications in information theory (signal compression), numerical integration, and mathematical models in economics (optimal location of service centers). In order to explain the meaning of the equation (1.1), we now briefly recall the gradient flow approach to the quantization problem introduced in [3], and further investigated in [4].
Given r ≥ 1, consider µ = ρ(x) dx a probability measure on an open set Ω ⊂ R n . Given N points x 1 , . . . , x N ∈ Ω, one wants to find the best approximation of µ, in the Wasserstein distance W r , by a convex combination of Dirac masses centered at x 1 , . . . , x N . Hence one minimizes where γ varies among all probability measures on Ω × Ω, and π i : Ω × Ω → Ω (i = 1, 2) denotes the canonical projection onto the i-th factor. See [1,15] for more details on Wasserstein distances.
As explained in [7, Chapter 1, Lemmas 3.1 and 3.4], this problem is equivalent to minimizing the functional To find a minimizer to this function, in [3] the authors introduce a dynamical approach where they study the dynamics of the gradient flow induced by F N,r . Since the main goal is to understand the structure of minimizers in the limit as N tends to infinity, in [3,Introduction] and in [4, Sections 2 and 3] the authors are able to find a formula for the "limit" of F N,r when N → ∞.
As shown in [3], when n = 1 this limit is given by the functional and its L 2 -gradient flow is given by the following non-linear parabolic equation coupled with the Dirichlet boundary condition. This equation provides a Lagrangian description of the evolution of our system of particles in the limit N → ∞. We can also study the Eulerian picture for (1.3). Indeed, if we denote by f (t, x) the image of the Lebesgue measure through the map X, i.e., then the PDE satisfied by f takes the form (see [1]) with periodic boundary conditions, and in view of the results in [7,3] we expect the following long time behavior: More precisely, the results in [3] show the validity of the limit only when r = 2 and under the assumption that ρ is close to 1 in C 2 . The goal here is to generalize and improve this result.
Our starting point for studying the asymptotic behavior of (1.1) is the observation that this equation can be seen as the gradient flow of the functional (1.4) with respect to the W 2 distance. In a first step, by exploiting a modulated L 2 energy method, we obtain exponential convergence to equilibrium under minimal assumptions on the density ρ. Then, we investigate the displacement convexity of the functional F ρ . Notice that, as we shall prove in Proposition 2.1 below, if ρ and f (0) are bounded away from zero, then f (t) remains uniformly away from zero for all t ≥ 0. In particular (1.1) is uniformly parabolic, and f (t) is smooth if ρ is so.
Since our focus is on the asymptotic behavior, we shall assume that ρ is of class C 2,α for some α ∈ (0, 1), so that parabolic regularity theory ensures that f (t) is of class C 2,α for all times, hence f is a classical solution. However our results are independent of the smoothness of ρ and can be thought as a priori estimates. In particular, we believe one could extend them to the setting of weak solutions by using the general theory of minimizing movements in [2] (See also [5]). Since our main goal is to understand the general asymptotic properties of the equation (1.1), we shall not investigate this here.
Our first result is the following: and there exist positive constants C 0 , c 0 , depending only on λ, a 1 , and A 1 , such that The result above shows the exponential convergence to equilibrium with a rate independent of the smoothness of ρ. However, it does not say anything about stability of solutions. For this, we investigate the convexity of the functional F ρ with respect to the 2-Wasserstein distance W 2 . In particular, we show that if ρ ′ ∞ + ρ ′′ ∞ is small enough, then F ρ is uniformly convex.
, and let f 1 , f 2 solve (1.1) with periodic boundary conditions, and let a, A > 0 be as in Theorem 1. Then The arguments used to prove Theorems 1.1 and 1.2 are very general, and could be applied also to the n-dimensional version of (1.1). However, since the connection between this equation and the quantization problem is valid only in 1D, we have decided to state and prove these results only on the 1 dimensional case.

Maximum principle
The goal of this section is to prove a maximum-type principle for (1.1) which shows that, if ρ and f (0) are bounded away from zero, then f (t) remains uniformly away from zero for all t ≥ 0. In particular (1.1) is uniformly parabolic, and f (t) is smooth if ρ is so. Note that, the following Proposition corresponds to the first part of Theorem 1.1. 1], and assume that ρ : [0, 1] → [λ, 1/λ] is periodic and of class C k,α for some k ≥ 0 and α ∈ (0, 1). Let f (0, ·) : [0, 1] → R be a periodic function of class C k,α satisfying 0 < a 1 ≤ f (0, ·) ≤ A 1 , and let f solve (1.1) with periodic boundary conditions. Then is of class C k,α for all t ≥ 0, and there exists a constant C, depending only on λ, ρ C k,α , k, α, a 1 , and Proof. It is enough to prove the bound for all t ≥ 0, since then, once these bounds are proved, the rest of the proposition follows by standard parabolic regularity.
To prove the result, we set With these new unknowns (1.1) becomes with periodic boundary conditions. The advantage of this form is that constants are solutions and we can prove a comparison principle with them. More precisely, we set c 0 := λ 1/(r+1) a 1 and C 0 := A 1 λ 1/(r+1) . Recalling the notation s + = max{s, 0} and s − = max{−s, 0}, we claim that the maps t → Hence, thanks to the claim m(x) and that λ 1/(r+1) ≤ m(x) ≤ λ −1/(r+1) , this proves the result. Hence, we only need to prove the claim.
To this aim, we only show that is nonincreasing (the other statement being analogous).
Since constants are solutions of (2.1), it holds We now multiply the above equation by −m φ ε , with φ ε a smooth approximation of the indicator function of R + satisfying φ ′ ε ≥ 0. Integrating by parts we get proving the result.
3 Exponential convergence to equilibrium: proof of Theorem 1.1 We begin by observing that, thanks to Proposition 2.1, f (t) satisfies (1.5). Also, recalling the definition of F ρ (see (1.4)), a direct computation gives then Given x ∈ [0, 1], let us define the function Then, with γ the renormalization constant of the stationary solution (so that γρ 1/(r+1) is a probability density). Now we will to introduce a function G x [f ] that, up to translation, has the same integral of F x [f ], and such that G x [f ] can be used to perform an L 2 Gronwall estimate. We define Then, By Proposition 2.1 we have that f is bounded away from zero and infinity, see for all times, with b, B positive constants. Moreover, thus, since f and γρ(x) 1/r+1 are two probability densities, G x and F x have the same integral up to an additive constant: Thus, denoting by c and C positive constants depending only on λ, a, A, r, and that c and C may change from line to line, we have: is bounded away from zero and infinity for all times (thanks to (1.5) and the bound λ ≤ ρ ≤ 1/λ): Now, the problem is that α(t) a priori does not coincide with γ. For this reason we use the following trick: Therefore, by Gronwall Lemma, there exists a constantĉ such that Since G x [f (t, x)] is comparable to f (t, x) − γρ 1/r+1 2 , this Gronwall estimate implies the exponential convergence of f to the stationary solution γρ 1/r+1 , namely as desired 4 Stability in W 2 : proof of Theorem 1.2 To prove Theorem 1.2, we shall first compute the Hessian of F ρ [f ] at a fixed probability density f , and then we apply this estimate to prove the contraction along two solutions of (1.1). Since, under our assumptions, solutions are of class C 2,α , in the next section we assume that f ∈ C 2 .

Hessian of F ρ [f ]
In this section we compute the Hessian of with respect to W 2 . For this, we use the Riemannian formalism introduced in [8].
Our state space M is the space of positive functions f : (0, 1) → (0, ∞) with unit integral: We may think of infinitesimal perturbations δf ∈ T f M of a state f ∈ M as functions δf : (0, 1) → R with 1 0 δf dx = 0. (4.1) For given f ∈ M we define the scalar product g f on T f M via where, up to additive constants, the functions φ i : (0, 1) → R are definite by Note that, since the variational derivative of F ρ [f ] is given by Now, given a periodic probability density f : [0, 1] → (0, ∞) of class C 2 , let the function δf satisfy (4.1), and let φ be related to δf by (4.2).
We compute the first derivative of F ρ [f ]. Using that we have: Now, to compute the Hessian of F ρ , we consider a geodesic f : [0, 1] → M such that f (0) = f . Then the Hessian of F ρ at f is computed by considering Recall that the geodesic equation is given by the system (see for instance [9, Sections 2 and 3.2]) and that, with this notation, where δf (s) is related to φ(s) by (4.2).
We now compute the second derivative of F ρ [f (s)]: Using again that ∂xf f r+1 = − 1 r ∂ x 1 f r , and integrating by parts, we get We now notice that We now want to investigate the µ-convexity of the functional F ρ in terms of the assumptions on ρ and f . Assume that ρ is a periodic probability density of class C 2,α with ρ ′ ∞ ≤ η 1 , and ρ ′′ ∞ ≤ η 2 . We assume also that 0 < λ ≤ ρ ≤ 1/λ, and that 0 < a ≤ f ≤ A. Then By Young inequality we have, for any ε > 0, Choosing ε = rλa r 2η 1 A r , we get Using Poincaré inequality on [0, 1] (recalling that the Poincaré constant is 1/2), we obtain where µ := 1 A r(r + 1)λ A r − 2η 2 1 (r + 1)A r rλa 2r − η 2 a r . This proves that the Hessian of F ρ at f is bounded from below by µ.

Application to stability of solutions to (1.1)
As we shall explain in the next section, to ensure that the above convexity results can be applied to equation (1.1), one needs to know that if f 1 (t, x), f 2 (t, x) are solutions of (1.1), and if is a Wasserstein geodesic such that f 0 (t, x) = f 1 (t, x) and f 1 (t, x) = f 2 (t, x), then there exist constants a, A > 0 such that Noticing that we obtain the validity of (4.6). In the next subsection, we briefly summarize the general consequences of µ-convexity and we conclude the proof of Theorem 1.2.

W 2 -stability
In this section we use Otto's formalism to deduce convergence and stability of solutions. Although these computations are formal, we present them as they show in a very elegant way why convexity of F implies such stability. For a rigorous proof, the reader may look at the paper [9, Section 4].
Recall that, formally, our equation (1.1) can be written aṡ where Now, given two solutions f 1 and f 2 as in the statement of the theorem, and denoting by f s the geodesic connecting them, we compute d dt Now, since f s is a geodesic, where in the last inequality we used again that f s is a geodesic. Hence, combining these two equations we get d dt which gives the result.