Nonlinear aggregation-diffusion equations with Riesz potentials

We consider an aggregation-diffusion model, where the diffusion is nonlinear of porous medium type and the aggregation is governed by the Riesz potential of order s. The addition of a quadratic diffusion term produces a more precise competition with the aggregation term for small s, as they have the same scaling if s=0. We prove existence and uniqueness of stationary states and we characterize their asymptotic behavior as s goes to zero. Moreover, we prove existence of gradient flow solutions to the evolution problem by applying the JKO scheme.

(1. 2) The natural free energy associated with the nonlocal PDE (1.1) is given by We notice that (1.1) has the structure of a continuity equation ∂ t ρ + ∇ • (ρu) = 0, where the velocity vector field is a gradient u = −∇ψ and the velocity potential is the functional derivative of F s [ρ] with respect to ρ.For this reason, the evolution equation 1.1 is formally the gradient flow of functional F s with respect to the Wasserstein distance.
Our first objective is the analysis of stationary states of the dynamics, with most emphasis on their behavior as s becomes small.In fact, in our first result we show that for any given mass M > 0, F s has a unique minimizer over coinciding with the unique radial stationary state of the dynamics with mass M and center of mass at the origin.Properties of stationary states have been thoroughly investigated for β = 0 and for different ranges of m, s, which are usually classified as follows: by considering the homogeneity property of the terms of functional F s , diffusion and aggregation are in balance if m is equal to the critical exponent m c := 2 − 2s/d, which is the so called fair competition regime that is analyzed in [8,9].The diffusion dominated regime m > m c was investigated in [12].Uniqueness of stationary states have also been proved in the different regimes [10,15,17].
Since the diffusion exponents in (1.1) are greater than 2, here we are considering a diffusion dominated model.However, the competition between the additional quadratic diffusion term and the aggregation term becomes crucial in the small s regime, since the limiting critical exponent is exactly 2.
Let us introduce the precise notion of stationary state.We will check in Section 2 that the assumptions on ρ in the next definition entail ρ∇ψ ∈ L 1 loc (R d ), where ψ is given by (1.4).
We stress that this definition differs from the one appearing in [8] and in later works.The definition that we propose is better suited to treat the small s regime.Indeed, as we explain in Section 2, minimizers of F s always satisfy the new definition.We have the following Theorem 1.1 Let β ≥ 0. If β = 0 and s < 1/2 assume in addition that m < 2−2s 1−2s .For any mass M > 0, there exists a unique stationary state of mass M and center of mass 0. Such steady state is radially decreasing, compactly supported, Hölder on R d and smooth inside its support.It coincides with the unique minimizer of the energy functional F s in the class Y M .
If β = 0 and s < 1/2, without the additional restriction m < 2−2s 1−2s we are not able to apply the radiality result from [11] and [12], and for this reason we do not get the same conclusion, but we shall still obtain uniqueness in the class of radial stationary states.
We are mostly interested in the limiting behavior of stationary states as s → 0. In this perspective, since K s → δ 0 , the limit functional is formally given by It is clear that the minimization problem min Y M F 0 is strongly influenced by the sign of the coefficient β − χ/2.Indeed, it has solutions if and only if β < χ/2, and in such case we will check that there is a unique radially decreasing minimizer, given by the characteristic function of a ball.Our second main result is the following.It will be proven in section (3), where some illustration of stationary states from numerical simulations will also be provided.
Theorem 1.2 For any s ∈ (0, 1/2), let ρ s ∈ Y M be the unique minimizer of F s over Y M .If 0 ≤ β < χ/2, there exists ρ ∈ Y M such that ρ s → ρ strongly in L m (R d ) as s ↓ 0, and moreover ρ is the unique radially decreasing minimizer of the functional (3.1) over Y M .Else if β ≥ χ/2, we have lim s↓0 F s [ρ s ] = 0 and ρ s → 0 uniformly on R d .
We next focus on the gradient flow structure of evolution problem (1.1).In this case, the initial datum ρ 0 is supposed to belong to Y M,2 , being Y M,2 the set of all densities in Y M with finite second moment: The analysis of Keller-Segel models with Newtonian potential as Wasserstein gradient flows is found in [4,3,6,5].More generally, there are many studies about gradient flow approach for interaction-driven evolutions.The case of the Riesz potential appears in [22], in the analysis of the porous medium equation with fractional pressure introduced in [7].Problem (1.1) is formally preserving mass, positivity and center of mass, and a solution is naturally seen as a trajectory in the space Y M .A narrowly continuous curve [0, +∞) ∋ t → ρ(t, •) ∈ Y M (i.e., t → R d ϕ(x)ρ(t, x) dx is continuous for every continuous bounded function ϕ on R d ), is a weak solution to (1.1) if ρ(0) = ρ 0 and for every ϕ ∈ C ∞ c (R d ) and every η ∈ C ∞ c ((0, +∞)) η(t) ∇ϕ(x) − ∇ϕ(y) • (x − y) |x − y| d+2−2s ρ(t, x)ρ(t, y) dx dy dt. (1.5) This notion of solution was introduced in [30] for the case of the interaction with the logarithmic potential, see also [4].We shall construct weak solutions to problem (1.1) by applying the Jordan-Kinderlehrer-Otto [18] scheme.Therefore, denoting by W 2 the Wasserstein distance of order 2, for a discrete time step τ > 0, we shall solve the recursive minimization problems and we shall prove that piecewise constant in time interpolations of minimizers do converge to a weak solution to (1.1) as τ → 0 along a suitable vanishing sequence (τ n ) n∈N .A weak solution that is constructed in this way, that is, as a limit of the JKO scheme applied to F s , will be called a gradient flow solution.We have the following existence result Theorem 1.3 Let β ≥ 0 and 0 < s < min{1, d/2}.If β = 0, assume in addition that d ≥ 2 and 1/2 ≤ s < 1.Let ρ 0 ∈ Y M,2 .Then there exists a gradient flow solution to problem (1.1).
Solutions are global-in-time, as expected in a diffusion dominated model.Further properties of solutions will be described in Section 4, along with some numerical analysis of the problem in Section 5. We shall also discuss an interesting feature: radial decreasing initial data do not necessarily preserve radial monotonicity during the dynamics.
We finally turn the attention to the behavior of solutions as s → 0. The formal limiting equation reads and its behavior is again crucially depending on the sign of the coefficient β − χ/2.Here, we limit ourselves to treat the case β ≥ χ/2, in which (1.6) is a standard degenerate parabolic equation.We have the following Theorem 1.4 Let β ≥ χ/2.Let ρ 0 ∈ Y M,2 .Let (s n ) {n∈N} ⊂ (0, 1/2) be a vanishing sequence, and for every n ∈ N let ρ n be a gradient flow solution to (1.1) with s = s n .Then the sequence (ρ n ) n∈N admits strong L 2 loc ((0, +∞); L 2 (R d )) limit points.If ρ is one of such limit points, then [0, +∞) ∋ t → ρ(t, •) is narrowly continuous with values in Y M,2 , ρ(0, •) = ρ 0 and ρ is a distributional solution to the nonlinear diffusion equation (1.6), i.e.,

Plan of the paper
In Section 2 we discuss Definition 1.1 and prove uniqueness of stationary states along with some regularity properties.Asymptotic behavior of stationary states as s approaches 0 is investigated in Section 3. In Section 4 we prove the theorems about the evolution problem.
In Section 5 we provide some numerical examples demonstrating other phenomena, and in Section 6 we provide a discussion on some open problems.

Stationary states and minimizers of the free energy
We start this section by discussing the definition of stationary states for the evolution equation in (1.1), see Definition 1.1.We begin with the following lemma about Riesz potentials of bounded continuous where ψ is the velocity potential defined by (1.4).The lemma includes the equivalence of two formulations for the equation that governs stationary states in Definition 1.1.
if and only if We postpone the technical proof of the above lemma to the appendix.Here, we focus on its consequences in relation with the definition of stationary states and on further properties that follow from Definition 1.1.
Remark 2.2 By virtue of Lemma (2.1), we have the equivalence between the stationary version of (1.5) and (2.1).Therefore, Definition 1.1 agrees with the natural definition of steady solutions of the evolution equation in (1.1) in the formulation (1.5), i.e., time independent solutions.Remark 2.3 Definition 1.1 provides a notion of stationary state which is weaker than the ones previously used in [8,9,10,12,11,17], see [8,Definition 2.1].Indeed, the latter definition requires ρ∇ψ = 0 in D ′ (R d ) along with more regularity properties of ρ.We point out that the W 1,1 (R d ) regularity of the density ρ and the W 1,∞ regularity of the velocity potential ψ in the interior of the support of ρ do always agree with the properties of the minimizers of the free energy functional F s that we shall discuss later on.On the other hand, such minimizers do not match the notion of stationary states from [8, Definition 2.1] if s is small.
Our aim is now to show that if ρ is a steady state, then we can obtain the natural zerodissipation identity for ψ defined by (1.4).
Proposition 2.4 Assume that ρ is a stationary state for the evolution equation in (1.1), according to Definition 1.1.Then R d |∇ψ| 2 ρ dx = 0.In particular ψ is constant in each connected component of Ω.
), and then there exists a sequence for every n ∈ N. Since ρ∇ψ ∈ L ∞ (Ω), by taking the limit as n → ∞ we find for every k ∈ R. The first term in the right hand side vanishes as k → +∞, since and since ∇η k → 0 in L ∞ (R d ).Therefore from (2.4), by applying the monotone convergence theorem to the second term in right hand side, we obtain R d |∇ψ| 2 ρ dx = 0. Since ρ > 0 in Ω, this implies ∇ψ = 0 a.e. in Ω, thus ψ is constant in each connected component of Ω.
Stationary states are closely related to minimizers of the free-energy functional F s .Indeed, by adapting the arguments of [12], we shall prove that for every M > 0 a minimizer of functional F s over Y M does exist, that it is necessarily radially decreasing and it satisfies a suitable Euler-Lagrange equation.Thanks to these properties, a minimizer of F s over Y M will be immediately seen to be a stationary state of mass M .In the analysis of minimizers we shall make use of the following notation ) , as a consequence of the Hardy-Littlewood-Sobolev inequality (see [20,Theorem 4.3]), that reads (notice that due to the condition m > 2 we always have where the optimal constant H d,s is given by As a direct consequence, we check that for any ρ ∈ Y M we have, for some constant C only depending on χ, m, s, d, Indeed, By the sharp Hardy-Littlewood-Sobolev inequality (2.6), letting there holds . (2.9) By taking advantage of the interpolation inequality and of Young inequality (taking conjugate exponents p, p ′ with p = 2θ), from (2.9) we deduce where C = C(χ, m, s, d) is defined by Then we have We have the following result about existence of minimizers, which employs the classical concentration compactness theorem of Lions [21].
Lemma 2.5 The functional F s admits a minimizer over Y M .If ρ s ∈ argmin Y M F s , then ρ s is radially decreasing, compactly supported and it satisfies ) In particular, we have (2.15) For the proof of the above lemma we follow the concentration compactness argument as applied in Appendix A.1 of [19].Indeed, the proof is based on [21, Theorem II.1, Corollary II.1], that are next recalled, denoting by M p (R d ) the Marcinkiewicz space or weak L p space.Theorem 2.6 [21, Theorem II.1] Suppose K ∈ M p (R d ), 1 < p < ∞, and consider the problem Here, and the nonlinearity j : R + → R + is a strictly convex nonnegative function such that Then there exists a minimizer of problem (I M ) if the following holds: (2.17) Proof of Lemma 2.5.The proof is similar to [12,Theorem 5], but we sketch the main lines of the arguments for the sake of completeness.Let p = d d−2s and q = p+1 p .We first notice that our potential ) and it is clear that K s verifies the homogeneity assumption in Proposition 2.7 for λ = d − 2s.Moreover the nonlinearity verifies the properties of Theorem 2.6, since m > 2 implies m > q.Then we just have to show that there exists some density ρ ∈ Y q,M such that F s [ρ] < 0. We fix R > 0 and define where B R denotes the ball centered at zero and of radius R > 0, and where is the surface area of the d-dimensional unit ball.Then Therefore we find Since m > 2 we have d(1 − m) < −d < 2s − d, then we can choose R > 0 large enough such that F s [ρ * ] < 0, and hence condition (2.17) is verified.Then Proposition 2.7 and Theorem 2.6 implies that there exists a minimizer ρ s of F in Y q,M Now, the Hardy-Littlewood-Sobolev inequality (2.6) implies that ρ s ∈ L m (R d ).Moreover, the Riesz rearrangement inequality [20,Theorem 3.7] yields that ρ s is radially decreasing (then K s * ρ s is also radially decreasing and vanishing at infinity).
The fact that ρ s satisfies (2.12) can be obtained by computing the first variation of F s at ρ s , as done in [14,Theorem 3.1].Then (2.13) follows by taking into account the equation (2.12) satisfied by ρ s , multiplying it by ρ s and integrating over R d .Since C s in (2.12) is positive, and since both ρ s and K s * ρ s are radially decreasing and vanishing at infinity, from (2.12) we deduce that ρ s is compactly supported.In particular Eventually we prove (2.14) and (2.15).Using the mass invariant dilation ρ s,λ (x) = λ d ρ s (λ x) we easily find By the minimality of ρ s , the function h(λ) has its unique minimizer at λ = 1, which implies by differentiation Then where the nonlinearity g is defined as ), by Hölder inequality and the radial monotonicity there is a constant C ≥ 0 such that .
Then we can proceed exactly as in [15,Proposition 2.8] in order to conclude that v 0 ∈ L ∞ (R d ), which implies in turns by (2.12) that ρ 0 ∈ L ∞ (R d ).Now we turn to the regularity properties of ρ 0 and we refer mainly to [12,Theorem 8].Let us first consider the easiest case s > 1/2.We have in this case u 0 ∈ W 1,∞ (R d ) from [12, Lemma 1], thus by (2.18) we have therefore (since g is Lipschitz) , by the first part of the proof of [12,Theorem 8] it follows that u 0 ∈ C 0,γ for any γ < 2s, then equation (2.19) gives ρ 0 ∈ C 0,γ for any γ < 2s.By the Hölder regularity of the Riesz potential (see again [12,Eq. 3.24]) we have u 0 ∈ C 0,γ for any γ < 4s.Bootstrapping, we finally find u 0 , ρ 0 ∈ W Proof.By Lemma 2.5, Lemma 2.8 and Remark 2.9, ρ s is continuous, compactly supported, radially decreasing and smooth on B R , where R is the radius of its support.Therefore its radial profile is absolutely continuous in [0, R], and belongs to the weighted Sobolev space W 1,1 ((0, R), r d−1 dr).This implies where ψ is defined by (1.4), thus ρ∇ψ ∈ L 1 (R d ).But the validity of (2.12) implies that ψ is constant on B R .In particular ρ∇ψ = 0 a.e. in R d and (2.1) follows.
The complete characterization of stationary states (according to Definition 1.1) is finally given by the next theorem.
Theorem 2.11 Let β > 0 (resp.β = 0).For any mass M > 0, there exists a unique stationary state (resp.a unique radial stationary state) of mass M and center of mass 0. Such steady state is radially decreasing, compactly supported, Lipschitz on R d (resp.Hölder on R d ) and smooth inside its support.Moreover, it coincides (up to translation) with the unique minimizer of the energy functional F s in the class Y M .
Let as usual Ω = {ρ > 0}.Since ρ is continuous, Ω is open and thus Ω = ∪ ∞ n=1 Θ n , where the Θ n 's are the countably many open connected components of Ω.Let us introduce the continuous functions From Proposition 2.4, for each n ∈ N there is a constant Q n ∈ R d such that there holds where f (t) := m m−1 t m−1 + 2βt, t ≥ 0. The constant Q n is nonnegative, since ρ and v are nonnegative continuous and ρ vanishes on ∂Θ n .Therefore we have where g : [0, +∞) → [0, +∞) is the inverse function of f , and we notice that g is Lipschitz on [0, +∞) since β > 0. Hence, we have ρ n ∈ C 0,γ (Θ n ), and (2.20) implies ρ n ∈ C 0,γ (R d ).We stress that the Hölder constant of ρ n , that we denote by c, is independent of n, since it only depends on the Hölder constant of v and the Lipschitz constant of f .Now, for every two distinct points x ∈ R d and y ∈ R d , there exist n ∈ N and m ∈ N such that ρ(x) = ρ n (x) and ρ(y) = ρ m (y) and for every arbitrarily small ε.Then we may bootstrap this argument and in a finite number of steps we get v ∈ C 0,1 (R d ) and ρ ∈ C 0,1 (R d ).Since m > 2 and ρ is bounded, we conclude that ρ m−1 is Lipschitz on R d as well.We recall that by Proposition (2.4) the the velocity potential ψ defined by (1.4) is constant on each connected component of ρ.
Now, an easy modification of [12, Theorem 3], which crucially exploits the Lipschitz regularity of ρ m−1 , shows the radiality of the steady state ρ: indeed, the actual entropy decreases under the modified continuous Steiner symmetrization introduced in the proof therein.
All the regularity properties of the steady states can be argued by the proof of Lemma 2.8.The uniqueness follows by the results of [17], namely by the Remark under the statement of [17,Remark 1.2], because the nonlinearity Φ : R + → R + , defined by Φ(ρ) = 1 m−1 ρ m + βρ 2 entering in the nonlinear diffusion is a strictly increasing smooth convex function.Finally, the identification (up to translation) with the unique minimizer of F s over Y M follows from Proposition 2.10.
Finally, if β = 0, we only consider the class of radial stationary states, and the uniqueness in this class follows from the results in [17].Again the unique radial stationary states is the unique minimizer of F s over Y M , see Proposition 2.10, and the other properties follow from Lemma 2.5 and [12,Theorem 8].
Remark 2.12 If β = 0 the general uniqueness result of Theorem 2.11 (without assuming radiality) is still an open problem, mainly due to the fact that the inverse function of 1) for m > 2 but not Lipschitz, preventing to obtain that ρ m−1 is Lipschitz for a stationary state ρ, which is crucial for applying the radiality result of [11].However, we still have radiality of every stationary states, thus uniqueness, as long as we assume in addition that m < m * , because in this case ρ m−1 can be proven to be Lipschitz, see [12,Theorem 8].
Proof of Theorem 1.1.The result follows from Theorem 2.11 and Remark 2.12.
3 Asymptotic behavior of stationary states as s → 0 We investigate the asymptotic behavior of ρ s as s approaches 0, where ρ s is, for each small s, the unique stationary state of given mass M and center of mass 0 for the equation in 1.1, provided by Theorem 2.11.Thanks to the identifications with minimizers of the free-energy, this will be done by showing that functionals F s Γ-converge to the limit energy functional F 0 defined by whose minimization is governed by the following proposition.
Proposition 3.1 (Minimization of the limit functional F 0 .)Suppose that 0 ≤ β < χ/2.Then functional F 0 admits a unique radially decreasing minimizer over Y M , given by where Proof.Through the proof, we let for simplicity γ := −β + χ/2.For every ρ ∈ Y M and every with value But writing thus for every ρ ∈ Y M there holds Therefore, if there is a density ρ achieving the constant at the right-hand side of (3.4), then ρ is a minimizer of F 0 .However, the above Hölder inequality is an equality if and only ρ is a multiple of a characteristic function, i.e., ρ(x) = t½ Ω (x) for some measurable subset Ω of R d , and the condition R d ρ(x) dx = M implies |Ω| > 0 and t = M |Ω| −1 .In particular, the second inequality in (3.4) is an equality if and only if ρ(x) = M |Ω| −1 ½ Ω (x).On the other hand, the first inequality in (3.4) is an equality if and only if ρ = ρ λ * , i.e., λ * = 1, We conclude that both inequalities in (3.4) are equalities if and only , implying that this family of functions coincides with argmin Y M F 0 up to translations.In this family there is a unique radially decreasing function ρ 0 , obtained by letting Ω = B R and by choosing Hence, ρ 0 is given by (3.2).The statement concerning the case β ≥ χ/2 (i.e., γ ≤ 0) follows by letting λ → 0 in (3.3). is also plotted for reference.
The steady states ρ s in one dimension for different values of s are shown in Figure 1, with m = 3, χ = 1, obtained by iterating the governing equation (2.12), i.e., given by Eq. (2.13), followed by a spatial scaling ρ new) x such that the total mass is exactly M .The convergence towards the limit ρ 0 in Proposition 3.1 as s goes to zero is illustrated, for both β = 0 (left figure) and β > 0 (right figure).The regularizing effect with β > 0 is obvious, especially near the boundary of the support.The next step is the investigation of the behavior of F s on characteristic functions.
where B R is the ball of radius R, centered at the origin.Assume that 0 ≤ β < χ/2.Then there exists a unique positive number R s such that In particular, for β = 0 its value is . Moreover, the map (0, 1/2) ∋ s → F s [ρ Rs ] is continuous, it has negative value for any s ∈ (0, 1/2) and there holds In particular, the value (3.5) is exactly F 0 (ρ R 0 ), where R 0 is given by (3.2).
Proof.By the proof of Lemma 2.5 we argue that for large R we have F s [ρ R ] < 0.Moreover, by the proof of [15, Lemma 6.1] one has and .
Since m > 2 and β < χ/2, it is then readily seen that for all s ∈ [0, 1/2), the map (0, +∞) ∋ R → F s [ρ R ] admits a unique minimizer R s , which is the unique solution to the algebraic equation Due to the structure of the coefficients, observe that the map h(•, R) is continuous for s ∈ [0, 1/2).We claim that the map s → R s is continuous up to s = 0 in the interval (0, 1/2).To prove the claim, for any s 0 ∈ [0, 1/2), let (s 0 , R s 0 ) be the unique solution to (3.6).Then we have ∂h ∂R Then the Implicit Function Theorem assures that the map s → R s is continuous in a neighborhood of any point s 0 and the claim follows.This implies in particular that the map s → F s (ρ Rs ) is continuous at s = 0 and we have then the only issue is to compute the value R 0 , which is obtained by equation (3.6) letting s → 0. Such computation shows that the value R 0 is the one in (3.2).Then, inserting ρ R 0 in the expression (3.1) of F 0 we have as desired.
Remark 3.3 Since the limit value as s → 0 of F s in ρ R is given by Next we investigate some asymptotic properties of minimizers as s ↓ 0.
where C s is defined in (2.15).
Proof.By (2.15) and (2.14) we obtain and the result follows by the minimality of ρ s and Lemma (3.2).
Proof.We a slight abuse of notation we still denote by ρ s the radial profile of ρ s and we notice that since ρ s is radially non-increasing there holds for any R > 0 Let supp(ρ s ) =: B Rs .Form (2.12) we deduce Assuming by contradiction that lim sup s↓0 R s = +∞, the above computation shows, recalling from (1.2) that c d,s /s is bounded on (0, s 0 ), that This contradicts (3.8), since lim inf s↓0 C s > 0 by Lemma 3.5.
Proof.We have ρ s ∈ W 1,1 (R d ), by reasoning as done in the proof of Proposition 2.10.We still denote by ρ s the radial profile of ρ s and we notice that ∇ρ s (0) = 0 and ∇ρ s The existence of such R, s 0 is due to Lemma 3.6.We have Therefore we always have where the finiteness is due to Lemma 3.4.The above uniform BV (R d ) estimate and the usual compact embedding ) compactness of the family (ρ s ), which can be extended to the whole L 1 (R d ) by the tightness due to Lemma 3.6.If ρ is a limit point along a vanishing sequence s n , we also have ρ ∈ L ∞ (R d ) and ρ sn → ρ strongly in L p (R d ) for any p ∈ [1, +∞), since the sequence (ρ sn ) is also equi-bounded by Lemma 3.4.
Proof.By Plancherel theorem we have About the first term in the right hand side, for any R > 1 we have, since As s ↓ 0, the first term in the right hand side goes to zero by dominated convergence (as dominating function we take |ξ| −2s 0 + 1 for |ξ| ≤ 1 and 2 for 1 < |ξ| ≤ R).Therefore, by the strong L 2 (R d ) convergence of ρ s to ρ, from (3.9) we get The result follows, since ρ ∈ L 2 (R d ) and R is arbitrary.Now we are in the position to state the main result about convergence of minimizers ρ s towards ρ 0 , where ρ 0 is defined in (3.2).
Theorem 3.9 Assume 0 ≤ β < χ/2.For any s ∈ (0, 1/2), let ρ s ∈ Y M be the unique minimizer of F s over Y M .Then, there exists ρ ∈ Y M such that ρ s → ρ strongly in L m (R d ) as s ↓ 0.Moreover, ρ is the unique radially decreasing minimizer of the functional (3.1) over Y M , given by (3.2).
Proof.Let (s n ) ⊂ (0, 1/2) be a vanishing sequence.Let ρ ∈ Y M be such that ρ sn → ρ strongly in L m (R d ) as n → +∞.The existence of such a limit point follows from Lemma 3.7, and the convergence holds in L p (R d ) for any p ∈ [1, +∞).Given ρ ∈ Y M , by the strong L m (R d ) convergence and by Lemma 3.8 we have By the arbitrariness of ρ, we conclude that ρ is a minimizer of F 0 over Y M .Finally, the whole family (ρ s ) converges to ρ in L m (R d ) as s ↓ 0.
For every ρ ∈ Y M and every λ > 0, let ρ λ (x) = λ d ρ(λx), x ∈ R d .We have Similarly to the proof of Proposition 3.1, we minimize the right hand side with respect to λ and find a unique optimal value λ * given by and by inserting such a value in (3.10) we get .
But the Hardy-Littlewood-Sobolev inequality (2.6) along with interpolation of L p norms, entails therefore we get the estimate for every ρ ∈ Y M .It is not difficult to check that (c d,s H d,s ) d 2s converges to a finite limit as s ↓ 0, therefore the above right hand side is negative and converges to 0 as s ↓ 0 due to the condition β ≥ χ/2.If ρ s denotes the unique minimizer of F s over Y M , we deduce from (2.14) that lim s↓0 F s [ρ s ] = 0.This implies from (2.13) and (2.14) that ρ s → 0 in L m (R d ) and C s → 0 as s ↓ 0.
Eventually, we prove that ρ s ∞ → 0 as s ↓ 0. Since ρ s is continuous and radially decreasing we have ρ s ∞ = ρ s (0), and as seen in the proof of Lemma 3.4 we may take advantage of (2.12) and get Since m > 2 and β ≥ χ/2, this forces ρ = 0 which is the desired result.

Weak solutions for the aggregation-diffusion problem
The first objective of this section is the proof of the main existence result stated in Theorem (1.3).We mention that an alternative existence proof for s > 1/2 and β = 0 is found in [35].
We fix M > 0. For ρ 0 ∈ Y M,2 , and we consider the Cauchy problem (1.1)We shall construct a weak solution to problem (1.1) by an application of the JKO scheme to the functional F s defined by (1.3).Therefore, for a discrete time step τ > 0, we consider the minimization problem for every ε > 0. On the other hand, by Cauchy-Schwarz and Young inequality we have for every n ∈ N and every ε > 0, where C is a constant that does not depend on n, since (ρ n ) is a bounded sequence in L 2 (R d ).A combination of the two above relations yields for every ε > 0. By dominated convergence, the two terms in the right hand side vanish as ε → 0, so that the arbitrariness of ε entails By the weak lower semicontinuity of the L m (R d ) and of the L 2 (R d ) norms, by the narrow lower semicontinuity of W 2 (•, ρ 0 ) (see [1,Proposition 7.1.3])and thanks to (4.2) we conclude that Since (ρ n ) is a minimizing sequence, we conclude that ρ * is a solution to problem (4.1).
Once existence of a discrete solution is established, we perform a recursive minimization and apply standard arguments from the theory of minimizing movements to obtain convergence of the scheme and existence of a limit curve, as summarized in the next two statements.We let ρ 0 τ := ρ 0 and for every k ∈ N, we take recursively thus defining a sequence (ρ k τ ) k≥1 of discrete minimizers, whose existence is ensured by Proposition 4.1.For every k ∈ N there hold and where C = C(χ, m, s, d, M ) is the constant given by (2.11),only depending on χ, m, s, d, M .
The curve t → ρ(t, •) that was obtained in Proposition 4.3 will be shown to be a weak solution to (1.1).The first step towards this goal is to obtain a first order optimality condition for discrete minimizers of the JKO scheme.
Here, T * is the unique optimal transport map (for the quadratic cost) from ρ * (x) dx to ρ 0 (x) dx.we get where I is the identity matrix and ∇ 2 is the Hessian operator.Therefore we have d dε which is of course still true if m is replaced by 2. On the other hand, the definition of push-forward entails

It is clear that
For the derivative of the Wasserstein distance, by a standard result (see [34,Theorem 8.13]) we get d dε Since ρ * is a minimizer, the derivative with respect to ε of F s [ρ ε ] + 1 2τ W 2 2 (ρ ε , ρ 0 ) needs to vanish at ε = 0. We obtain the result.
If we wish to remove the compatibility condition (4.9), we just replace ζ with hence ζ satisfies (4.9).Inserting ζ in (4.8) and taking into account that we have that (4.8) holds for any test function The next result is based on a different perturbation of ρ * , which gets perturbed along the solution of the heat equation originating from it.For nonnegative L 1 (R d ) functions u with finite second moment on R d we introduce the entropy functional which is a displacement convex functional in the sense of McCann [24].We recall that the solution u of the heat equation ∂ t u = ∆u with initial datum ρ 0 ∈ Y M,2 is the Wasserstein gradient flow of G and it satisfies the evolution variational inequalities for every w ∈ Y M,2 and every t > 0, see [1,Chapter 11].This allows to take advantage of the flow interchange lemma introduced in [23], as we do in the next proof.
Proof.Let us introduce the Cauchy problem The unique solution to the heat equation with initial datum in ρ * ∈ Y M,2 is given by u(t, •) = Γ t * ρ * , where Γ t (x) := (4πt) −d/2 exp{−|x| 2 /(4t)} is the Gaussian kernel.u(t, •) is smooth, positive for every t > 0, and moreover a direct computation by means of integration by parts shows that for any t > 0 there holds and in particular Similarly, thanks to [22, Lemma 4.5] we have for any t > 0 Here, Ḣr (R d ) denotes the homogeneous Sobolev space of order r ∈ R, i.e., the completion of and by Young inequality we deduce, for given α > 0, The choice α = (χ(1 − s)/(2β)) s−1 in (4.14) entails, together with (4.11) and (4.12), for every t > 0.Moreover, the maps are continuous up to t = 0.They are also differentiable at any t > 0 with derivatives given by (4.11), (4.12) and (4.13), therefore by Lagrange mean value theorem, for every t > 0 there exists θ(t) ∈ (0, t) such that , so that by applying (4.15) we obtain Since the L 2 (R d ) norm decreases along the solution to the heat equation (4.10), we deduce showing that Therefore, we can apply the flow interchange lemma from [23], in its version from [22,Proposition 4.3] and deduce thus showing that the spatial gradient of u(θ(t), Thus Sobolev embedding shows that u(θ(t), •) m/2 is in fact bounded in H 1 (R d ) as t ↓ 0. Since θ(t) → 0 as t ↓ 0, and since u(θ(t), •) → ρ * pointwise a.e. as t ↓ 0, by the weak lower semicontinuity of the H 1 (R d ) norm we finally deduce which is the desired result.
Moreover, for every T > 0 there holds the time integrated estimate where C * i , i = 1, 2, 3, are a suitable explicit constants, only depending on χ, M, m, s, d, β, and on ρ 0 .
Proof.The first estimate in the statement is a direct consequence of Proposition 4.5, and it implies that for every T > 0 we have From (4.5) we also have for every We insert the latter two estimates, combined with (4.4)-(4.5),into (4.17):since ⌈T /τ ⌉τ ≤ T +τ we deduce , where where C * 2 is twice the right hand side of (4.4) and where C * 3 is defined in (4.18).The proof is concluded.

Proof. Let us introduce the auxiliary functional
and the Cauchy problem This is a standard porous media equation with initial datum in Y M,2 and it enjoys the following properties, for which we refer to [33, Theorem 9.12, Proposition 9.13]: there exists a unique strong solution u (meaning that the equation ∂ t u = ∆u m−1 is satisfied pointwise a.e. in spacetime) such that u ∈ C 0 ([0, +∞); L 1 (R d )) and ∇u m−1 ∈ L 2 ((0, T ) × (R d )) for every T > 0.
We have shown the absolute continuity of the map t → F s [u(t, •)], which together with (4.20) and (4.22) entails for every t > 0 By applying (4.23), since the L 2d d+4s−2 (R d ) norm decreases along the solution to the porous media equation (4.19), we deduce showing that Therefore, we can apply the flow interchange lemma, in its version from [22,Proposition 4.3] and deduce .
By the absolute continuity of the map t → t 0 R d |∇u(r, x) m−1 | 2 dx dr, we may apply l'Hospital rule and get By taking a suitable vanishing sequence (t n ) n∈N of positive numbers, the above bound shows that ∇u(t n , •) m−1 → u * weakly in L 2 (R d ) as n → +∞.But u(t n , •) strongly converge to ρ * as n → +∞, hence up to subsequences we also have that

and weakly in
This allows to conclude that u * = ∇ρ m−1 * , and the weak lower semicontinuity of the L 2 (R d ) norm yields the desired estimate.Since , ∈ (0, 1).
Proof.Let T k+1 τ be the unique optimal transport map (for the quadratic cost) from ρ k+1 τ (x) dx to ρ k τ (x) dx.Let ϕ ∈ W 2,∞ (R d ) be smooth, let h > 0, and notice that where we used the Taylor expansion formula where Let us separately treat the two terms in the right hand side.For the first, by (4.8) we have where K is an explicit constant, only depending on M, m, s, d, β, χ and ρ 0 , which can be obtained by applying Proposition 4.2, and in particular by combining (2.10) and (4.5).Concerning the second, we have By inserting the latter two estimates in (4.25) we deduce that for every smooth function where H m is defined by (2.5) and C is defined by (2.11), and where the latter inequality follows again from Proposition 4.2 together with (2.10), and from the basic inequalities ⌈ t+h τ ⌉ ≤ t+h τ +1, ⌈ t τ ⌉ ≥ t τ .Therefore, fixing q ∈ N with q > 2+d/2, so that the continuous embedding given by Morrey's theorem In particular lim sup The conclusion is similar to the one in [6, Proposition 14].Let (τ n ) and ρ be the vanishing sequence and the limit function in the statement.Let us consider the set of functions {ρ τn (t, •) : t ∈ [0, T ], n ∈ N}.This is a set of functions having uniformly bounded second moments and uniformly bounded L 1 ∩ L 2 (R d ) norm, thanks to the estimates (4.5) and (4.6).Hence, it is relatively compact in H −q (R d ) by Lemma 4.11 below.Thanks to this fact and to the equicontinuity estimate (4.26), we may apply [1, Proposition 3.3.1]and deduce that there exists a vanishing subsequence (τ n k ) and ρ ∈ C([0, T ]; By uniqueness of the limit we have ρ ≡ ρ and the above convergence holds along the original sequence (τ n ).
The conclusion of the proof is split in two cases.We first consider the case β > 0, and we will take advantage of Corollary (4.7).We observe that for every for a suitable constant c.Therefore, from (4.16) and Jensen inequality we deduce that where C T is a constant depending only on M, m, s, d, β, χ, T and the initial datum ρ 0 .This shows that the sequence ρ τn is bounded in L 2 ((0, T ); X m ).Since L 2 (R d ) continuously embeds in H −q (R d ), by the uniform L 2 (R d ) bound deduced from (4.5) and by (4.27) we may apply the dominated convergence theorem and obtain the convergence of ρ τn to ρ in L 2 ((0, T ); The latter convergence, together with (4.28) allows for an application of the compactness result in the space L 2 ((0, T ); L 2 (R d )) from [31, Lemma 9] so that we conclude that Here, [31, Lemma 9] is applied by using the Banach triple Proof.Let us give the proof for Y 1 (the argument for Y 2 is analogous).Let us consider a sequence for every ball in R d , so that there is u ∈ L 2 (R d ) and a not relabeled subsequence (u n ) such that u n → u strongly in L 2 (B) for every ball B and weakly in L m (R d ).Let ε > 0 and choose B = B ε to be a large enough ball, such that for every n ∈ N: this is possible thanks to the tightness of the sequence (u n ), which has uniformly bounded second moments by assumption.We have We also have and since u n → u strongly in L 1 (B), the arbitrariness of ε shows that u n → u strongly in L 1 (R d ) as well.Therefore, the boundedness of (u n ) in L m (R d ) and (4.29) show that u n → u strongly in L 2 (R d ).
Similarly, let (u n ) n∈N be a bounded sequence in L 2 (R d ) ∩ L 1 (R d , (1 + |x| 2 ) dx), and given ε > 0 let as above B = B ε such that R d \B |u| + R d \B |u n | < ε for every n ∈ N. By Sobolev embedding, H q (B) compactly embeds into L 2 (B) and then (by Schauder's theorem) L 2 (B) compactly embeds in the dual space H q (B) * , therefore up to subsequences we have u n → u weakly in L 2 (R d ) and strongly in H q (B) * .We have where we have also used the continuous embedding Taking the limit as n → +∞, since u n → u strongly in H q (B) * and since ε is arbitrary, we deduce that u n − u H −q (R d ) → 0.
We are ready to prove Theorem (1.3), recalling that by a gradient flow solution ρ = ρ(t, x) to (1.1) we mean a weak solution according to (1.5) which is a limit of the JKO scheme, i.e., ρ is a limit function (obtained from Proposition 4.3 along a vanishing sequence (τ n ) n∈N ⊂ (0, 1)) of the piecewise constant interpolations ρ τ (defined by (4.7)) constructed from a sequence (ρ k τ ) k≥0 of discrete minimizers from (4.3).Proof of Theorem 1.3.We apply Proposition 4.10: we have ρ τn → ρ strongly in L 2 ((0, T )× R d ).In particular, up to extracting a further subsequence, we have the pointwise a.e.spacetime convergence of ρ τn to ρ and thus of ρ m τn to ρ m .Thanks to the Sobolev embedding we deduce from (4.16) if β > 0 and from (4.24) if β = 0 that the sequence (ρ m τn ) n∈N is also bounded in , hence by interpolation we also find that (ρ 2 τn ) n∈N is bounded in the same space.Thus up the the extraction of one more subsequence we get x) ρ(t, y) dx dy for every t, by making use of the same argument of the proof of Proposition 4.1.By dominated convergence, the associated time integrals on (0, T ) also converge.Finally, we have for every η ∈ C ∞ c (0, T ), by the convergence properties of minimizing movements, see for instance [1,Theorem 11.1.6], where T τn (t, •) is the unique optimal transport map from ρ τn (t, x) dx to ρ ⌈t/τn⌉−1 τn (x) dx.Recalling the definition of ρ τ as the piecewise constant interpolation of discrete minimizers, by writing (4.8) for ρ τ we have x) ρ τn (t, y) dx dy for every t > 0. By multiplying the latter by η ∈ C ∞ c (0, T ) and by integrating on (0, T ), we may therefore pass to the limit along the sequence (τ n ) n and conclude that ρ is a weak solution to problem (1.1).
Let us collect some properties of the constructed solution.
(ii) The following L m (R d ) estimate holds for every t > 0 where C is defined by (2.11) and (2.8).
(iii) if β > 0, then ρ m/2 ∈ L 2 ((0, T ); H 1 (R d )) for every T > 0 along with the estimate where C * i are the explicit constants defined in the proof of Proposition 4.7.
(iv) for every T > 0 there holds for every T > 0 along with the estimate where C * * i are the explicit constants appearing in Corollary 4.9.
Proof.Point (i) was shown in Proposition 4.3.Point (ii) follows from the uniform L m (R d ) bound from (4.5) along with the narrow lower semicontinuity of the L m norm.Points (iii) and (iv) respectively follow from the uniform bounds (4.16) and (4.6), again by lower semicontinuity properties.Similarly, point (v) follows from (4.24).
We conclude this section by proving Theorem 1.4.Before giving the proof, we include a couple of technical lemmas, whose proofs are postponed to the Appendix.
A generalization of the previous lemma is the following We are ready for the proof of our last result.
Proof of Theorem 1.4.
•) is absolutely continuous with respect to W 2 as recalled in point (i) of Proposition 4.12, the map t → R d ϕ(x)ρ n (t, x) dx is absolutely continuous on [0, +∞) and we may write the time integrated version of (1.5), i.e., for every 0 ≤ t 1 < t 2 < +∞.We estimate the last term as done in (2.9)-(2.10),obtaining where C is defined by (2.11) and (2.8), therefore we have by It is immediate to check that sup s∈(0,1/2) C(χ, m, s, d, M ) < +∞, therefore by including the estimate in point (ii) of Proposition 4.12 we deduce that where C is a suitable constant, depending on m, d, χ, β, M and ρ 0 , but not on n.We take q ∈ N, q > 2 + d/2, so that we have the continuous embedding H q (R d ) ⊂ W 2,∞ (R d ) with embedding constant Q, and we deduce the time equi-Lipschitz estimate for every n ∈ N.
Letting T > 0, we repeat the same arguments of the proof of Proposition 4.10: indeed, the family of functions {ρ n (t, •) : t ∈ [0, T ], n ∈ N} has uniformly bounded second moments and L 1 ∩ L 2 (R d ) norms, which is seen by applying to ρ n the estimates of points (ii) and (iv) of Proposition 4.12 (as already noticed, the right hand side of (ii) can be estimated uniformly with respect to n ∈ N).Therefore by Lemma 4.11 such a family of functions is relatively compact in H −q (R d ), so that in view of the above equi-Lipschitz estimate we may apply By the dominated convergence theorem, we also get ρ n → ρ ∈ L 2 ((0, T ); H −q (R d )) as n → +∞: here, the dominating function for showing that ) and the estimate in point (ii) of Proposition 4.12, where again the right hand side is uniformly bounded with respect to n.The same estimate and estimate in point (iv) of the same Proposition implies that ρ n (t, •) converges weakly to ρ(t, •) in L 1 ∩ L m , therefore the weak lower semicontinuity of the L m norm also shows that ρ ∈ L ∞ ((0, +∞); L m (R d )).
The estimate in point (iii) of Proposition 4.12, applied to ρ n , implies , and the supremum in the right hand side is finite, thanks to the crucial assumption β ≥ χ/2 (as observed in Remark 4.6).Therefore, we may reason as done in the proof of Proposition (4.10) to get that the sequence (ρ n ) enjoys a uniform L 2 ((0, T ); W 2/m,m (R d )) bound.Similarly, in view of point (iv) of Proposition 4.12, the sequence (ρ n ) is also uniformly bounded in L 2 ((0, T ); L 1 (R d , (1 + |x| 2 ) dx)).By the same argument at the end of the proof of Proposition 4.10, we have ρ n → ρ strongly in L 2 ((0, T ) × R d ).
Let us conclude by passing to the limit in the equation.
. By definition of weak solution, for each n ∈ N we have that ρ n satisfies After multiplying by η and integrating on (0, T ), the time integral passes to the limit by dominated convergence: a dominating function is obtained by the usual estimates of the form (2.9)-(2.10),yielding for a.e.t ∈ (0, T ) where we have also used point (ii) of Proposition 4.12, and C = C(χ, m, s n , d, M ) stays bounded as n → +∞.
Eventually, we take advantage of the previously obtained uniform L 2 ((0, T ); W 2/m,m (R d )) estimate: as in the proof of Theorem 1.3, by Sobolev embedding it implies that (ρ m n ) n∈N is also uniformly bounded in ) which allow to pass to the limit in the other two terms of (4.30).

Some qualitative properties of solutions
In this section we present some numerical simulations about the evolution problem (1.1), using the scheme developed in [13].The time evolution is shown in Figure 2 and 3 for different initial data in one dimension, providing a numerical illustration of the expected asymptotic behaviors, i.e., solutions approaching the unique stationary states.In general, if s is not too close to zero, the stationary states are reached quickly, otherwise the convergence may take longer with the appearance of "disturbances" near the boundary of the support as in Figure 3 below.A big open problem concerning the Cauchy problem (1.1) is the uniqueness of the solution.Assuming that this property holds true, by the rotationally invariant property of the main equation (1.1) it follows that for a given radial initial datum ρ 0 = ρ 0 (|x|) we have that the density solution ρ is radial w.r.t. x, i.e. ρ = ρ(|x|, t).But the property of being radially decreasing may not be preserved during the evolution, even with initial data (and limiting steady states) sharing this property.The following counterexample is an adaptation of the one contained in [19,Proposition 4.3].Set (5.1) being ε > 0, ϕ a mollifier with mass 1 supported in the ball B(0, 1), being ϕ ε Assume that there exists a radial solution ρ(x, t) to (1.1) with datum ρ 0,ε and suppose we know that the solution ρ(x, t) is smooth enough up to t = 0. Then it is possible to show that ρ is not radially decreasing.Indeed, it is immediate to see that for ε small and ε Now, observe that since and ∆δ 0 is supported at 0, taking a cutoff function η ε such that η ε = 1 in a B(0, ε) we find as ε → 0 and an easy computation shows that ∆ we have and by Young inequality  as ε → 0. This implies that for ε small, ∆(K s * ρ 0,ε )(x 1 ) > ∆(K s * ρ 0,ε )(x 2 ), hence (5.2) gives (recalling that ρ 0,ε is radially decreasing) ∂ t ρ(x 1 , 0) < ∂ t ρ(x 2 , 0), meaning that the radially decreasing monotonicity is not preserved for small times.The non-monotonicity of the solution is shown in the simulation from Figure 4.
Figure 5 shows a further simulation, which takes into account a characteristic function of a symmetric interval as initial datum: also in this case the radial monotonicity is not preserved.

Comments, extensions and open problems
• As mentioned in the previous section, an open problem concerns the uniqueness of solutions, which would give radiality of solutions with radial initial data as a direct consequence.
• A second open problem is to rigorously prove that every solution to the evolution problem (1.1) does converge to the unique stationary state provided by Theorem 1.1.We mention that a similar result is available in the two dimensional setting, in the case of aggregation with the Newtonian potential instead of the Riesz potential, with β = 0 and m > 1 (i.e., diffusion-dominated regime), see [11].• Concerning Theorem 1.4, uniqueness of the distributional solution for the Cauchy problem for (1.6) with β ≥ χ/2 is known under additional conditions.For instance, according to the classical result by [26], uniqueness holds among distributional solutions that are essentially bounded on any strip R d × (τ, T ), for all T > 0 and τ ∈ (0, T ).Therefore, in order to obtain a unique limit as s → 0 for families of gradient flow solutions ρ s to (1.1), further a-priori L ∞ bounds (uniformly in s) should be established for ρ s .
• Another interesting open problem is to show that the family of solutions ρ s to problem (1.1) converges as s → 0 to a solution (in an appropriate sense) to the equation 1.6 even in the case β < χ/2.We notice that such equation has the form ∂ t ρ = ∆ϕ(ρ), and if β < χ/2 the nonlinearity ϕ is nonmonotone and equation (1.6) is of forward-backward type, with the unstable phase given by the interval [0, ( χ−2β m ) 1/(m−2) ] and the stable phase by [( χ−2β m ) 1/(m−2) , +∞).The nontrivial zero of ϕ is which coincides exactly with the height of the minimizer of the free energy limit functional F 0 given in (3.2).If β < χ/2, we would like to consider equation (1.6) as a singular limit as s → 0 of the main equation in (1.1).An existence theory for equation (1.6) supplemented with an initial condition ρ(x, 0) = ρ 0 (x) could be given in the setting of Young measure solutions, see for instance [27], where such notion of solution is recovered for cubic-like nonlinearities ϕ as vanishing limit as ε → 0 of a third order pseudo-parabolic regularization ∂ t ρ = ∆ϕ(ρ) + ε∆ρ t .
It would be interesting to show that even a weak limit ρ as s → 0 of a family of densities ρ s solving the equation (1.1) in the sense of Theorem 1.3 fits the above mentioned existence theory.The result is true if ρ ∈ C ∞ c (R d ), since in this case we may apply (6.4), and we may integrate by parts and take advantage of the fact that K s → δ 0 in the sense of distributions to get where the right hand side vanishes as s ↓ 0 thanks to the assumptions on the family (ρ s ).

1 m− 1 ,
and where the first embedding is compact by Lemma 4.11.The proof for the case β > 0 is concluded Eventually, let us consider the caseβ = 0, d ≥ 2, 1/2 ≤ s < 1.In this case we change the definition of the Sobolev space X m and we let X m = W 2m−2 (R d ), and by invoking Corollary (4.9) instead of Corollary (4.7), we conclude by repeating the same argument that we have used for β > 0. Lemma 4.11 Let m > 2. The spaces Y 1

Figure 2 :
Figure 2: The evolution of the solution starting with two bumps with parameters m = 3, χ = 1, s = 0.1 and β = 0.2, reaching the stationary state reasonably fast.

Figure 3 :
Figure 3: The evolution starting with a rescaled Gaussian, with m = 3, s = 0.08, χ = 1 and β = 0.4, where the solution does converge to the expected stationary state.

Figure 4 :
Figure 4: Numerical demonstration of the fact that radially decreasing initial data does not necessarily remain radially decreasing.The initial condition is the one in (5.1), with the parameters m = 3, χ = 1, s = 0.1 and β = 0.
[1,is clear from Lemma 2.5 that the functional to be minimized over Y M,2 is bounded from below.Let (ρ n ) ⊂ Y M,2 be one of its minimizing sequences.The sequence (ρ n ) has uniformly bounded L m (R d ) norm by inequality (2.7).It also has uniformly bounded second moment, thanks to the uniform bound for W 2 (ρ n , ρ 0 ), which follows from the fact that ρ n is a minimizing sequence and again from (2.7) and (2.10).Hence, up to subsequences, it converges to ρ * weakly in L p (R d ) for every p ∈ [1, m] and narrowly by Prokhorov's theorem (see e.g.[1, Theorem 5.1.3]).This implies that ρ * has mass M and by [1, Lemma 5.17] that ρ * has 0 center of mass, therefore ρ * ∈ Y M .Furthermore, by the lower semicontinuity of the second moments with respect to the narrow topology we have ρ Let ε > 0 and let η ε : R d → [0, 1] be a smooth cutoff function such that η ε [2, Proposition 4.1 (Existence of discrete minimizers) Let τ > 0 and ρ 0 ∈ Y M,2 .The minimization problem (4.1) admits solutions.Proof.*∈Y M 2 .We notice that also the sequence of product measures ρ n (x) dx ρ n (y) dy is narrowly converging to ρ * (x) dx ρ * (y) dy, see for instance[2, Theorem 2.8].