Ground states in the diffusion-dominated regime

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller–Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric non-increasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document}C∞ inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.


Introduction
We are interested in the diffusion-aggregation equation for a density ρ(t, x) of unit mass defined on R + × R N , and where we define the mean-field potential S k [ρ](x) := W k (x) * ρ(x) for some interaction kernel W k . The parameter χ > 0 denotes the interaction strength. Since (1.1) conserves mass, is positivity preserving and invariant by translations, we work with solutions ρ in the set The interaction W k is given by the Riesz kernel Let us write k = 2s − N with s ∈ 0, N 2 . Then the convolution term S k [ρ] is governed by a fractional diffusion process, For k > 1 − N the gradient ∇ S k [ρ] := ∇ (W k * ρ) is well defined locally. For k ∈ (−N , 1 − N ] however, it becomes a singular integral, and we thus define it via a Cauchy principal value, Here, we are interested in the porous medium case m > 1 with N ≥ 1. The corresponding energy functional writes Given ρ ∈ Y, we see that H m and W k are homogeneous by taking dilations ρ λ (x) := λ N ρ(λx). More precisely, we obtain In other words, the diffusion and aggregation forces are in balance if N (m − 1) = −k. This is the case for choosing the critical diffusion exponent m c := 1 − k/N called the faircompetition regime. In the diffusion-dominated regime we choose m > m c , which means that the diffusion part of the functional (1.3) dominates as λ → ∞. In other words, concentrations are not energetically favourable for any value of χ > 0 and m > m c . The range 0 < m < m c is referred to as the attraction-dominated regime. In this work, we focus on the diffusiondominated regime m > m c .
Further, we define below the diffusion exponent m * that will play an important role for the regularity properties of global minimisers of F : The main results in this work are summarised in the following two theorems: Diffusion-aggregation at the top equations of the form (1.1) are ubiquitous as macroscopic models of cell motility due to cell adhesion and/or chemotaxis phenomena while taking into account volume filling constraints [10,29,45]. The non-linear diffusion models the very strong localised repulsion between cells while the attractive non-local term models either cell movement toward chemosubstance sources or attractive interaction between cells due to cell adhension by long filipodia. They encounter applications in cancer invasion models, organogenesis and pattern formation [18,24,28,42,46].
The archetypical example of the Keller-Segel model in two dimensions corresponding to the logarithmic case (m = 1, k = 0) has been deeply studied by many authors [2,3,5,6,15,19,23,[30][31][32]43,44,47], although there are still plenty of open problems. In this case, there is an interesting dichotomy based on a critical parameter χ c > 0: the density exists globally in time if 0 < χ < χ c (diffusion overcomes self-attraction) and expands self-similarly [14,27], whereas blow-up occurs in finite time when χ > χ c (self-attraction overwhelms diffusion), while for χ = χ c infinitely many stationary solutions exist with intricated basins of attraction [3]. The three-dimensional configuration with Newtonian interaction (m = 1, k = 2 − N ) appears in gravitational physics [20,21], although it does not have this dichotomy, belonging to the attraction-dominated regime. However, the dichotomy does happen for the particular exponent m = 4/3 of the non-linear diffusion for the 3D Newtonian potential as discovered in [4]. This was subsequently generalised for the fair-competition regime where m = m c for a given k ∈ (−N , 0) in [12,13].
In fact, as mentioned before two other different regimes appear: the diffusion-dominated case when m > m c and the attraction-dominated case when m < m c . In Figure 1, we make a sketch of the different regimes including cases related to non-singular kernels for the sake of completeness. Note that non-singular kernels k > 0 allow for values of m < 1 corresponding to fast-diffusion behaviour in the diffusion-dominated regime m > m c . We refer to [12,13] and the references therein for a full discussion of the state of the art in these regimes.
In the diffusion-dominated case, it was already proven in [16] that global minimisers exist in the particular case of m > 1 = m c for the logarithmic interaction kernel k = 0. Their uniqueness up to translation and mass normalisation is a consequence of the important symmetrisation result in [17] asserting that all stationary states to (1.1) for 2 − N ≤ k < 0 are radially symmetric. We will generalise this result to our present framework for the range For m = m c , attractive and repulsive forces are in balance (i.e. in fair-competition). For m c < m < m * in the diffusion-dominated regime, global minimisers of F are stationary states of (1.1), see Theorem 1, a result which we are not able to show for m ≥ m * (striped region) −N < k < 2 − N not included in [17] due to the special treatment needed for the arising singular integral terms. This is the main goal of Sect. 2 where we remind the reader the precise definition and basic properties of stationary states for (1.1). In short, we show that stationary solutions are continuous compactly supported radially non-increasing functions with respect to their centre of mass. Some of these results are in fact generalisations of previous results in [12,17] and we skip some of the details.
Let us finally comment that the symmetrisation result reduces the uniqueness of stationary states to uniqueness of radial stationary states that eventually leads to a full equivalence between stationary states and global minimisers of the free energy (1.3). This was used in [17] to solve completely the 2D case with m > 1 = m c for the logarithmic interaction kernel k = 0, and it was the new ingredient to fully characterise the long-time asymptotics of (1.1) in that particular case.
In view of the main results already announced above, we show in Sect. 3 the existence of global minimisers for the full range m > m c and k ∈ (−N , 0) which are steady states of the Eq. (1.1) as soon as m < m * . This additional constraint on the range of non-linearities appears only in the most singular range −N < k < 1 − N and allows us to get the right Hölder regularity on the minimisers in order to make sense of the singular integral in the gradient of the attractive non-local potential force (1.2).
Besides existence of minimisers, Sect. 3 contains some of the main novelties of this paper. First, in order to prove boundedness of minimisers, we develop a fine estimate on the interaction term based on the asymptotics of the Riesz potential of radial functions, and show that this estimate is well suited exactly for the diffusion dominated regime (see Lemma 2 and Theorem 7). Moreover, thanks to the Schauder estimates for the fractional Laplacian, we improve the regularity results for minimisers in [12] and show that they are smooth inside their support, see Theorem 10. This result applies both to the diffusion-dominated and fair-competition regime.
These global minimisers are candidates to play an important role in the long-time asymptotics of (1.1). We show their uniqueness in one dimension by optimal transportation techniques in Sect. 4. The challenging open problems remaining are uniqueness of radially non-increasing stationary solutions to (1.1) in its full generality and the long-time asymptotics of (1.1) in the whole diffusion-dominated regime, even for non-singular kernels within the fast diffusion case.
Plan of the paper: In Sect. 2 we define and analyse stationary states, showing that they are radially symmetric and compactly supported. Section 3 is devoted to global minimisers. We show that global minimisers exist, are bounded and we provide their regularity properties. Eventually, Sect. 4 proves uniqueness of stationary states in the one-dimensional case.

Stationary states
Let us define precisely the notion of stationary states to the diffusion-aggregation equation (1.1).
In fact, as shown in [12] via a near-far field decomposition argument of the drift term, the function S k [ρ] and its gradient defined in (1.2) satisfy even more than the regularity ∇ S k [ρ] ∈ L 1 loc R N required in Definition 1: . Then the following regularity properties hold: Lemma 1 implies further regularity properties for stationary states of (1.1). For precise proofs, see [12].

where C[ρ](x) is constant on each connected component of supp (ρ).
It follows from Proposition 1 thatρ ∈ W 1,∞ R N in the case m c < m ≤ 2.

Radial symmetry of stationary states
The aim of this section is to prove that stationary states of (1.1) are radially symmetric. This is one of the main results of [17], and is achieved there under the assumption that the interaction kernel is not more singular than the Newtonian potential close to the origin. As we will briefly describe in the proof of the next result, the main arguments continue to hold even for the more singular Riesz kernels W k .
Theorem 3 (Radiality of stationary states) Let χ > 0 and m > m c . Ifρ ∈ L 1 + (R N )∩L ∞ (R N ) with ρ 1 = 1 is a stationary state of (1.1) in the sense of Definition 1, thenρ is radially symmetric non-increasing up to a translation.
Proof The proof is based on a contradiction argument, being an adaptation of that in [17,Theorem 2.2], to which we address the reader the more technical details. Assume thatρ is not radially non-increasing up to any translation. By Proposition 1, we have for some positive constant c in supp(ρ). Let us now introduce the continuous Steiner symmetrisation S τρ in direction e 1 = (1, 0, · · · , 0) ofρ as follows. For any is the continuous Steiner symmetrisation of the U h x (see [17] for the precise definitions and all the related properties). As in [17], our aim is to show that there exists a continuous family of functions μ(τ, x) such that μ(0, ·) =ρ and some positive constants C 1 > 0, c 0 > 0 and a small δ 0 > 0 such that the following estimates hold for all τ ∈ [0, δ 0 ]: for some sufficiently small constant h 0 > 0 to be determined. Note that this choice of the velocity is different to the one in [17, Proposition 2.7] since we are actually keeping the level sets ofS τρ (·, x ) frozen below the layer at height h 0 . Next, we note that inequality (2.3) and the Lipschitz regularity ofS k (Lemma 1) are the only basic ingredients used in the proof of [17,Proposition 2.7] to show that the family μ(τ, ·) satisfies (2.5) and (2.6). Therefore, it remains to prove (2.4). Since different level sets ofS τρ (·, x ) are moving at different speeds Then, in order to establish (2.4), it is enough to show As in the proof of [17, Proposition 2.7], proving (2.7) reduces to show that for sufficiently small h 0 > 0 one has To this aim, we write and we splitS τρ similarly, taking into account that v(h) = 1 for all h > h 0 : Note that can be controlled by ||ρ|| ∞ and the α-Hölder seminorm ofρ. Hence, we can apply the argument in [ which is a contradiction with (2.4) for small τ .

Stationary states are compactly supported
In this section, we will prove that all stationary states of Eq. (1.1) have compact support, which agrees with the properties shown in [16,17,33]. We begin by stating a useful asymptotic estimate on the Riesz potential inspired by [50, § 4]. For the proof of Proposition 2, see Appendix 1.
Here, C 1 > 0 and C 2 > 0 are explicit constants depending only on k and N .
From the above estimate, we can derive the expected asymptotic behaviour at infinity.

Corollary 1
Let ρ ∈ Y be radially non-increasing. Then W k * ρ vanishes at infinity, with decay not faster than that of |x| k .
Proof Notice that Proposition 2(i) entails the decay of the Riesz potential at infinity for 1) and notice that |y| k ≤ |y| k+r if |y| ≥ 1, so that if B 1 is the unit ball centered at the origin we have The first term in the right hand side vanishes as |x| → ∞, since y → |y| k is integrable at the origin, and since ρ is radially non-increasing and vanishing at infinity as well. The second term goes to zero at infinity thanks to Proposition 2(i), since the choice of r yields On the other hand, the decay at infinity of the Riesz potential can not be faster than that of |x| k . To see this, notice that there holds As a rather simple consequence of Corollary 1, we obtain: Corollary 2 Letρ be a stationary state of (1.1). Thenρ is compactly supported.
Proof By Theorem 3 we have thatρ is radially non-increasing up to a translation. Since the translation of a stationary state is itself a stationary state, we may assume thatρ is radially symmetric with respect to the origin. Suppose by contradiction thatρ is supported on the whole of R N , so that Eq. (2.2) holds on the whole R N , with C k [ρ](x) replaced by a unique constant C. Then we necessarily have C = 0. Indeed,ρ m−1 vanishes at infinity since it is radially non-increasing and integrable, and by Corollary 1 we have that S k [ρ] = W k * ρ vanishes at infinity as well. Thereforē .
But Corollary 1 shows that W k * ρ decays at infinity not faster than |x| k and this would entail, since m > m c , a decay at infinity of ρ not faster than that of |x| −N , contradicting the integrability of ρ.

Global minimisers
We start this section by recalling a key ingredient for the analysis of the regularity of the drift term in (1.1), i.e. certain functional inequalities which are variants of the Hardy-Littlewood-Sobolev (HLS) inequality, also known as the weak Young's inequality [36,Theorem 4.3]: for The optimal constant C H L S is found in [35]. In the sequel, we will make use of the following variations of the above HLS inequality: Proof The inequality is a direct consequence of the standard sharp HLS inequality and of Hölder's inequality. It follows that C * is finite and bounded from above by the optimal constant in the HLS inequality.

Existence of global minimisers
Then there exists a minimiser of problem (I M ) if the following holds:

) holds if and only if
where B R denotes the ball centered at zero and of radius R > 0, and where σ N = 2π (N /2) /Γ (N /2) denotes the surface area of the N -dimensional unit ball. Then We conclude that Since we are in the diffusion-dominated regime N (1 − m) < k < 0, we can choose R > 0 large enough such that F [ρ * ] < 0, and hence condition (3.4) is satisfied. We conclude by Proposition 3 and Theorem 6 that there exists a minimiserρ of F in Y q,M with q = It can easily be seen that in factρ ∈ L m (R N ) using the HLS inequality (3.1): Translatingρ so that its centre of mass is at zero and choosing M = 1, we obtain a minimiser ρ of F in Y. Moreover, by Riesz's rearrangement inequality [36, Theorem 3.7], we have where ρ # is the Schwarz decreasing rearrangement of ρ. Thus, ifρ is a global minimiser of F in Y, then so isρ # , and it follows that We conclude from [36, Theorem 3.7] thatρ =ρ # , and so all global minimisers of F in Y are radially symmetric non-increasing. Optimising over λ, we find a unique λ * > 0 such that g (λ * ) = 0: Substitution the optimal dilation ρ λ * of ρ into the energy functional F , we obtain  N , 0) and m > m c . If ρ is a global minimiser of the free energy functional F in Y, then ρ is radially symmetric and non-increasing, satisfying Here, we denote

Boundedness of global minimisers
This section is devoted to showing that all global minimisers of F in Y are uniformly bounded. In the following, for a radial function ρ ∈ L 1 (R N ) we denote by M ρ (R) := B R ρ dx the corresponding mass function, where B R is a ball of radius R, centered at the origin. We start with the following technical lemma:  We fix q ∈ [0, m/N ) and H > H 0 as above from here on. Let us make use of Proposition 2, which we apply to the compactly supported function Case 1 − N < k < 0: Proposition 2(i) applied to ρ H gives the estimate and hence, integrating against ρ on B C A H and using ρ ≤ H on B C A H , which conludes the proof in that case. Case −N < k ≤ 1 − N : In this case, we obtain from Proposition 2(ii) applied to ρ H the estimate We split the integral in the right hand side as I 1 + I 2 , where Let us first consider I 2 , where we have |x| ≥ H −q ≥ 2 A H on the integration domain. Since the map |x| → |x|+A H |x|−A H is monotonically decreasing to 1 in (A H , +∞), it is bounded above by 3 on (2 A H , +∞). We conclude from (2.9) that T k (|x|, A H ) ≤ 3 for |x| ∈ (H −q , +∞). This entails where we used once again |x| ≥ H −q , recalling that k < 0.
Concerning I 1 , we have ρ ≤ H on B C A H which entails (3.10) Combining (3.8), (3.9), (3.10) we conclude These information together with the estimate (3.7) can be inserted into (3.6) to conclude.
We are now in a position to prove that any minimiser of F is bounded.

Theorem 7 Let χ > 0, k ∈ (−N , 0) and m > m c . Then any global minimiser of F over Y is uniformly bounded and compactly supported.
Proof Since ρ is radially symmetric non-increasing by Proposition 4, it is enough to show ρ(0) < ∞. Let us reason by contradiction and assume that ρ is unbounded at the origin. We will show that F [ρ] − F [ρ] > 0 for a suitably chosen competitorρ, where B A H and q are defined as in Lemma 2, B C A H denotes the complement of B A H and 1 D r is the characteristic function of a ball D r := B r (x 0 ) of radius r > 0, centered at some x 0 = 0 and such that D r ∩ B A H = ∅. Note that A H ≤ H −q /2 < H −q 0 /2 < 1/2. Hence, we can take r > 1 and D r centered at the point x 0 = (2r , 0, . . . , 0) ∈ R N . Notice in particular that since ρ is unbounded, for any H > 0 we have that B A H has non-empty interior. On the other hand, B A H shrinks to the origin as H → ∞ since ρ is integrable.
As D r ⊂ B C A H and ρ =ρ on B C A H \ D r , we obtain We bound where we use the convexity identity (a + b) m ≥ |a m − b m | for a, b > 0. Hence, ε r goes to 0 as r → ∞. Summarising we have for any r > 1, with ε r vanishing as r → ∞.
To estimate the interaction term, we split the double integral into three parts: (3.12) Let us start with I 3 . By noticing once again that ρ =ρ on B C A H \ D r for any r > 0, we have By the HLS inequality (3.1), we have if a > 1, b > 1 and 1/a + 1/b − k/N = 2. We can choose b ∈ (1, min {m, N /(k + N )}), which is possible as −N < k < 0, m > 1, and then we get a > 1, ρ ∈ L b (R N ) as 1 < b < m, and The latter vanishes as r → ∞. For the term I 31 , we have With the same choice of a, b as above, the HLS inequality implies which vanishes as r → ∞ since a > 1 and b > 1. We conclude that I 3 (r ) → 0 as r → ∞.
The integral I 1 can be estimated using Theorem 4, and the fact that ρ ≥ H > 1 on B A H together with m > m c , On the other hand, the HLS inequalities (3.1) and (3.2) do not seem to give a sharp enough estimate for the cross-term I 2 , for which we instead invoke Lemma 2, yielding (3.14) for given q ∈ [0, m/N ) and large enough H as specified in Lemma 2.
In order to conclude, we join together (3.11), (3.12), (3.13) and (3.14) to obtain for any r > 1 and any large enough H , Now we choose q. On the one hand, notice that for a choice η > 0 small enough such that m > m c + η, we have On the other hand, −N < k < 0 implies 1 − k/N > 2N / (2N + k). Since m > m c , this gives the inequality m > 2N / (2N + k). Hence, for small enough η > 0 such that Thanks to (3.16) and (3.17) we see that we can fix a non-negative q such that Inserting the last two estimates in (3.15) we get for some η > 0 for any r > 1 and any large enough H . First of all, notice that B A H ρ m dx is strictly positive since we are assuming that ρ is unbounded. We can therefore fix H large enough such that the constant in front of B A H ρ m is strictly positive. Secondly, we have already proven that ε r and I 3 (r ) vanish as r → ∞, so we can choose r large enough such that , we get a contradiction with the minimality of ρ. We conclude that minimisers of F over Y are bounded.
Finally, we can just use the Euler-Lagrange Eq. (3.5) and the same argument as for Corollary 2 to prove that ρ is compactly supported.

Regularity properties of global minimisers
This section is devoted to the regularity properties of global minimisers. With enough regularity, global minimisers satisfy the conditions of Definition 1, and are therefore stationary states of Eq. (1.1). This will allow us to complete the proof of Theorem 1.
We begin by introducing some notation and preliminary results. As we will make use of the Hölder regularising properties of the fractional Laplacian, see [48,51], the notation is better adapted to the arguments that follow, fixing s = (k + N )/2, and we will therefore state the results in this section in terms of s. One fractional regularity result that we will use repeatedly in this section follows directly from the HLS inequality (3.1) applied with k = 2s − N : for any we have Further, for 1 ≤ p < ∞ and s ≥ 0, we define the Bessel potential space L 2s, p (R N ) as made by all functions f ∈ L p (R N ) such that (I − Δ) s f ∈ L p (R N ), meaning that f is the Bessel potential of an L p (R N ) function (see [52, pag. 135]). Since we are working with the operator (−Δ) s instead of (I − Δ) s , we make use of a characterisation of the space L 2s, p (R N ) in terms of Riesz potentials. For 1 < p < ∞ and 0 < s < 1 we have (3.20) see [49,Theorem 26.8,Theorem 27.3], see also Exercise 6.10 in Stein's book [52, pag. 161]. Moreover, for 1 ≤ p < ∞ and 0 < s < 1/2 we define the fractional Sobolev space We have the embeddings  (3.23) with the convention that if α ≥ 1 for any open set U in R N , then C 0,α (U ) := C α ,α (U ), where α + α = α, α ∈ [0, 1) and α is the greatest integer less than or equal to α. With this notation, we have C 0,1 (R N ) = C 1,0 (R N ) = W 1,∞ (R N ). In particular, using (3.23) it follows that for α > 0, s ∈ (0, 1) and α + 2s not an integer, Moreover, rescaling inequality (3.23) in any ball B R (x 0 ) where R = 1 we have the estimate where α 1 , α 2 are the greatest integers less than α and α + 2s respectively. In Let us begin by showing that global minimisers of F enjoy the good Hölder regularity in the most singular range, as long as diffusion is not too slow. χ > 0 and s ∈ (0, N /2). If m c < m < m * , then any global minimiser ρ ∈ Y of F satisfies S k [ρ] = W k * ρ ∈ W 1,∞ (R N ), ρ m−1 ∈ W 1,∞ (R N ) and ρ ∈ C 0,α (R N ) with α = min{1, 1 m−1 }. Proof Recall that the global minimiser ρ ∈ Y of F is radially symmetric non-increasing and compactly supported by Theorem 5 and Theorem 7. Since ρ ∈ L 1 R N ∩ L ∞ R N by Theorem 7, we have ρ ∈ L p R N for any 1 < p < ∞. 2s , ∞). Then, if s ∈ (0, 1), since S k is the Riesz potential of the density ρ in L p , by the characterisation (3.20) of the Bessel potential space, we conclude that S k [ρ] ∈ L 2s, p (R N ) for all p > N N −2s . Let us first consider s < 1/2, as the cases 1/2 < s < N /2 and s = 1/2 follow as a corollary. 0 < s < 1/2: In this case, we have the embedding (3.21) and so S k [ρ] ∈ W 2s, p (R N ) for all p ≥ 2 > N N −2s if N ≥ 2 and for all p > max{2, 1 1−2s

Theorem 8 Let
Hence ρ m−1 ∈ C 0,β R N for the same choice of β using the Euler-Lagrange condition (3.5) since ρ m−1 is the truncation of a function which is S k [ρ] up to a constant.
Note that m c ∈ (1, 2) and m * > 2. In what follows we split our analysis into the cases m c < m ≤ 2 and 2 < m < m * , still assuming s < 1/2. If m ≤ 2, the argument follows along the lines of [12, Corollary 3.12] since ρ m−1 ∈ C 0,α (R N ) implies that ρ is in the same Hölder space for any α ∈ (0, 1). Indeed, in such case we bootstrap in the following way. Let us fix n ∈ N such that 1 n + 1 < 2s ≤ 1 n (3.26) and let us define Form (3.26) and (3.27) we see that by choosing large enough p there hold 1 − 2s < β n < 1.

Remark 2
If m ≥ m * and s < 1/2, we recover some Hölder regularity, but it is not enough to show that global minimisers of F are stationary states of (1.1). More precisely, m ≥ m * means 2s(m−1) m−2 ≤ 1, and so it follows from (3.28) that ρ ∈ C 0,γ R N for any γ < 2s m−2 . Note that m ≥ m * also implies 2s m−2 ≤ 1 − 2s, and we are therefore not able to go above the desired Hölder exponent 1 − 2s.

Remark 3
In the arguments of Theorem 8 one could choose to directly bootstrap on fractional Sobolev spaces. In fact, for 0 < s < 1/2 and m > 2 we have that thus |u| α ∈ W αs, p/α (R N ). This property is also valid for Sobolev spaces with integer order, see [41]. In particular, thanks to this property, in case m ≥ m * we may obtain ρ m−1 ∈ W α, p (R N ) for any α < 2s(m−1) m−2 and any large enough p, hence (3.22) implies that ρ has the Hölder regularity stated in Remark 2.
We are now ready to show that global minimisers possess the good regularity properties to be stationary states of equation (1.1) according to Definition 1. In fact, we can show that global minimisers have even more regularity inside their support.
Theorem 10 Let χ > 0, m c < m and s ∈ (0, N /2). If ρ ∈ Y is a global minimiser of F , then ρ is C ∞ in the interior of its support.
Proof By Theorem 8 and Remark 2, we have ρ ∈ C 0,α (R N ) for some α ∈ (0, 1). Since ρ is radially symmetric non-increasing, the interior of supp (ρ) is a ball centered at the origin, which we denote by B. Note also that ρ ∈ L 1 (R N ) ∩ L ∞ (R N ) by Theorem 7, and so S k [ρ] ∈ L ∞ (R N ) by Lemma 1. Assume first that s ∈ (0, 1) ∩ (0, N /2). Applying (3.25) with B R centered at a point within B and such that B R ⊂⊂ B, we obtain S k [ρ] ∈ C 0,γ (B R/2 ) for any γ < α + 2s. It follows from the Euler-Lagrange condition (3.5) that ρ m−1 has the same regularity as S k [ρ] on B R/2 , and since ρ is bounded away from zero on B R/2 , we conclude ρ ∈ C 0,γ (B R/2 ) for any γ < α + 2s. Repeating the previous step now on B R/2 , we get the improved regularity S k [ρ] ∈ C 0,γ (B R/4 ) for any γ < α + 4s by (3.25), which we can again transfer onto ρ using (3.5), obtaining ρ ∈ C 0,γ (B R/4 ) for any γ < α+4s. Iterating, any order of differentiability for S k (and then for ρ) can be reached in a neighborhood of the center of B R . We notice that the argument can be applied starting from any point x 0 ∈ B, and hence ρ ∈ C ∞ (B). When N ≥ 3 and s ∈ [1, N /2), we take numbers s 1 , . . . , s l such that s i ∈ (0, 1) for any i = 1, . . . , l and such that l i=1 s i = s. We also let Therefore we may recursively apply (3.25), starting from S 1 where the ball B R is centered at a point within B such that B R ⊂⊂ B, and using the iteration rule We obtain S l+1 k [ρ] = S k [ρ] ∈ C 0,γ (B R/(2 l ) ) for any γ < α + 2s, and as before, the Euler-Lagrange Eq. (3.5) implies that ρ ∈ C 0,γ (B R/(2 l ) ) for any γ < α + 2s. If we repeat the argument, we gain 2s in Hölder regularity for ρ each time we divide the radius R by 2 l . In this way, we can reach any differentiability exponent for ρ around any point of B, and thus ρ ∈ C ∞ (B).

Remark 4
We observe that the smoothness of minimisers in the interior of their support also holds in the fair-competition regime m = m c . In such case global Hölder regularity was obtained in [12].

Optimal transport tools
Optimal transport is a powerful tool for reducing functional inequalities onto pointwise inequalities. In other words, to pass from microscopic inequalities between particle locations to macroscopic inequalities involving densities. This sub-section summarises the main results of optimal transportation we will need in the one-dimensional setting. They were already used in [11] and in [13], where we refer for detailed proofs.
Letρ and ρ be two probability densities. According to [7,39], there exists a convex function ψ whose gradient pushes forward the measureρ(a)da onto ρ(x)dx: ψ # (ρ(a)da) = ρ(x)dx. This convex function satisfies the Monge-Ampère equation in the weak sense: for any test function ϕ ∈ C b (R), the following identity holds true The convex map is unique a.e. with respect to ρ and it gives a way of interpolating measures using displacement convexity [40]. On the other hand, regularity of the transport map is a complicated matter. Here, as it was already done in [11,13], we will only use the fact that ψ (a)da can be decomposed in an absolute continuous part ψ ac (a)da and a positive singular measure [53,Chapter 4]. In one dimension, the transport map ψ is a non-decreasing function, therefore it is differentiable a.e. and it has a countable number of jump singularities. For any measurable function U , bounded below such that U (0) = 0 we have [40] The following Lemma proved in [11] will be used to estimate the interaction contribution in the free energy.

Lemma 3
Let K : (0, ∞) → R be an increasing and strictly concave function. Then, for any where the convex combination of a and b is given by [a, b] s = (1 − s)a + sb. Equality is achieved in (4.2) if and only if the distributional derivative of the transport map ψ is a constant function.

Functional inequality in one dimension
In what follows, we will make use of a characterisation of stationary states based on some integral reformulation of the necessary condition stated in Proposition 4. This characterisation was also the key idea in [11,13] to analyse the asymptotic stability of steady states and the functional inequalities behind.
Lemma 4 (Characterisation of stationary states) Let N = 1, χ > 0 and k ∈ (−1, 0). If m > m c with m c = 1 − k, then any stationary stateρ ∈ Y of system (1.1) can be written in the formρ The proof follows the same methodology as for the fair-competition regime [ Proof For a given stationary stateρ ∈ Y and a given ρ ∈ Y, we denote by ψ the convex function whose gradient pushes forward the measureρ(a)da onto ρ(x)dx: ψ # (ρ(a)da) = ρ(x)dx. Using (4.1), the functional F [ρ] rewrites as follows: In fact, the result in Theorem 11 implies that all critical points of F in Y are global minimisers. Further, we obtain the following uniqueness result: where, for u ∈ [0, 1), Using the change of variables t = cos 2 θ 2 , we get from the integral formulation (A.4), The function F(a, b; c; z) is increasing in z and then for any z ∈ (0, 1) there holds  a). Inserting this into (A.5) concludes the proof of (i). If |x| > R, we have (|x|+η)(|x|−η) −1 ≤ (|x|+ R)(|x|− R) −1 for any η ∈ (0, R), therefore we can put R in place of η in the right hand side of (A.9) and (A.10), insert into (A.11) and conclude.