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 decreasing 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 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.
We are interested in the diffusion-aggregation equation
for a density of unit mass defined on , and where we define the mean-field potential for some interaction kernel . The parameter denotes the interaction strength. Since (1.1) conserves mass, is positivity preserving and invariant by translations, we work with solutions in the set
The interaction is given by the Riesz kernel
Let us write with . Then the convolution term is governed by a fractional diffusion process,
For the gradient is well defined locally. For 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 with . The corresponding energy functional writes
Given , we see that and are homogeneous by taking dilations . More precisely, we obtain
In other words, the diffusion and aggregation forces are in balance if . This is the case for choosing the critical diffusion exponent called the fair-competition regime. In the diffusion-dominated regime we choose , which means that the diffusion part of the functional (1.3) dominates as . In other words, concentrations are not energetically favourable for any value of and . The range is referred to as the attraction-dominated regime. In this work, we focus on the diffusion-dominated regime .
Further, we define below the diffusion exponent that will play an important role for the regularity properties of global minimisers of :
The main results in this work are summarised in the following two theorems:
Let , and . All stationary states of equation (1.1) are radially symmetric decreasing. If , then there exists a global minimiser of on . Further, all global minimisers are radially symmetric non-increasing, compactly supported, uniformly bounded and inside their support. Moreover, all global minimisers of are stationary states of (1.1), according to Definition 3, whenever . Finally, if , we have .
Aggregation-diffusion 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 [29, 44, 10]. 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 [28, 24, 45, 41, 18].
The archetypical example of the Keller-Segel model in two dimensions corresponding to the logarithmic case has been deeply studied by many authors [31, 32, 43, 30, 42, 23, 6, 46, 5, 2, 3, 15, 19], although there are still plenty of open problems. In this case, there is an interesting dichotomy based on a critical parameter : the density exists globally in time if (diffusion overcomes self-attraction) and expands self-similarly [14, 27], whereas blow-up occurs in finite time when (self-attraction overwhelms diffusion), while for infinitely many stationary solutions exist with intricated basins of attraction . The three-dimensional configuration with Newtonian interaction 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 of the non-linear diffusion for the 3D Newtonian potential as discovered in . This was subsequently generalised for the fair-competition regime where for a given in [12, 13].
In fact, as mentioned before two other different regimes appear: the diffusion-dominated case when and the attraction-dominated case when . 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 allow for values of corresponding to fast-diffusion behaviour in the diffusion-dominated regime . 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  that global minimisers exist in the particular case of for the logarithmic interaction kernel . Their uniqueness up to translation and mass normalisation is a consequence of the important symmetrisation result in  asserting that all stationary states to (1.1) for are radially symmetric. We will generalise this result to our present framework for the range not included in  due to the special treatment needed for the arising singular integral terms. This is the main goal of Section 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  to solve completely the 2D case with for the logarithmic interaction kernel , 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 Section 3 the existence of global minimisers for the full range and which are steady states of the equation (1.1) as soon as . This additional constraint on the range of non-linearities appears only in the most singular range 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, Section 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 15 and Theorem 16). Moreover, thanks to the Schauder estimates for the fractional Laplacian, we improve the regularity results for minimisers in  and show that they are smooth inside their support, see Theorem 21. 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 Section 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 Section 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, Section 4 proves uniqueness of stationary states in the one-dimensional case.
2. Stationary states
Let us define precisely the notion of stationary states to the diffusion-aggregation equation (1.1).
Given with and letting , we say that is a stationary state for the evolution equation (1.1) if , , and it satisfies
in the sense of distributions in . If , we further require for some .
Let with and . Then the following regularity properties hold:
, assuming additionally with in the range .
Let and . If is a stationary state of equation (1.1) and , then is continuous on , , and
where is constant on each connected component of .
It follows from Proposition 5 that in the case .
2.1. 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 , 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 .
Theorem 6 (Radiality of stationary states).
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 that is not radially decreasing up to any translation. By Proposition 5, we have
for some positive constant in . Let us now introduce the continuous Steiner symmetrisation in direction of as follows. For any , let
and is the continuous Steiner symmetrisation of the (see  for the precise definitions and all the related properties). As in , our aim is to show that there exists a continuous family of functions such that and some positive constants , and a small such that the following estimates hold for all :
Following the arguments of the proof in [17, Proposition 2.7], if we want to construct a continuous family for (2.5) to hold, it is convenient to modify suitably the continuous Steiner symmetrisation in order to have a better control of the speed in which the level sets are moving. More precisely, we define as
with defined as
for some sufficiently small constant 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 of frozen below the layer at height . Next, we note that inequality (2.3) and the Lipschitz regularity of (Lemma 4) are the only basic ingredients used in the proof of [17, Proposition 2.7] to show that the family satisfies (2.5) and (2.6). Therefore, it remains to prove (2.4). Since different level sets of are moving at different speeds , we do not have for all , but it is still possible to prove that (see [17, Proposition 2.7])
Then, in order to establish (2.4), it is enough to show
To this aim, we write
and we split similarly, taking into account that for all :
where is the truncation at height of . Since for , we have
If we are in the singular range , we have for some . Since the continuous Steiner symmetrisation decreases the modulus of continuity (see [8, Theorem 3.3] and [8, Corollary 3.1]), we also have . Further, Lemma 4 and the arguments of [17, Proposition 2.7] guarantee that the expressions
can be controlled by and the -Hölder seminorm of . Hence, we can apply the argument in [17, Proposition 2.7] to conclude for the estimate (2.8). Now it is possible to proceed exactly as in the proof of [17, Theorem 2.2] to show that for some positive constant , we have the quadratic estimate
which is a contradiction with (2.4) for small . ∎
2.2. Stationary States are Compactly Supported
In this section, we will prove that all stationary states of equation (1.1) have compact support, which agrees with the properties shown in [33, 16, 17]. We begin by stating a useful asymptotic estimate on the Riesz potential inspired by [49, §4]. For the proof of Proposition 7, see Appendix A.
Proposition 7 (Riesz potential estimates).
Let and let be radially symmetric.
If , then on .
If and if is supported on a ball for some , then
Here, and are explicit constants depending only on and .
From the above estimate, we can derive the expected asymptotic behaviour at infinity.
Let be radially non-increasing. Then vanishes at infinity, with decay not faster than that of .
Notice that Proposition 7(i) entails the decay of the Riesz potential at infinity for . Instead, let . Let and notice that if , so that if is the unit ball centered at the origin we have
The first term in the right hand side vanishes as , since is integrable at the origin, and since is radially non-increasing and vanishing at infinity as well. The second term goes to zero at infinity thanks to Proposition 7(i), since the choice of yields .
On the other hand, the decay at infinity of the Riesz potential can not be faster than that of . To see this, notice that there holds
with since is radially non-increasing. ∎
As a rather simple consequence of Corollary 8, we obtain:
Let be a stationary state of (1.1). Then is compactly supported.
By Theorem 6 we have that is radially non-increasing up to a translation. Since the translation of a stationary state is itself a stationary state, we may assume that is radially symmetric with respect to the origin. Suppose by contradiction that is supported on the whole of , so that equation (2.2) holds on the whole , with replaced by a unique constant . Then we necessarily have . Indeed, vanishes at infinity since it is radially decreasing and integrable, and by Corollary 8 we have that vanishes at infinity as well. Therefore
But Corollary 8 shows that decays at infinity not faster than and this would entail, since , a decay at infinity of not faster than that of , contradicting the integrability of . ∎
3. 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 all , there exists an optimal constant such that
The optimal constant is found in . In the sequel, we will make use of the following variations of above HLS inequality:
Let , and . For , we have
where is the best constant.
The inequality is a direct consequence of the standard sharp HLS inequality and of Hölder’s inequality. It follows that is finite and bounded from above by the optimal constant in the HLS inequality. ∎
3.1. Existence of Global Minimisers
Theorem 11 (Existence of Global Minimisers).
For all and , there exists a global minimiser of in . Moreover, all global minimisers of in are radially non-increasing.
(see [37, Theorem II.1]) Suppose , , and consider the problem
Then there exists a minimiser of problem if the following holds:
Proof of Theorem 11.
First of all, notice that our choice of potential is indeed in with . Further, it can easily be verified that Proposition 13 applies with . Hence we are left to show that there exists a choice of such that . Let us fix and define
where denotes the ball centered at zero and of radius , and where denotes the surface area of the -dimensional unit ball. Then
We conclude that
Since we are in the diffusion-dominated regime , we can choose large enough such that , and hence condition (3.4)
is satisfied. We conclude by Proposition 13 and Theorem 12 that there exists a minimiser of in with
It can easily be seen that in fact using the HLS inequality (3.1):
where . Using Hölder’s inequality, we find
Hence, since ,
Translating so that its centre of mass is at zero and choosing , we obtain a minimiser of in . Moreover, by Riesz’s rearrangement inequality [36, Theorem 3.7], we have
where is the Schwarz decreasing rearrangement of . Thus, if is a global minimiser of in , then so is , and it follows that
We conclude from [36, Theorem 3.7] that , and so all global minimisers of in are radially symmetric non-increasing. ∎
Let and . If is a global minimiser of the free energy functional in , then is radially symmetric and non-increasing, satisfying
Here, we denote
3.2. Boundedness of Global Minimisers
This section is devoted to showing that all global minimisers of in are uniformly bounded. In the following, for a radial function we denote by the corresponding mass function, where is a ball of radius , centered at the origin. We start with the following technical lemma:
Let , , and . Assume is radially decreasing. For a fixed , the level set is a ball centered at the origin whose radius we denote by . Then we have the following cross-range interaction estimate: there exists , depending only on , such that, for any ,
and is a constant depending only on and .
Notice that the result is trivial if is bounded. The interesting case here is unbounded, implying that for any .
First of all, since and on , the estimate
implies that is vanishing as as soon as , and in particular that we can find , depending only on , such that
We fix and as above from here on.
Let us make use of Proposition 7, which we apply to the compactly supported function .
7(i) applied to gives the estimateProposition
and hence, integrating against on and using on ,
which conludes the proof in that case.