Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics
We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as .
The evolution of interacting particles and their equilibrium configurations has attracted the attention of many applied mathematicians and mathematical analysts for years. Continuum description of interacting particle systems usually leads to analyze the behavior of a mass density of individuals at certain location and time . Most of the derived models result in aggregation-diffusion nonlinear partial differential equations through different asymptotic or mean-field limits [59, 12, 23]. The different effects reflect that equilibria are obtained by competing behaviors: the repulsion between individuals/particles is modeled through nonlinear diffusion terms while their attraction is integrated via nonlocal forces. This attractive nonlocal interaction takes into account that the presence of particles/individuals at a certain location produces a force at particles/individuals located at proportional to where the given interaction potential is assumed to be radially symmetric and increasing consistent with attractive forces. The evolution of the mass density of particles/individuals is given by the nonlinear aggregation-diffusion equation of the form:
with initial data . We will work with degenerate diffusions, , that appear naturally in modelling repulsion with very concentrated repelling nonlocal forces [59, 12]. These models are ubiquitous in mathematical biology where they have been used as macroscopic descriptions for collective behavior or swarming of animal species, see [53, 13, 54, 55, 67, 16] for instance, or more classically in chemotaxis-type models, see [60, 44, 40, 39, 11, 10, 20] and the references therein.
On the other hand, this family of PDEs is a particular example of nonlinear gradient flows in the sense of optimal transport between mass densities, see [2, 26, 27]. The main implication for us is that there is a natural Lyapunov functional for the evolution of (1.1) defined on the set of centered mass densities given by
being the last integral defined in the improper sense. Therefore, if the balance between repulsion and attraction occurs, these two effects should determine stationary states for (1.1) including the stable solutions possibly given by local (global) minimizers of the free energy functional (1.2).
Many properties and results have been obtained in the particular case of Newtonian attractive potential due to its applications in mathematical modeling of chemotaxis [60, 44] and gravitational collapse models . In the classical 2D Keller-Segel model with linear diffusion, it is known that equilibria can only happen in the critical mass case  while self-similar solutions are the long time asymptotics for subcritical mass cases [11, 18]. For supercritical masses, all solutions blow up in finite time . It was shown in [48, 19] that degenerate diffusion with is able to regularize the 2D classical Keller-Segel problem, where solutions exist globally in time regardless of its mass, and each solution remain uniformly bounded in time. For the Newtonian attraction interaction in dimension , the authors in  show that the value of the degeneracy of the diffusion that allows the mass to be the critical quantity for dichotomy between global existence and finite time blow-up is given by . In fact, based on scaling arguments it is easy to argue that for , the diffusion term dominates when density becomes large, leading to global existence of solutions for all masses. This result was shown in  together with the global uniform bound of solutions for all times.
However, in all cases where the diffusion dominates over the aggregation, the long time asymptotics of solutions to (1.1) have not been clarified, as pointed out in . Are there stationary solutions for all masses when the diffusion term dominates? And if so, are they unique up to translations? Do they determine the long time asymptotics for (1.1)? Only partial answers to these questions are present in the literature, which we summarize below.
To show the existence of stationary solutions to (1.1), a natural idea is to look for the global minimizer of its associated free energy functional (1.2). For the 3D case with Newtonian interaction potential and , Lions’ concentration-compactness principle  gives the existence of a global minimizer of (1.2) for any given mass. The argument can be extended to kernels that are no more singular than Newtonian potential in at the origin, and have slow decay at infinity. The existence result is further generalized by  to a broader classes of kernels, which can have faster decay at infinity. In all the above cases, the global minimizer of (1.2) corresponds to a stationary solution to (1.1) in the sense of distributions. In addition, the global minimizer must be radially decreasing due to Riesz’s rearrangement theorem.
Regarding the uniqueness of stationary solutions to (1.1), most of the available results are for Newtonian interaction. For the 3D Newtonian potential with , for any given mass, the authors in  prove uniqueness of stationary solutions to (1.1) among radial functions, and their method can be generalized to the Newtonian potential in with . For the 3D case with ,  show that all compactly supported stationary solutions must be radial up to a translation, hence obtaining uniqueness of stationary solutions among compactly supported functions. The proof is based on moving plane techniques, where the compact support of the stationary solution seems crucial, and it also relies on the fact that the Newtonian potential in 3D converges to zero at infinity. Similar results are obtained in  for 2D Newtonian potential with using an adapted moving plane technique. Again, the uniqueness result is based on showing radial symmetry of compactly supported stationary solutions. Finally, we mention that uniqueness of stationary states has been proved for general attracting kernels in one dimension in the case , see . To the best of our knowledge, even for Newtonian potential, we are not aware of any results showing that all stationary solutions are radial (up to a translation).
Previous results show the limitations of the present theory: although the existence of stationary states for all masses is obtained for quite general potentials, their uniqueness, crucial for identifying the long time asymptotics, is only known in very particular cases of diffusive dominated problems. The available uniqueness results are not very satisfactory due to the compactly supported restriction on the uniqueness class imposed by the moving plane techniques. And thus, large time asymptotics results are not at all available due to the lack of mass confinement results of any kind uniformly in time together with the difficulty of identifying the long time limits of sequences of solutions due to the restriction on the uniqueness class for stationary solutions.
If one wants to show that the long time asymptotics are uniquely determined by the initial mass and center of mass, a clear strategy used in many other nonlinear diffusion problems, see  and the references therein, is the following: one first needs to prove that all stationary solutions are radial up to a translation in a non restrictive class of stationary solutions, then one has to show uniqueness of stationary solutions among radial solutions, and finally this uniqueness will allow to identify the limits of time diverging sequences of solutions, if compactness of these sequences is shown in a suitable functional framework. Let us point out that comparison arguments used in standard porous medium equations are out of the question here due to the lack of maximum principle by the presence of the nonlocal term.
In this work, we will give the first full result of long time asymptotics for a diffusion dominated problem using the previous strategy without smallness assumptions of any kind. More precisely, we will prove that all solutions to the 2D Keller-Segel equation with converge to the global minimizer of its free energy using the previous strategy. The first step will be to show radial symmetry of stationary solutions to (1.1) under quite general assumptions on and the class of stationary solutions. Let us point out that standard rearrangement techniques fail in trying to show radial symmetry of general stationary states to (1.1) and they are only useful for showing radial symmetry of global minimizers, see . Comparison arguments for radial solutions allow to prove uniqueness of radial stationary solutions in particular cases [50, 46]. However, up to our knowledge, there is no general result in the literature about radial symmetry of stationary solutions to nonlocal aggregation-diffusion equations.
Our first main result is that all stationary solutions of (1.1) are radially decreasing up to translation by a fully novel application of continuous Steiner symmetrization techniques for the problem (1.1). Continuous Steiner symmetrization has been used in calculus of variations  for replacing rearrangement inequalities [14, 49, 56], but its application to nonlinear nonlocal aggregation-diffusion PDEs is completely new. Most of the results present in the literature using continuous Steiner symmetrization deal with functionals of first order, i.e. functionals involving a power of the modulus of the gradient of the unknown, see [43, Section II] and [42, 15], while in our case the functional (1.2) is purely of zeroth order. The decay of the attractive Newtonian potential interaction term in follows from [15, Corollary 2] and , which is the only result related to our strategy. We will construct a curve of measures using continuous Steiner symmetrization such that the functional (1.2) decays strictly along that curve unless the base point is radially symmetric. This fact together with the assumption that the base point is a stationary state leads to a contradiction unless the stationary state is radially symmetric. This first main result is the content of Section 2 in which we specify the assumptions on the interaction potential and the notion of stationary solutions in details. We point out that the variational structure of (1.1) is crucial to show the radially decreasing property of stationary solutions.
The result of radial symmetry for general stationary solutions to (1.1) is quite striking in comparison to other gradient flow models in collective behavior based on the competition of attractive and repulsive effects via nonlocal interaction potentials. Actually, it is demonstrated both numerically and asymptotically in [47, 6, 3] that there should be stationary solutions of these fully nonlocal interaction models which are not radially symmetric despite the radial symmetry of the interaction potential. Our first main result shows that this break of symmetry does not happen whenever nonlinear diffusion is chosen to model very strong localized repulsion forces, see . Another consequence of our radial symmetry results is the lack of non-radial local minimizers of the free energy functional (1.2) which is not all obvious.
We next study more properties of particular radially decreasing stationary solutions. We make use of the variational structure to show the existence of global minimizers to (1.2) under very general hypotheses on the interaction potential . In section 3, we show that these global minimizers are in fact compactly supported radially decreasing continuous functions. These results fully generalize the results in [62, 22]. Putting together Sections 2 and 3, the uniqueness and full characterization of the stationary states is reduced to uniqueness among the class of radial solutions. This result is known in the case of Newtonian attraction kernels .
Finally, we make use of the uniqueness among translations for any given mass of stationary solutions to (1.1) to obtain the second main result of this work, namely to answer the open problem of the long time asymptotics to (1.1) with Newtonian interaction in 2D. This is accomplished in Section 4 by a compactness argument for which one has to extract the corresponding uniform in time bounds and a careful treatment of the nonlinear terms and dissipation while taking the limit . We do not know how to obtain a similar result for Newtonian interaction in due to the lack of uniform in time mass confinement bounds in this case. We essentially cannot show that mass does not escape to infinity while taking the limit . However, the compactness and characterization of stationary solutions is still valid in that case.
The present work opens new perspectives to show radial symmetry for stationary solutions to nonlocal aggregation-diffusion problems. While the hypotheses of our result to ensure existence of global radially symmetric minimizers of (1.2), and in turn of stationary solutions to (1.1), are quite general, we do not know yet whether there is uniqueness among radially symmetric stationary solutions (with a fixed mass) for general non-Newtonian kernels. We even do not have available uniqueness results of radial minimizers beyond Newtonian kernels. Understanding if the existence of radially symmetric local minimizers, that are not global, is possible for functionals of the form (1.2) with radial interaction potential is thus a challenging question. Concerning the long-time asymptotics of (1.1), the lack of a novel approach to find confinement of mass beyond the usual virial techniques and comparison arguments in radial coordindates hinders the advance in their understanding even for Newtonian kernels with . Last but not least, our results open a window to obtain rates of convergence towards the unique equilibrium up to translation for the Newtonian kernel in 2D. The lack of general convexity of this variational problem could be compensated by recent results in a restricted class of functions, see . However, the problem is quite challenging due to the presence of free boundaries in the evolution of compactly supported solutions to (1.1) that rules out direct linearization techniques as in the linear diffusion case .
2. Radial Symmetry of stationary states with degenerate diffusion
Throughout this section, we assume that , and satisfies the following four assumptions:
is attracting, i.e., is radially symmetric
and for all with .
is no more singular than the Newtonian kernel in at the origin, i.e., there exists some such that for .
There exists some such that for all .
Either is bounded for or there exists such that for all :
As usual, denotes the positive and negative part of such that . In particular, if is the attractive Newtonian potential, where is the fundamental solution of operator in , then satisfies all the assumptions.
We denote by the set of all nonnegative functions in . Let us start by defining precisely stationary states to the aggregation equation (1.1) with a potential satisfying (K1)-(K4).
Given we call it a stationary state for the evolution problem (1.1) if , , and it satisfies
in the sense of distributions in .
Let us first note that is globally bounded under the assumptions (K1)-(K3). To see this, a direct decomposition in near- and far-field sets yields
where we split the integrand into the sets and , and apply the assumptions (K1)-(K3).
Under the additional assumptions (K4) and , we will show that the potential function is also locally bounded. First, note that (K1)-(K3) ensures that for all with some , where
Hence we can again perform a decomposition in near- and far-field sets and obtain
Our main goal in this section is the following theorem.
Before going into the details of the proof, we briefly outline the strategy here. Assume there is a stationary state which is not radially decreasing under any translation. To obtain a contradiction, we consider the free energy functional associated with (1.1),
We first observe that the energy of the steady state is finite since the potential function satisfies (2.4) with . Using the assumption that is not radially decreasing under any translation, we will apply the continuous Steiner symmetrization to perturb around and construct a continuous family of densities with , such that for some and any small . On the other hand, using that is a stationary state, we will show that for some and any small . Combining these two inequalities together gives us a contradiction for sufficiently small .
Let us characterize first the set of possible stationary states of (1.1) in the sense of Definition 2.1 and their regularity. Parts of these arguments are reminiscent from those done in [62, 22] in the case of attractive Newtonian potentials.
with right hand side belonging to for all . As a consequence, is in fact a weak solution in for all of (2.6) by classical elliptic regularity results. Sobolev embedding shows that belongs to some Hölder space , and thus . Let us define the set . Since , then is an open set and it consists of a countable number of open possibly unbounded connected components. Let us take any bounded smooth connected open subset such that . Since , then is bounded away from zero in and thus due to the assumptions on , we have that holds in the distributional sense in . We conclude that wherever is positive, (2.1) can be interpreted as
where is a constant in each connected component of the support of , and its value may differ in different connected components. Due to , we deduce that if and for . Putting together (2.7) and (2.4), we conclude the desired estimate. ∎
2.1. Some preliminaries about rearrangements
Now we briefly recall some standard notions and basic properties of decreasing rearrangements for nonnegative functions that will be used later. For a deeper treatment of these topics, we address the reader to the books [37, 5, 41, 45, 49] or the papers [64, 65, 66, 57]. We denote by the Lebesgue measure of a measurable set in . Moreover, the set is defined as the ball centered at the origin such that .
A nonnegative measurable function defined on is called radially symmetric if there is a nonnegative function on such that for all . If is radially symmetric, we will often write for by a slight abuse of notation. We say that is rearranged if it is radial and is a nonnegative right-continuous, non-increasing function of . A similar definition can be applied for real functions defined on a ball .
We define the distribution function of by
Then the function defined by
will be called the Hardy-Littlewood one-dimensional decreasing rearrangement of . By this definition, one could interpret as the generalized right-inverse function of .
Making use of the definition of , we can define a special radially symmetric decreasing function , which we will call the Schwarz spherical decreasing rearrangement of by means of the formula
where is the volume of the unit ball in .
It is clear that if the set of has finite measure, then is supported in the ball
One can show that (and so ) is equidistributed with (i.e. they have the same distribution function). Thus if , a simple use of Cavalieri’s principle (see e.g. [65, 45]) leads to the invariance property of the norms:
In particular,using the layer-cake representation formula (see e.g. ) one could easily infer that
Among the many interesting properties of rearrangements, it is worth mentioning the Hardy-Littlewood inequality (see [37, 5, 45] for the proof): for any couple of nonnegative measurable functions on , we have
Since in Section 4 we will use estimates of the solutions Keller-Segel problems in terms of their integrals, let us now recall the concept of comparison of mass concentration, taken from , that is remarkably useful.
Let be two nonnegative, radially symmetric functions on . We say that is less concentrated than , and we write if for all we get
The partial order relationship is called comparison of mass concentrations. Of course, this definition can be suitably adapted if are radially symmetric and locally integrable functions on a ball . The comparison of mass concentrations enjoys a nice equivalent formulation if and are rearranged, whose proof we refer to [1, 69]:
Let be two nonnegative rearranged functions. Then if and only if for every convex nondecreasing function with we have
From this Lemma, it easily follows that if and are rearranged and non-negative, then
Let us also observe that if are nonnegative and rearranged, then if and only if for all we have
If , we denote by the second moment of , i.e.
In this regard, another interesting property which will turn out useful is the following
Let with . If additionally is rearranged and , then .
Let us consider the sequence of bounded radially increasing functions , where is the truncation of the function at the level and define the function
Then passing to the limit as we find the desired result. ∎
2.2. Continuous Steiner symmetrization
Although classical decreasing rearragement techniques are very useful to study properties of the minimizers and for solutions of the evolution problem (1.1) in next sections, we do not know how to use them in connection with showing that stationary states are radially symmetric. For an introduction of continuous Steiner symmetrization and its properties, see [14, 15, 49]. In this subsection, we will use continuous Steiner symmetrization to prove the following proposition.
Let , and assume it is not radially decreasing after any translation. Moreover, assume that there exist some and , such that in . Then there exist some (depending on and ) and a function with , such that satisfies the following for a short time , where is as given in (2.5):
2.2.1. Definitions and basic properties of Steiner symmetrization
Let us first introduce the concept of Steiner symmetrization for a
measurable set . If , the Steiner symmetrization of is the symmetric interval . Now we want to define the Steiner symmetrization of with respect to a direction in for . The direction we symmetrize corresponds to the unit vector , although the definition can be modified accordingly when considering any other direction in .
Let us label a point by , where and . Given any measurable subset of we define, for all , the section of with respect to the direction as the set
Then we define the Steiner symmetrization of with respect to the direction as the set which is symmetric about the hyperplane and is defined by
In particular we have that .
Now, consider a non-negative function , for . For all , let us consider the distribution function of , i.e. the function
Then we can give the following definition:
We define the Steiner symmetrization (or Steiner rearrangement) of in the direction as the function such that is exactly the Schwarz rearrangement of i.e. (see (2.9))
As a consequence, the Steiner symmetrization is a function being symmetric about the hyperplane and for each the level set
is equivalent to the Steiner symmetrization
which implies that and are equidistributed, yielding the invariance of the norms when passing from to , that is for all we have
Moreover, by the layer-cake representation formula, we have
Now, we introduce a continuous version of this Steiner procedure via an interpolation between a set or a function and their Steiner symmetrizations that we will use in our symmetry arguments for steady states.
For an open set , we define its continuous Steiner symmetrization for any as below. In the following we abbreviate an open interval by , and we denote by the sign of (which is for positive , for negative , and if ).
If , then
If (where all are disjoint), then for , where is the first time two intervals share a common endpoint. Once this happens, we merge them into one open interval, and repeat this process starting from .
If (where all are disjoint), let for each , and define .
See Figure 1 for illustrations of in the cases (1) and (2). Also, we point out that case (3) can be seen as a limit of case (2), since for each one can easily check that for all . Moreover, according to , the definition of can be extended to any measurable set of , since
being open sets and a nullset.
In the next lemma we state four simple facts about . They can be easily checked for case (1) and (2) (hence true for (3) as well by taking the limit), and we omit the proof.
Given any open set , let be defined in Definition 2.9. Then
for all .
If , we have for all .
has the semigroup property: for any and open set .
Once we have the continuous Steiner symmetrization for a one-dimensional set, we can define the continuous Steiner symmetrization (in a certain direction) for a non-negative function in .
Given , we define its continuous Steiner symmetrization (in direction ) as follows. For any , let
where is defined in (2.17).
For an illustration of for , see Figure 2.
Furthermore, it is easy to check that for all if and only if is symmetric decreasing about the hyperplane . Below is the definition for a function being symmetric decreasing about a hyperplane:
Let . For a hyperplane (with normal vector ), we say is symmetric decreasing about if for any , the function is rearranged, i.e. if .
The continuous Steiner symmetrization in Definition 2.11 has the following properties:
For any , . As a result, for all .
has the semigroup property, that is, for any and non-negative .
2.2.2. Interaction energy under Steiner symmetrization
In this subsection, we will investigate . It has been shown in [15, Corollary 2] and [49, Theorem 3.7] that is non-increasing in . Indeed, in the case that is a characteristic function , it is shown in  that is strictly decreasing for small enough if is not a ball. However, in order to obtain (2.14) for a strictly positive , some refined estimates are needed, and we will prove the following:
Let . Assume the hyperplane splits the mass of into half and half, and is not symmetric decreasing about . Let be given in (2.5), where satisfies the assumptions -. Then is non-increasing in , and there exists some (depending on ) and (depending on and ), such that
The building blocks to prove Proposition 2.14 are a couple of lemmas estimating how the interaction energy between two one-dimensional densities changes under continuous Steiner symmetrization for each of them. That is, we will investigate how
changes in for a given one dimensional kernel to be determined. We start with the basic case where are both characteristic functions of some open interval.
By definition of , we have for and all . If , the two intervals are moving towards the same direction for small enough , during which their interaction energy remains constant, implying . Hence it suffices to focus on and prove (2.20).
Without loss of generality, we assume that , so that is either 2 or 1. The definition of gives