Inhomogeneous Patlak-Keller-Segel models and Aggregation Equations with Nonlinear Diffusion in
Aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with nonlinear diffusion are popular models for nonlocal aggregation phenomenon and are a source of a number of interesting mathematical problems in nonlinear PDE. The purpose of this work is twofold. First, we continue our previous work , which focused on nonlocal aggregation, modeled with a convolution. The goal was to unify the local and global theory of these convolution-type models, including the identification of a sharp critical mass; however, some cases involving unbounded domains were left open. In particular, the biologically relevant case was not treated. In this paper, we present an alternative proof of local existence, which now applies to for all and give global results that were left open . The proof departs from [8, 5] in that it uses a more direct and intuitive regularization that constructs approximate solutions on instead of on sequences of bounded domains. Second, this work develops the local, subcritical, and small data critical theory for a variety of Patlak-Keller-Segel models with spatially varying diffusion and decay rate of the chemo-attractant.
In this paper we study several types of aggregation models with nonlinear diffusion and nonlocal self-attraction.
The primary focus is to develop and extend the relevant local, subcritical, and small data critical/supercritical theory.
These results exist in the perturbative regime as they all fundamentally treat the PDE as a nonlinear perturbation of the diffusion equation (see below for more information).
Furthermore, we also present several non-perturbative global existence results for a class of critical problems as well.
The first general class of systems we study are those where the nonlocal self-attraction arises as the result of a convolution operator
Equation (1) is of interest in mathematical biology as it models the competition between a species’ desire to aggregate and to disperse.
Dispersal is modeled via the, potentially nonlinear, diffusion term and the aggregation is modeled via the nonlocal advective term .
The most well-known example of (1) is the parabolic-elliptic Patlak-Keller-Segel model, based on the original models of Patlak  and Keller and Segel .
For more information on the modeling aspects, see the [26, 25] for reviews of various chemotaxis models and [44, 15, 34, 24, 16] for more general swarming and aggregation models.
In this paper, we extend our recent work  to provide a more satisfactory and complete local and global theory for (1) on for .
In , we
studied the local and global existence and uniqueness of bounded, integrable solutions to (1) in bounded domains for and all space for .
The primary goal of the previous
work was to unify the existing Patlak-Keller-Segel global existence theory [41, 42, 43, 14, 31, 17, 11] with the local existence and uniqueness theory for less singular versions of (1) .
In that work , the case was not treated due to several technical difficulties.
Since this case is very important for mathematical biology, we make specific effort to treat this case and discuss the difficulties in more detail below.
We present an alternative proof of local existence of (1) for a wide range of and which applies on , , for solutions with bounded second moment.
The new proof is based on a regularization directly
on , in contrast to .
One of the benefits of this regularization is that it allows one to rigorously justify the application of homogeneous Sobolev embeddings in formal arguments, which are crucial in deducing small data global existence and uniform boundedness in supercritical cases. We prove such results below following iteration techniques similar to those employed in [20, 41, 3].
We also expand the global existence results of  to estimate the critical mass for kernels with a logarithmic
singularity at the origin in .
The second class of nonlocal aggregation models we study is the variable-coefficient parabolic-elliptic Patlak-Keller-Segel model,
We assume is strictly positive and is non-negative.
This PDE system is in many ways similar to (1), however they are not of the same form, as the solution to
the equation for the chemical concentration, , cannot be written in convolution form unless and are both constant.
In this paper we develop the local, subcritical and small data critical/supercritical theory for (2).
The proofs are analogous to those of (1) with additional complications arising due to the different nature of the estimates for in terms of (see Appendix).
Analysis of the critical case and the identification of the critical mass has been completed by I. Kim and one of the authors in a separate work .
The entropy, , and the interaction energy (potential energy), , are given by
with if the system is of convolution type. The entropy density (internal energy density), , is a strictly convex function satisfying
In fact, both (1) and (2) are formally the gradient flows for (3) in the Euclidean Wasserstein metric (see e.g. ). For (1) and (2) there is no fully developed theory for making this precise; however, some aspects can be recovered and have proven very useful [9, 10]. In any case, the free energy (3) plays an important role, especially in the global theory as in for example [41, 14, 17, 11, 12, 5].
Notation and Conventions
We work on for . For notational simplicity we denote the parabolic domain by and the standard norm by We also introduce the following notation for the moments:
We let with for , for , and for be our canonical cut-off function and denote the standard mollifer .
We use to denote the Newtonian potential:
By ‘weighted Young’s inequality’ we mean for and and ,
Since we will be working with many largely irrelevant constants, we use the notation to denote , where is a generic constant that depends on etc.
1.1 Definitions and Assumptions
We consider the general class of kernels introduced in , which includes fundamental solutions to elliptic PDEs and other commonly considered attractive kernels.
Definition 1 (Admissible Kernel).
We say a kernel is admissible if and the following holds:
is radially symmetric, and is non-increasing.
and are monotone on for some .
This definition ensures that is attractive, well-behaved at the origin, and has second derivatives that define bounded distributions on for .
The obvious example of an admissible kernel is the Newtonian potential, which is effectively the most singular admissible kernel both at the origin and at infinity (in the sense that it decays the slowest).
We remark that many of our results (§2.1 and §3) still hold if we replace condition (KN) with the assumption , allowing for non-radially symmetric and general attractive/repulsive type kernels.
We limit ourselves to diffusions that do not spread mass faster than linear diffusion; however, using the techniques of  one could also treat cases with fast diffusion. This is more general than the diffusion considered in [8, 5], which were restricted to degenerate diffusion.
Definition 2 (Admissible Diffusion Functions).
We say that the function is an admissible diffusion function if:
for some .
when for some .
In particular (D3) implies as and
for all .
Following [5, 8, 7] we use the following notion of weak solution, which is stronger than traditional distribution solutions. In , test functions are taken in , whereas in minor adjustments must be made, as discussed below and in . By density arguments, this is basically the same as taking test functions in and requiring various regularity assumptions on the solution. However, but we prefer the current statement of the definition to emphasize the kind of test functions that we are interested in. Taking test functions in these spaces is important for the proof of uniqueness, which is based on an stability estimate [8, 7, 5, 2].
Definition 3 (Weak Solution in , ).
Due to the regularity we are imposing on our solution, we could equivalently require for all for a.e.
with the modification in .
The additional complication in is due to the fact that the norm is not well-behaved in , since scales like in . Indeed, there exists a sequence of Schwartz functions with and point-wise a.e. (consider and scaling , ). In order to remove such pathologies from our space of test functions we follow , which also requires test functions to lie in .
Following , we now define a notion of criticality for (1), which in general has no scaling symmetries. However, a kind of scaling symmetry can be recovered in the limit of mass concentration, which in turn is expected to govern blow-up (see Theorem 3 below). Suppose as , with and let . Then if we have,
From here we see the limit is independent of unless . If the limit is we expect aggregation to dominate near mass concentration and if the limit is we expect diffusion to dominate. As mass concentration should occur on vanishing length-scales, we may use this scaling heuristic to define a notion of criticality. The limiting case of occurs when considering as and is discussed more below. Of course this corresponds to the 2D classical parabolic-elliptic PKS model, but we can consider the same singularity in higher dimensions (as done in for example ).
Definition 4 (Critical Exponent).
Suppose is admissible such that for some (with the convention that if ) we have as . Then the critical exponent associated to is given by
For the variable-coefficient Patlak-Keller-Segel system (2) we take .
The variable coefficient system (2) should be roughly as singular as the constant coefficient case, hence the corresponding definition in this case.
Now we define the notion of criticality by relating the critical exponent of the kernel to the diffusion, again focusing on the limit of mass concentration. It is easier to define this notion in terms of the quantity , as opposed to using directly. This is not so surprising as is precisely the local coefficient of diffusivity, and directly measures the strength of the diffusion relative to the mass density.
Definition 5 (Criticality).
We say that the problem is subcritical if
and supercritical if
The following lemma, from , enumerates several important aspects of admissible kernels. Part (c) in particular provides a useful characterization of kernels with .
Let be admissible. Then each of the following is true:
, and if then . If , then .
For all , , such that for all , . Moreover, as .
Let and be such that . Then, if and only if . If , then if and only if and for some if and only if . Note by (BD), if for some , then, .
Lemma 2 (Logarithmic Hardy-Littlewood-Sobolev inequality ).
Let and be such that . Then,
One also has for all ,
Also recall the well-known convolution inequality (see ): for all , and for satisfying ,
As noted above for the admissible kernels are generally only in and hence can grow logarithmically at infinity. This introduces a number of complications for the local and global well-posedness. To begin with, in the proof of the energy dissipation inequality, one must ensure that the interaction energy of the approximate solutions converges to the interaction energy of the weak solution being constructed. However, will be unbounded for general hence more care must be taken than in . The dual of is the Hardy space , a strict subset of , which we define via duality,
Accordingly, we have the natural analogue of Hölder’s inequality 
which in particular implies whenever and . The following lemma found in  provides sufficient conditions for such that and a useful estimate of the norm that supplies the convergence of the interaction energies.
Lemma 3 ().
Let for some and satisfy , . Then and
1.2 Statement of Results
Theorem 1 (Local Existence and Energy Dissipation for Convolution-type Systems).
Let , be admissble and . Then there exists a and a weak solution of (1) which satisfies and . Moreover, and satisfies the energy dissipation inequality,
For a rigorous interpretation of the free energy dissipation,
see for example . As mentioned above, the local existence for general models (1) for was proved in , which extended the existence results of [41, 14, 8]. We present an alternative which is specialized to treating that has certain advantages. In particular we treat . We also prove the corresponding theorem for variable-coefficient Patlak-Keller-Segel systems.
Theorem 2 (Local Existence and Energy Dissipation for Variable-Coefficient Systems).
Let , be strictly positive such that and let be non-negative. In , further suppose that is strictly positive. If then there exists a and a weak solution of (2) which satisfies and . Moreover, and satisfies the energy dissipation inequality
Uniqueness for convolution-type systems is proven in  for ; the same proof works for (2) using the elliptic estimates found in the Appendix. More recent work undertaken by J. Azzam and one of the authors  proves uniqueness for the case. For completeness, we state a continuation theorem proved in , which extends previous theorems stated in [27, 17, 11]. The extension to cover (2) is straightforward and is briefly discussed below in §3.1.
Theorem 3 (Continuation ).
One of the primary tools in the proofs of Theorems 1-3 is the use of Alikakos iteration methods commonly used in the study of these PDEs as for example [27, 31, 17, 5, 3]. These methods are fundamentally perturbative in nature (as are the related methods of [38, 13]), depending on relatively crude Gagliardo-Nirenberg inequalities to overpower the nonlinear aggregation with diffusion only in certain regimes. In subcritical regimes this is sufficient and these methods prove global existence and uniform boundedness in , as in [31, 17]. In the critical and supercritical cases, one can prove the same provided that the initial condition is small in the corresponding critical norm and that the nonlinear diffusion compares favorably with the homogeneous diffusion even at low densities. As shown in for example [20, 41, 38, 3], stronger decay estimates may also be deduced using various refinements of similar iteration methods.
Theorem 4 (Subcritical and Small Data Theory).
(subcritical) Suppose that for sufficiently large there exists some such that
for . Then the solution is global and .
then the solution is global () and . Furthermore, if is not strictly positive and then there exists a constant such that if
then the solution is global and .
One can check that if and then if , (1) has a scaling symmetry which leaves the norm invariant. It is in this sense that the norm is critical. Hence, even when , part (ii) is still a small data critical result, which is expected. However the proof is more difficult in this case, as (1) has no controlled quantity which controls the a priori.
We are unsure if the additional requirement on the mass in (ii) when is not strictly positive in supercritical problems is necessary. For the Alikakos iteration methods to work, the singularity of the advective term is quantified in terms of the elliptic estimates of Lemmas 11 and 12 in the Appendix. In order to prove part (ii) for supercritical cases , one must use these estimates in a bootstrap argument to also control the critical norm ; however, Lemma 11 introduces a ‘subcritical’ lower-order term that seems to require the additional assumption in order to control.
For critical problems, deducing global bounds and decay estimates for larger data requires fully non-pertubative techniques that depend heavily on the energy dissipation inequalities (10) and (11). The use of sharp functional inequalities to determine when mixed-sign energies such as (3) are coercive is the classical standard method of treating large data and determining the sharp critical mass, for example for the focusing nonlinear Schrödinger equations, marginal unstable thin film equations and, of course, PKS [45, 46, 30, 14, 17, 11, 5]. Our previous work of  treated the case for critical convolution-type systems in , and in this work we complete the case, for example, now covering the variants of the classical Patlak-Keller-Segel system in . For Patlak-Keller-Segel, the proof that the critical mass is sharp follows easily from a standard Virial argument (see e.g. [14, 41, 43, 11, 5] or the more classical ), which can be modified in a straightforward manner to treat more general problems . The corresponding program for the variable-coefficient systems (2) is a more difficult problem, completed by I. Kim and one of the authors elsewhere . We also stress that (ii) is already known, e.g. [14, 11], but we restate and prove it to provide comparison with the other problems that do not have scaling symmetries.
Theorem 5 (Global Existence in , Critical case, ).
Let be admissible and suppose and let be the local-in-time weak solution to (1) with and define .
Suppose the problem is critical. Then there exists a critical mass such that if then exists globally. The estimate of is given below by (14). If is bounded from below and
then we may additionally assert .
Suppose , and , the Newtonian potential. Furthermore, suppose that . Then exists globally and satisfies
Let satisify for some . Then, the critical mass satisifies
Note the additional requirements in part (i) of Theorem 5 in order to assert global boundedness. Although we believe that solutions are bounded (and in many cases likely decay as in (ii)) our proof cannot rule out an unbounded increase of entropy or interaction energy as the solution spreads without the additional assumptions.
2 Local Existence and Energy Dissipation
2.1 Local Existence for Convolution-type Systems
This section focuses on the proof of local existence of weak solutions with bounded second moment. The proofs included here are simplifications of our work in  and combines techniques from the PKS model found in  and the non-singular aggregation-diffusion equations found in . One advantage of the new techniques used here is the treatment of . Furthermore, in contrast to the work in , we obtain solutions directly in , removing the need for the intermediate step of finding solutions in bounded domains. Consider the regularized aggregation-diffusion equation
where is a smooth function, such that . We denote,
Hence by Definition 1 .
Proposition 2 (Local Existence for the Regularized System).
Let be fixed and . Then (15) has a classical solution on for all with .
To prove the above proposition we need some preliminary definitions. Define the Hilbert space
with the inner product defined via .
Note by the Rellich-Khondrashov compactness theorem, is compactly embedded in .
We will construct a weak solution to (15) with an analogous definition of weak solutions to Definition 3.
For the remainder of the paper we denote the mass of the initial data, , by , i.e. .
We prove Proposition 2 using the Schauder fixed point theorem, see e.g. . The necessary compactness for the application is obtained via the Aubin-Lions Lemma . Now we state and prove some a priori estimates that will be of used in the proof of Proposition 2. Some of these estimates are the same or closely related to estimates proved elsewhere (e.g. [14, 8, 5]).
Lemma 4 (A priori bounds with linear advection).
For fixed let be given. Let be the global strong solution to
with initial data with . Then,
In what follows denote . By (17) and integration by parts, once on the diffusion term and twice on the aggregation term we obtain
which gives (i). The bound (ii) follows similarly by estimating the growth of and passing to the limit . To continue, we need a bound on the norm of . Condition (D3) of Definition 2 implies that for some and sufficiently small . Hence by Chebyshev’s inequality,