Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in with Measure-valued Initial Data
We establish a new local well-posedness result in the space of finite Borel measures for mild solutions of the parabolic-elliptic Patlak-Keller-Segel (PKS) model of chemotactic aggregation in two dimensions. Our result only requires that the initial measure satisfy the necessary assumption . This work improves the small-data results of Biler  and the existence results of Senba and Suzuki . Our work is based on that of Gallagher and Gallay , who prove the uniqueness and log-Lipschitz continuity of the solution map for the 2D Navier-Stokes equations (NSE) with measure-valued initial vorticity. We refine their techniques and present an alternative version of their proof which yields existence, uniqueness and Lipschitz continuity of the solution maps of both PKS and NSE. Many steps are more difficult for PKS than for NSE, particularly on the level of the linear estimates related to the self-similar spreading solutions.
The primary focus of this work is establishing a large-data local well-posedness result in the space of finite Borel measures for the parabolic-elliptic Patlak-Keller-Segel model in two dimensions:
This system is generally considered the fundamental mathematical model for the study of aggregation by chemotaxis of certain microorganisms [59, 49, 45, 44]. From now on we will refer to (1.1) as Patlak-Keller-Segel (PKS). The first equation describes the motion of the microorganism as a random walk with drift up the gradient of the chemo-attractant . The second equation describes the production and (instantaneous) diffusion of the chemo-attractant. PKS and related variants have received considerable mathematical attention over the years, for example, see the review  or some of the following representative works [24, 46, 54, 4, 42, 63, 7, 13, 62, 11, 8, 9].
It has been known for some time that (1.1) possesses a critical mass: if then classical solutions exist for all time (see e.g. [63, 13, 11, 9, 8]) and if then all classical solutions with finite second moment blow up in finite time [46, 54, 13] and are known to concentrate at least mass into a single point at blow-up  (see also [42, 62]). Another important property of (1.1) that plays a decisive role in our work is the existence (and uniqueness) of self-similar spreading solutions for all mass . These are known to be global attractors for the dynamics if the total mass is less than  and for the purposes of our analysis, should be thought of as analogous to the Oseen vortices of the Navier-Stokes equations. When studied in higher dimensions, (1.1) is supercritical and the dynamics are quite different, see for example [4, 26, 2, 41, 40]. Variants of (1.1) involving nonlinear diffusion which are critical in higher dimensions have also been studied [51, 14, 65, 66, 10, 50] (see also the related ). The parabolic-parabolic version of (1.1) has also been analyzed in various contexts (see e.g. [62, 16, 43, 57, 6]). We should also mention that variants of (1.1) have been studied in the context of astrophysics (referred to as Smoluchowski-Poisson) as a simplified model for the collapse of overdamped self-gravitating particles undergoing Brownian motion (see e.g. [5, 22, 23, 64]).
The goal of the present work is to prove the most general local well-posedness result known for (1.1). We work with so-called mild solutions, motivated by similar notions used in fluid mechanics (other authors have also used this notion for (1.1)). See below for the full definition and discussion (Definition 1), but the main idea is that these solutions satisfy (1.1) as the integral equation
and satisfy the optimal hypercontractive estimate (the self-similar spreading solutions show that the rate cannot generally be better). We show that there exists a unique mild solution to (1.1) given initial data which is a non-negative, finite Borel measure that satisfies . Moreover, we also show that the solution map is locally Lipschitz continuous with respect to the total variation norm of the initial data. This is the most general well-posedness result possible in this space without considering weaker notions of solutions which can be extended past blow up (such solutions do exist , see below for a discussion). The mild solutions we construct are smooth for at least for some short time, which is not possible if there already exists a concentration with critical mass. Biler proved  using a contraction mapping argument that if the initial measure has a small atomic part then one can construct a unique mild solution. Senba and Suzuki  construct weak solutions only under the assumption , however, their solutions are not a priori mild solutions and it is far from clear that more general solutions would necessarily agree with the mild solution. Hence, our proof also yields existence, which was to our knowledge open.
Much of our work and motivation is a result of the similarities (1.1) shares with the Navier-Stokes equations in vorticity-transport form
Existence of mild solutions to (1.2) with measure-valued initial data was proved earlier in [27, 38] and similar to (1.2), it is relatively easy to prove well-posedness if the initial data has only very small atoms. However, in , Gallagher and Gallay proved that given arbitrary initial vorticity in the space of finite Borel measures, there is a unique mild solution to (1.2) and the solution map is log-Lipschitz continuous with respect to the total variation norm (see also [36, 34] for a proof of uniqueness of the Oseen vortex with point measure initial data).
The proof of Gallagher and Gallay  uses an accurate approximate solution and an intelligent decomposition of the error between the approximate solution and the true solution, which is shown to be very small in an appropriate sense. A Grönwall-type estimate is used to prove that if two solutions have the same initial data then they must differ from the approximate solution in the same way and hence are equal. However, the argument is not quite a contraction mapping. Consequently, it requires the a priori existence of well-behaved mild solutions and yields log-Lipschitz dependence on initial data, but not Lipschitz. Our argument follows the same general principles set forth in , however, we use a different decomposition which allows stronger results. In particular, unlike , our argument is a true contraction mapping, and this allows us to prove existence of solutions as well as the Lipschitz continuity of the solution maps of both (1.1) and (1.2) (see Theorem 3).
As in , the approximate solution is constructed by guessing that near a large atomic concentration in the initial data, for short time, the solution to (1.1) or (1.2) should look like a self-similar spreading solution, and elsewhere can be approximated by a linear evolution. In order to close a contraction mapping argument, some knowledge about the linearization around the approximate solution is necessary. A ‘brute force’ linear analysis is likely intractable, however, it turns out that knowing good spectral properties of all the well-separated pieces of the approximate solution is sufficient to close the argument. In particular, we need good spectral properties of the linearization of (1.1) or (1.2) around the self-similar spreading solutions. For NSE, the nonlinearity vanishes for radially symmetric data, which is why the Oseen vortices are simply the self-similar solutions to the linear heat equation. Moreover, the linearization around the Oseen vortices is relatively easy to analyze, as it is a sum of the Fokker-Planck operator and an operator which is skew-symmetric in an appropriate Hilbert space. Nothing analogous to these properties hold in the case of the Patlak-Keller-Segel system: the self-similar spreading solutions solve a genuinely nonlinear elliptic system and the spectral properties of the linearization are far from trivial to analyze. One of the main tools for dealing with the linearization is a variant of the spectral gap recently obtained by J. Campos and J. Dolbeault . An independent proof of a weaker version specific to our needs is given in Appendix §A.1. This spectral gap needs to be further adapted to the spaces we are working in, similar to what is done in Gallay and Wayne  for NSE (see Proposition 5 below).
An additional technicality that appears here is the fact that the velocity field for PKS is not divergence free. This makes most of the results of Carlen and Loss  on the fundamental solutions of linear advection-diffusion equations inapplicable. Due to the singular nature of the velocity fields, the linear advection-diffusion equations we study cannot be treated as a perturbation of the heat equation locally in time (see  for a related issue) and hence even on the linear level we need to develop tools to carefully deal with questions such as uniqueness and continuity at the initial time.
Global measure-valued solutions of (1.1) in the sense of Poupaud’s weak solutions , which make sense even if there are mass concentrations, have been constructed by Dolbeault and Schmeiser in  by taking sequences of regularized problems and extracting a measure for along with an appropriate ‘defect measure’ to make sense of the nonlinear term. It appears that the resulting solution depends on the chosen regularization, as the formal dynamics derived by Velázquez [67, 68] are different from those constructed by Dolbeault and Schmeiser. Whether or not measure-valued solutions can be uniquely selected by physically or biologically relevant criteria remains an interesting open question. See  for some work in this direction.
Let us end this introduction by summarizing some of the main difficulties compared to the study of the NSE in :
For the PKS, the vector field is not divergence-free, hence we cannot use the results of  on the pointwise decay and localization for the fundamental solution of the linear advection-diffusion equation.
For NSE, the existence of a mild solution with strong a priori estimates was already known, a fact which Gallagher and Gallay exploit multiple times in combination with the results of . For the PKS, the existence of mild solutions with such general initial data was not known.
The self-similar profiles of (1.1) corresponding to the Oseen vortices are not linear in the mass. The critical mass will appear in many places in our analysis.
The linear operators we have to deal with are harder than those that arise in the study of NSE. For NSE, due to the divergence-free property, these linear operators are a skew-symmetric perturbation of a Fokker-Planck operator.
In order to overcome difficulties (a) and (b), we had to find a better decomposition of the error terms between the solution and the approximate solution. This gives better control of the error in norms which permit us to close a contraction mapping argument, allowing also the deduction of Lipschitz continuity of the solution map with respect to the initial data. To our knowledge, this is a new result even for NSE. To overcome difficulty (d), a compactness/rigidity argument is used to prove uniqueness for the singular linear equations and a variant of the spectral gap of Campos and Dolbeault  and known spectral properties of general Fokker-Planck equations both play important roles in the linear analysis.
The precise notion of weak solution we are using is that of a mild solution, which are motivated by similar notions in fluid mechanics and have also been used previously in the study of PKS (e.g. [4, 12]).
Definition 1 (Mild Solution).
Given , we define to be a mild solution to (1.1) with initial data on if the following are satisfied:
satisfies the following Duhamel integral equation for all
with in the sense
Recall the estimates on the heat kernel,
However, if the initial measure has a non-zero atom, the integral does not converge to zero in these norms as . That is, the solution cannot be approximated by the linear heat evolution in the critical norms by choosing small; the only option would be to impose that the atoms are small (see Theorem 1 and the results of Biler  below). In this general sense, the work here is related to the recent works on 3D NSE in the critical space [48, 47].
Here is the space of which take values in finite non-negative Borel measures continuously in time with respect to the weak topology.
Often in the sequel we will be studying singular advection-diffusion equations of the form with measure-valued initial data. For these we use a definition of mild solution exactly analogous to Definition 1 except that we will not impose a priori that the solution or initial data is non-negative and of course the velocity field is imposed externally and not derived from the solution itself.
where denotes the semi-norm on which measures the total variation of the atomic part:
Estimate (1.8) and related estimates play a key role in our analysis and the work of Gallagher and Gallay, as they show that size conditions for short time results should only depend on the atomic part of the initial data. Additional important facts about mild solutions are summarized in the following theorem.
Part (i), (to our knowledge new), shows that the condition is equivalent to the hypercontractivity estimate (the proof shows that all such estimates are equivalent). Accordingly, standard parabolic theory implies that all mild solutions are smooth and strictly positive after until (potentially) critical mass concentration. Part (ii) is due to Biler  and combined with (1.8) shows that given a measure with a sufficiently small atomic part, one can construct a unique mild solution local in time. Part (i) will play a role in the proof of the main results of the paper, although (ii) will not.
We now state our main results. For PKS we prove the following existence and uniqueness theorem:
Let with . Then there exists a unique, local-in-time mild solution to (1.1) with initial data .
As discussed above, our approach also yields the Lipschitz continuity of the solution maps for (1.1) and (1.2). Even for NSE, this is an improvement of the existing result of log-Lipschitz continuity, due to Gallagher and Gallay .
The solution maps of both NSE and PKS in two dimensions are locally Lipschitz continuous with respect to the total variation norm. That is, for all (in the case of PKS, we assume additionally and ) with associated mild solutions , there exists some constant and such that for all sufficiently small, if
Here, denotes the total variation norm on finite Borel measures.
Let us briefly discuss the energy structure of (1.1), which is important for characterizing the self-similar spreading solutions and for analyzing the global behavior, the former being crucially important for our work. Formally, the Patlak-Keller-Segel model (1.1) is a gradient flow in the Wasserstein metric for the free energy (see [8, 9]),
In particular, if the initial data has finite free energy, then for reasonable notions of weak solution we have the energy dissipation inequality,
for all until blow-up time. Using the sharp logarithmic Hardy-Littlewood-Sobolev inequality (see e.g. ) this implies global existence of any weak solution which has finite initial free energy provided that the total mass is less than [30, 13]. The energy dissipation inequality is actually stronger in similarity variables:
and . In these variables, (1.1) becomes the following,
which is formally a gradient flow for the self-similar free energy
As the second moment is now part of the energy, uniform control on the entropy from below can be obtained and the sharp logarithmic Hardy-Littlewood-Sobolev can then be used to show that any solution to (1.11) with finite self-similar free energy and mass strictly less than is uniformly bounded in time. In physical variables this is the optimal decay estimate as .
The free energy is important to characterize the self-similar solutions of (1.1). Biler et. al. show in  that for all there exists a unique, radially symmetric self-similar solution with mass , denoted here in self-similar variables by (existence had been previously established in [5, 56]). These solutions will play the role that the Oseen vortices play in  as the approximation for the solution near the large atomic pieces of the initial data. The following proposition collects the important properties of the self-similar solutions, which show that in many ways they are qualitatively similar to the Gaussian Oseen vortices of the NSE. While these results are trivial for NSE, they are more difficult for PKS, due to the fully nonlinear nature of the self-similar solutions . Parts (i-iii) are not new, but we sketch some aspects of the proof in Appendix §B for the readers’ convenience, as they are not all located in one place in the literature. Parts (iv) and (v) seem to be new and are both crucial in deducing the Lipschitz dependence in Theorem 3. Part (iv) is also necessary to show that many constants and linear estimates are uniform as , which is necessary to prove Theorem 2. We should point out that the result of (v) depends on the variant of the spectral study  in Appendix §A.1. For any , denotes the Riesz symmetric decreasing re-arrangement (see  for more information on this symmetrization technique). In what follows we denote the polynomial weighted space,
with the convention . We also define . Note that for , . For any we have the following, which will be useful later
Now we may state Proposition 1.
Proposition 1 (Properties of the self-similar solutions).
There exists a stationary solution to (1.11), denoted , which is smooth, strictly positive, satisfies , and denoting we have
in the sense of asymptotic expansion. Moreover, for all and we have
In physical variables, is the unique mild solution with finite self-similar free energy with initial data , where denotes the Dirac delta mass.
For sufficiently small, the following estimate holds for all and ,
For all and for , the estimate holds for all and .
The proof of Proposition 1 (iv) primarily shows that for sufficiently small, where is the standard Gaussian.
Due to the a priori estimates and the uniqueness of , a compactness argument shows that if , then for . These results are naturally analogous to the well-known results for the heat equation and for the 2D Navier-Stokes equations . The spectral gap-type inequality deduced by J. Campos and J. Dolbeault in  can also be used to deduce an exponential estimate on the rate of convergence; see also e.g. [36, 12].
An obvious question arises about whether or not Theorems 2 and 3 can be extended to more general models than PKS and NSE. If the nonlocal velocity law is a linear combination of the Biot-Savart law for NSE and the chemotactic gradient law for PKS then this generalization should be more or less straightforward since the will still be the self-similar solutions. However, if the velocity law is no longer homogeneous, for example if the equation for the chemo-attractant is replaced by or , then there are no longer exact self-similar solutions. If are still good short-time approximations for the evolution of atomic initial data, as should be the case in the examples just mentioned, then the stated results of Theorems 2 and 3 can likely be proved with similar arguments after some additional approximations. Such cases should also include models in which the chemo-attractant and/or the density is subjected to an external drift, provided the drift is sufficiently regular. If no longer provide a good short-time approximation to atomic initial data, more substantial changes would have to be made.
Notation and Conventions
We denote the norms by . If a measure other than Lebesgue measure is used to define the norm, this is denoted by (note that this is different than the definition of the polynomial weighted space ).
To avoid clutter in computations, function arguments (time and space) will be omitted whenever they are obvious from context. In formulas we will sometimes use the notation to denote a generic constant, which may be different from line to line or even term to term in the same computation. Moreover, to further reduce clutter in formulas, we make very frequent use of the notation to denote . We will generally suppress the dependencies which are not relevant for the estimate at hand and simply write . In most cases, universal constants from functional inequalities and parameters which are not important for the discussion are omitted. We will also usually suppress dependence from uniform estimates which have already been established, although we often alert the reader to the estimate being used.
Self-similar solutions of mass are denoted , and when the mass is given by , we will often shorten this to . Similarly, the velocity field associated with the self similar solutions are denoted or .
The following proposition collects the basic properties of the nonlocal velocity law of (1.1), which are essentially analogous to the properties of the Biot-Savart law for NSE.
Proposition 2 (Properties of the nonlocal velocity law).
Let for some . Then,
Let . Then,
If for some or for some . Then for all
2.1 Outline for the proof of Theorem 2
For chosen small later we define the decomposition of the initial data
for , and chosen such that . If the measure contains only finitely many point masses then is finite and independent of for sufficiently small. However, in general there may be infinitely many point masses and in this case it is important to note that is fixed large when is fixed small. Define the minimal distance between any two concentrations (which is generally forced small when is chosen small):
The goal of this decomposition is to construct an accurate approximate solution and use a perturbation argument to build a true solution which is very close to the approximate one. Analogously to , we construct a mild solution via a decomposition of the form
with the terms , defined below. However, our definition of is different than in .
In what follows we will explain the decomposition formally, assuming that we have a well-behaved mild solution already. In reality, we will construct this solution using the decomposition. The term is defined as the solution associated with the (approximately) non-atomic portion of the initial data, which formally satisfies the initial value problem
where still is given by the nonlocal velocity law associated with the full solution. Since has a small atomic part, (1.8) suggests that for short time will be small. Of course, it will take some work (Proposition 6) to make this convincing, as is very singular at time zero. On the other hand, the part of the solution associated with the large atomic parts of the initial data is not small in any relevant sense, so further decomposition is necessary. Consider the solutions of the advection-diffusion equation in physical variables:
In , the authors consider the difference between and the self-similar solution of mass centered at . This quantity turns out to be small as is an accurate approximation for for short time, however, proves difficult to correctly control in a contraction argument. Hence, we choose a different decomposition which still satisfies
and while each will be localized around , (although the proof will show they are close to being equal). In particular, (2.5) cannot be decoupled into separate expressions for .
Applying (2.5) to implies a more precise PDE for :
where and . For future convenience define
We now turn to the definition of for , which is somewhat more technical. For notational clarity define
the velocity fields induced by the perturbations in the coordinate system of and the velocity fields induced by the self-similar solutions of the concentrations written in the coordinate system of . Let be a smooth, non-negative, radially symmetric, non-increasing function such that for and for . We will define to be a solution of the following:
It is in the second and third line where our definition differs from . Our definition more naturally treats the dangerous terms when making estimates, but destroys the advection-diffusion structure of (2.8) and the coupling makes it trickier to prove that all mild solutions can be decomposed in this manner (proved below in Proposition 3). We re-write the equations for the perturbations as the corresponding Duhamel integral equations. Given some , define to be the mild solution to the following singular PDE
We now turn to the perturbations . Define the following linear operator, which is the linearization of the transport term around the self-similar solution,
and define the Fokker-Planck operator
Denote by the linear propagator for the PDE
The primary effort of proving Theorem 2 goes into showing that the system of integral equation (2.11),(2.13) has a unique solution in the relevant spaces, which is done using a contraction mapping argument. The perturbations are normed with defined as follows, which differs from the norm used in  by the presence of the constant to be chosen later. Let
and for ,
for some large constant to be chosen later. We use to enforce more control over than the other perturbations, which is very important for dealing with the potentially disruptive effect of on , .
By construction, the unique solution to the system (2.11),(2.13) can be re-constituted into a mild solution of (1.1) via (2.3). However, it is not a priori clear that every mild solution can be represented as a solution to the system. This nontrivial fact is stated in the following proposition. The proof mainly depends on a compactness argument and that are the unique self-similar solutions, analogous to Proposition 4.5 in . However, unlike , an additional step is required to construct which satisfy (2.13) since the cross-terms in (2.13) couple all the in a more subtle manner than in .
Proposition 3 (Equivalence of formulations).
Alternatively, in order to complete the proof of Theorem 2 we could fall back to a proof which more closely matches Gallagher and Gallay and use their decomposition to show that any second mild solution must agree with the one constructed with the integral equations (2.11),(2.13). This should work, however, we prefer to give a more self-contained proof by going through Proposition 3.
The rest of the paper is organized as follows. In Section §2.2 we state the main linear estimates which are required for the proof of both Theorem 2 and 3. In Section §3 we prove Theorem 2 in several steps. In Section §4 we establish Theorem 3, the proof of which is closely related to the main steps of Theorem 2. In Appendix §A we establish the linear estimates stated in §2.2 and in Appendix §B we sketch the proof of Proposition 1. Finally in Appendix §A.1 we include an independent proof of a version of the spectral gap estimate due to Campos and Dolbeault.
2.2 Requisite Linear Estimates
Proposition 4 (Properties of ).
Fix . Then,
defines a strongly continuous semigroup on and for all ,
for all and where .
If and , then
If then for all and ,
The following proposition is of crucial importance. It is the analogue to Proposition 4.6 in  and Proposition 4.12 in  but the proof deviates in several key places due to the different nature of the linear operator. Indeed, Recall that analyzing the spectral properties of the linearization around is more difficult for PKS than for NSE, as the operator is not skew-symmetric in any relevant Hilbert space. As mentioned previously, the key tool used is a variant of the spectral gap-type results recently obtained by J. Campos and J. Dolbeault . This spectral gap must be adapted to the polynomial weighted spaces , a procedure analogous to what is done in , which we carry out in Appendix §A.
Fix and .
defines a strongly continuous semigroup which is bounded on and satisfies
For some which depends on and for all ,
If then is a bounded operator from to and there exists a (the same as in (ii)) such that,
The following is the analogue of Proposition 4.3 in , but the proof must deviate from the corresponding one for NSE in a non-trivial manner, as the underlying linear operator no longer has as nice structure (carried out in Appendix §A.3). The first step is a general lemma (Lemma A.8) which exhibits at least one well-behaved mild solution to a class of singular advection-diffusion equations including (2.9). Next, uniqueness is proved for the case by a compactness/rigidity argument that requires the monotonicity of to localize potential pathologies in the solution as well as a decay estimate of Carlen and Loss . The extension to is straightforward following a similar argument of Gallagher and Gallay . The proof of (iii) below uses spectral properties of linear Fokker-Planck equations with general confining potentials.
There exists some sufficiently small such that
defines a weak continuous linear propagator (see Remark 7 below) on and for all and we have
For all and (uniformly in ),
There exists some independent of such that the following holds: for all , for all and for all ,
All of the implicit constants above are independent of ,, and . Moreover, it will suffice to choose such that
for some which is independent of and .
By ‘weak continuous linear propagator’ we mean that is linear and if satisfy and and , , with and we have
3 Existence and Uniqueness: Proof of Theorem 2
We proceed in several steps. First we prove the contraction mapping argument which establishes the existence and uniqueness of solutions to the integral equations (2.11) and (2.13) which is the core of the proof. Next we establish Theorem 1 (i) which is necessary to establish Proposition 3, which is carried out last. Finally we briefly summarize the full argument at the end of the section.
3.1 Contraction Mapping
Note that the ball is centered around , although given Proposition 6 (ii), this is a minor detail. For any , the corresponding constructed by (2.3) will be in (but is not small due to the presence of the large atomic pieces). It might be useful to bear in mind that the approximate solution we are perturbing around is,