# The two-dimensional Keller-Segel model after blow-up

## Transcript Of The two-dimensional Keller-Segel model after blow-up

The two-dimensional Keller-Segel model after blow-up

Jean Dolbeault1 and Christian Schmeiser2

Abstract. In the two-dimensional Keller-Segel model for chemotaxis of biological cells, blow-up of solutions in ﬁnite time occurs if the total mass is above a critical value. Blow-up is a concentration event, where point aggregates are created. In this work global existence of generalized solutions is proven, allowing for measure valued densities. This extends the solution concept after blow-up. The existence result is an application of a theory developed by Poupaud, where the cell distribution is characterized by an additional defect measure, which vanishes for smooth cell densities. The global solutions are constructed as limits of solutions of a regularized problem.

A strong formulation is derived under the assumption that the generalized solution consists of a smooth part and a number of smoothly varying point aggregates. Comparison with earlier formal asymptotic results shows that the choice of a solution concept after blow-up is not unique and depends on the type of regularization.

This work is also concerned with local density proﬁles close to point aggregates. An equation for these proﬁles is derived by passing to the limit in a rescaled version of the regularized model. Solvability of the proﬁle equation can also be obtained by minimizing a free energy functional.

Key words: Keller-Segel, chemotaxis, aggregation, blow-up, measure valued solutions, defect measure

AMS subject classiﬁcation: 35D05, 35D10, 35K55

Acknowledgement: This work has been supported by the Amadeus project no. 13785 UA and by the European network DEASE. The ﬁrst author has been supported by the ANR project IFO. The second author acknowledges the hospitality at the Universit´e Paris Dauphine as well as support by the Austrian Science Fund under grant No. W8.

1CEREMADE UMR CNRS no. 7534, Universit´e Paris Dauphine, Paris, France. 2Institut fu¨r Mathematik, Universit¨at Wien, Austria.

1

1 Introduction

The simplest description of a cell population, which produces a chemical signal and responds to it chemotactically, is the Keller-Segel model

∂t + ∇ · ( ∇S − ∇ ) = 0 ,

(1)

−∆S = .

(2)

The model is written in nondimensionalized form with the cell density (t, x) and the chemical concentration S(t, x) both depending on time t and position x. The ﬁrst equation is a convection-diﬀusion equation, where the drift term ∇S describes the chemotactic reaction of the cells, and the second equation is a quasistationary approximation of a reaction-diﬀusion equation for the chemical concentration. The reaction term models the production of the chemical by the cells. The model is based on the assumption that characteristic times for the dynamics of the chemical are much shorter than those for the cell dynamics. Also the chemotactic sensitivity, the cell diffusivity, the chemical diﬀusivity, and the reaction rate (coeﬃcients of ∇S, ∇ , ∆S, and, respectively, ) have been assumed constant and have been removed by an appropriate scaling.

Since its ﬁrst formulation [15] and ﬁrst mathematical investigation [13], this model (as well as variants of it) has received a considerable amount of attention in the mathematical literature. This is caused by the interesting nonlinear eﬀects it shows. In particular, it is well known that in general smooth solutions of initial(-boundary) value problems may only exist for ﬁnite time, with L∞-blow-up of the cell density at the end of the existence interval. This phenomenon strongly depends on the spatial dimension. It does not occur in one-dimensional problems, and it occurs conditionally in higher dimensional situations. For the two-dimensional whole space problem and initial data I satisfying

I ∈ L1+(IR2) ∩ L∞(IR2) ,

|x|2 I(x)dx < ∞ ,

IR2

the situation has recently been clariﬁed completely. The qualitative behaviour depends on the total mass

M=

I dx .

IR2

For M < 8π, a global bounded solution of the initial value problem with (t = 0) = I exists, which is dispersed for t → ∞. For M > 8π, blow-up

2

in ﬁnite time occurs. First results in this direction have been obtained in [12] and recently completed in [8] and [2]. Actually, even the critical case M = 8π is understood now [3]: A global solution exists in this case, which possibly becomes unbounded as t → ∞.

Blow-up scenarios have been investigated in [9], showing that at the blowup time, mass concentrates in a point. Biologically, this represents aggregation of cells, and the description of the dynamics of these aggregates and of their interaction with the non-aggregated cells is of natural interest.

This led to the study of regularized models which, in some cases, can be seen as more precise descriptions of the actual biological processes. Examples are the inclusion of volume ﬁlling eﬀects by a density dependent chemotactic sensitivity [10], [17], [18], [7], the inclusion of a ﬁnite sampling radius, which results in a regularization of the chemical concentration [11], and kinetic transport models, whose macroscopic limit is the Keller-Segel model [5]. All these models share the properties that they have global solutions and that they contain the Keller-Segel model as a formal limit.

The asymptotic behaviour of a class of regularized models with density dependent chemotactic sensitivities after blow-up has been investigated in [17], [18]. By formal asymptotics the dynamics of solutions of the regularized problem is studied under the assumption that in the limit the cell density is the sum of a smooth part and of a ﬁnite number of point aggregates, mathematically represented as Delta distributions. The result is a partial diﬀerential equation for the smooth part coupled to a system of ordinary diﬀerential equations for the dynamics of the masses and the positions of the aggregates. Well posedness of this formal limiting problem is proved locally in time. Actually, the system can only be expected to be valid on bounded time intervals between blow-up events and/or collisions of aggregates.

In this work we study the regularized model

∂t ε + ∇ · ( ε∇Sε[ ε] − ∇ ε) = 0 ,

(3)

where the Poisson equation ∆S = − is replaced by the regularized Newtonian potential solution

1 Sε[ ](x) = − 2π IR2 log(|x − y| + ε) (y)dy . (4) This regularization is similar to the ﬁnite-sampling-radius model [11] mentioned above. We consider the initial value problem

ε(t = 0) = I ∈ L1+(IR2) ∩ L∞(IR2) .

(5)

3

Our main result is a rigorous characterization of the limits of solutions as ε → 0 globally in time and for arbitrary initial mass. We use the framework developed by Poupaud in [16], which he applied to the two-dimensional incompressible Euler equations as well as to the system (3)–(4) without diffusion of cells. The limit of the cell density satisﬁes a generalized weak formulation of the Keller-Segel model (1)–(2) allowing for measure valued cell densities. Obviously, the main mathematical problem is an appropriate deﬁnition of the only nonlinear term, the convective ﬂux ∇S. Here the fact that the spatial dimension is two, is of essential importance.

The rest of this work is structured as follows: in the following section, a priori estimates for solutions of (3)–(5) are derived and the theory from [16] is outlined. In Section 3, the limit ε → 0 is carried out and the limiting problem is formulated. A strong formulation is derived under the assumption that the limiting cell density is the sum of a smooth part and a number of point aggregates. It turns out that the strong formulation is similar to the limiting model formally derived by Vela´zquez [17], [18], except one detail, showing that the limit actually depends on the type of regularization. The subject of Section 4 is the study of local density proﬁles of the regularized problem approximating point aggregates. After an appropriate rescaling, an equation for these proﬁles is rigorously derived. Finally, a free energy functional for the regularized problem is introduced and it is shown that solutions of the proﬁle equation can be obtained as minimizers.

2 A priori estimates and diagonal defect measures

Theorem 1 For every positive ε, the problem (3)–(5) has a global solution satisfying

ε(·, t) L1(IR2) = I L1(IR2) := M ,

(6)

and ε(t, ·) L∞(IR2) ≤ c 1 + ε12 , (7)

with an ε-independent constant c.

4

Proof: The existence of a local solutions and the mass conservation property (6) are well known results. As a consequence,

|∇Sε[ ε](x, t)| ≤ 1

ε(y, t)dy M ≤

2π IR2 |x − y| + ε 2πε

holds and the second a priori estimate (7) follows from Lemma 1 in [11]. Global existence is an immediate consequence.

Basic and important for what follows is the following representation of the distributional interpretation of the convective ﬂux: For ϕ ∈ C0∞(IR2),

1

(ϕ(x) − ϕ(y))(x − y)

IR2 ϕ ∇Sε[ ]dx = − 4π IR2 IR2 |x − y|(|x − y| + ε) (x) (y)dx dy (8)

holds, implying the uniform-in-ε estimate

ϕ ε∇Sε[ ε]dx ≤ M 2 |ϕ|1,∞ , (9)

IR2

4π

where |ϕ|k,∞ = maxk1+k2=k supIR2 |∂xk11∂xk22ϕ|. The form of the integral kernel in (8) suggests to introduce the family

mε(t, x) := Kε(x − y) ε(t, x) ε(t, y)dy , IR2

with Kε(x) = x⊗2 , (10) |x|(|x| + ε)

of matrix valued functions. Following [16] (to which we also refer to for some of the details omitted in the rest of this section), we consider ε(t, ·) and mε(t, ·) as time dependent measures ε(t) and mε(t).

Lemma 1 The families { ε(t)}ε>0 and {mε(t)}ε>0 are tightly bounded locally uniformly in t, and { ε(t)}ε>0 is tightly equicontinuous in t.

Proof: The proof is actually contained in the proof of Theorem 3.2 of [16] and repeated here only for completeness. We compute

d ϕ ε dx = ε(∆ϕ + ∇ϕ · ∇S [ ε])dx ,

dt IR2

IR2

ε

which, by (9), can be estimated by

d ϕ ε dx ≤ c |ϕ| .

dt IR2

2,∞

5

with c independent of ε and t. This implies equicontinuity in W 2,∞(IR2) :

ϕ ε(t, x) dx − ϕ ε(s, x) dx ≤ C(ϕ)|t − s| .

IR2

IR2

Now let ϕ ∈ Cb(IR2). Then for every δ > 0 there exists ϕδ ∈ W 2,∞(IR2)

such that ϕ − ϕδ L∞(IR2) ≤ δ. By the above inequality and by the uniform boundedness of ε, we have

ϕ ε(t, x) dx − ϕ ε(s, x) dx ≤ 2δM + C(ϕδ)|t − s| ,

IR2

IR2

implying, together with ε(t)(IR2) = M , the tight equicontinuity of ε(t). With a test function ϕR(x) = 1 − β(|x|2/R2) with β nonincreasing,

β(r) = 1 for 0 ≤ r ≤ 1/2, and β(r) = 0 for r ≥ 1, the above inequality

gives

ε(t)(IR2 \ BR) ≤ I (IR2 \ BR/2) + Rc t2 ,

which immediately implies the locally uniform tight boundedness. The result for mε is a consequence of the estimate |mε| ≤ M ε.

By the Prokhorov criterium, tight boundedness and equicontinuity of ε(t) provides compactness for ε(t) and mε(t) in the sense that there exist

nonnegative bounded time dependent measures (t) and m(t) such that, restricting to subsequences, ε(t) converges to (t) tightly and locally uniformly

in t, as ε → 0, and

t2

t2

ϕ(t, x)mε(t, x)dx dt →

ϕ(t, x)m(t, x)dx dt ,

t1 IR2

t1 IR2

for all ϕ ∈ Cb([t1, t2] × IR2). Actually, also

ϕ(x, y) ε(t, x) ε(t, y) dx dy →

ϕ(x, y) (t, x) (t, y) dx dy ,(11)

IR2 IR2

IR2 IR2

uniformly in t for all ϕ ∈ Cb(IR2 × IR2) (see [16]). However, by the discontinuity of the limiting kernel in (10), we cannot pass to the limit there, but we have to introduce the defect measure

ν(t, x) = m(t, x) − K(x − y) (t, x) (t, y)dy , IR2 6

with

K(x) = lim Kε(x) = ε→0

x|x⊗|22 for x = 0 , 0 for x = 0 .

The atomic support of the measure (t) will be denoted by Sat( (t)) := {a ∈ IR2 : (t)({a}) > 0} .

It is an at most countable set.

Lemma 2 ([16]) The defect measure ν is symmetric and nonnegative, and satisﬁes

tr(ν(t, x)) ≤

( (t)({a}))2δ(x − a) .

a∈Sat( (t))

Outline of a proof: Symmetry is obvious. For a test function ϕ ∈ Cb(IR2 × IR2), (ϕ(x, y)−ϕ(x, x))Kε(x−y) converges uniformly to the continuous func-

tion (ϕ(x, y) − ϕ(x, x))K(x − y). Therefore, by (11),

(ϕ(x, y) − ϕ(x, x))Kε(x − y) ε(t, x) ε(t, y) dx dy

IR2 IR2

→

(ϕ(x, y) − ϕ(x, x))K(x − y) (t, x) (t, y) dx dy .

IR2 IR2

By the deﬁnitions of m and ν this implies

ϕ(x, y)Kε(x − y) ε(t, x) ε(t, y) dx dy

IR2 IR2

→

ϕ(x, y)K(x − y) (t, x) (t, y) dx dy + ϕ(x, x)ν(t, x) dx .

IR2 IR2

IR2

Since Kε is nonnegative, so is the right hand side for a nonnegative test function. Choosing ϕ(x, y) = ψ(x)η(R(x − y)) ≥ 0 with an arbitrary nonnegative ψ and a nonnegative bounded η with compact support and η(0) = 1, the ﬁrst term on the right hand side tends to zero for R → ∞, proving nonnegativity of ν. The convergence is indeed a consequence of Lebesgue’s theorem of dominated convergence using the fact that K(x − y)η(R(x − y)) is bounded and converges to 0 pointwise.

7

For proving the second statement, note that tr(Kε) ≤ 1. Combined with the above this gives (again with a nonnegative test function)

ϕ(x, y) (t, x) (t, y) dx dy

IR2 IR2

≥

ϕ(x, y)tr(K(x − y)) (t, x) (t, y) dx dy + ϕ(x, x)tr(ν(t, x)) dx

IR2 IR2

IR2

=

ϕ(x, y)(1 − χD(x, y)) (t, x) (t, y) dx dy + ϕ(x, x)tr(ν(t, x)) dx ,

IR2 IR2

IR2

where χD denotes the characteristic function of the diagonal in IR2 × IR2. Since

χD(x, y) (t, x) (t, y) =

(t)({a})2 δ(x − a)δ(y − a) , (12)

a∈Sat( (t))

the desired result follows.

The limit of ε is thus characterized by the pair ( , ν) whose properties are collected in the following deﬁnition.

Deﬁnition 1 For an interval I ⊂ IR, the set of time dependent measures with diagonal defects is deﬁned by

DM+(I; IR2) = ( , ν) : (t) ∈ M+1 (IR2) ∀t ∈ I, ν ∈ M(I × IR2)2×2,

is tightly continuous with respect to t,

ν is a nonnegative, symmetric, matrix valued measure,

tr(ν(t, x)) ≤

( (t)({a}))2δ(x − a) ,

a∈Sat( (t))

where M denotes spaces of Radon measures and M+1 the subspace of nonnegative bounded measures.

3 Global measure valued solutions and a strong formulation

With the tools presented in the previous section, it is now not hard to pass to the limit in the regularized Keller-Segel model. We again follow along

8

the lines of [16]. Starting from the distributional formulation (8) for the regularized ﬂux, we observe that

(ϕ(x) − ϕ(y))(x − y) = Kε(x − y)∇ϕ(x) + Lε(ϕ)(x, y) , |x − y|(|x − y| + ε)

with

Lε(ϕ)(x, y) = (ϕ(x) − ϕ(y) − (x − y) · ∇ϕ(x))(x − y) , |x − y|(|x − y| + ε)

which converges uniformly to the continuous L0(ϕ)(x, y) for any test function ϕ ∈ Cb1(IR2). For any time interval (0, T ), we may therefore pass to the limit in

T ϕ(t, x) ε(t, x)∇Sε[ ε](t, x)dx dt = − 1 T mε(t, x)∇ϕ(t, x) dx dt

0 IR2

4π 0 IR2

1T −

ε(t, x) ε(t, y)Lε(ϕ)(t, x, y) dx dy dt .

4π 0 IR2 IR2

As a result, restricting to subsequences, ε∇Sε[ ε] converges to j[ , ν] in the sense of distributions, with the limiting ﬂux deﬁned by

T

ϕ(t, x)j[ , ν](t, x)dx dt

0 IR2

1T

=−

(ϕ(t, x) − ϕ(t, y))K(x − y) (t, x) (t, y)dx dy dt

4π 0 IR4

1T

−

ν(t, x)∇ϕ(t, x)dx dt .

(13)

4π 0 IR2

for ϕ ∈ Cb1((0, T ) × IR2) with

K(x) = |xx|2 for x = 0 , (14) 0 for x = 0 .

This actually completes the proof of our main result.

Theorem 2 For every T > 0, as ε → 0, a subsequence of solutions ε of (3)– (5) converges tightly and uniformly in time to a time dependent measure (t). There exists ν(t) such that ( , ν) ∈ DM+((0, T ); IR2) is a generalized solution of

∂t + ∇ · (j[ , ν] − ∇ ) = 0 ,

(15)

9

in the sense that the convective ﬂux j[ , ν] is given by (13)–(14) and that (15) holds in the sense of distributions. The initial condition (t = 0) = I is satisﬁed in the sense of tight continuity.

Note that, for a not charging points, ν = 0 and j[ , 0] = ∇S0[ ], implying that (15) is a generalization of the classical Keller-Segel model.

In order to derive a strong formulation of (15), we decompose the cell density as

= + ˆ, with ˆ(t, x) = Mn(t)δ(x − xn(t)) , δn(t, x) = δ(x − xn(t))

n∈N

where N ⊂ IIN, assuming is smooth and that t varies in a time interval, where the atomic support of consists of smooth paths xn(t) carrying smooth weights Mn(t). Then, by ( , ν) ∈ DM+((0, T ); IR2),

ν(t, x) = νn(t)δn(t, x) ,

n∈N

with nonnegative symmetric νn satisfying tr(νn) ≤ Mn2. The convective ﬂux can be written as

1

j[ , ν] = ∇S0[ + ˆ] + Mnδn∇S0 + Mmδm +

νn∇δn .

n

m=n

4π n

The equation (15) is then equivalent to

∂t + ∇ · ( ∇S0[ ] − ∇ ) + ∇ · ∇S0[ˆ]

+ δn(M˙ n − Mn)

n

−

Mn∇δn x˙ n − ∇S0 + Mmδm

n

m=n

+

1 ν

: ∇2δ

− M ∆δ

= 0.

n 4π n

n

nn

The terms in the ﬁrst row are in L∞ t L1x. The other three rows contain zeroth, ﬁrst and second order derivatives of δn. Therefore, all the coeﬃcients (in each row and for every n) have to vanish individually. Starting from the last

row this gives

νn = 4πMn id .

(16)

10

Jean Dolbeault1 and Christian Schmeiser2

Abstract. In the two-dimensional Keller-Segel model for chemotaxis of biological cells, blow-up of solutions in ﬁnite time occurs if the total mass is above a critical value. Blow-up is a concentration event, where point aggregates are created. In this work global existence of generalized solutions is proven, allowing for measure valued densities. This extends the solution concept after blow-up. The existence result is an application of a theory developed by Poupaud, where the cell distribution is characterized by an additional defect measure, which vanishes for smooth cell densities. The global solutions are constructed as limits of solutions of a regularized problem.

A strong formulation is derived under the assumption that the generalized solution consists of a smooth part and a number of smoothly varying point aggregates. Comparison with earlier formal asymptotic results shows that the choice of a solution concept after blow-up is not unique and depends on the type of regularization.

This work is also concerned with local density proﬁles close to point aggregates. An equation for these proﬁles is derived by passing to the limit in a rescaled version of the regularized model. Solvability of the proﬁle equation can also be obtained by minimizing a free energy functional.

Key words: Keller-Segel, chemotaxis, aggregation, blow-up, measure valued solutions, defect measure

AMS subject classiﬁcation: 35D05, 35D10, 35K55

Acknowledgement: This work has been supported by the Amadeus project no. 13785 UA and by the European network DEASE. The ﬁrst author has been supported by the ANR project IFO. The second author acknowledges the hospitality at the Universit´e Paris Dauphine as well as support by the Austrian Science Fund under grant No. W8.

1CEREMADE UMR CNRS no. 7534, Universit´e Paris Dauphine, Paris, France. 2Institut fu¨r Mathematik, Universit¨at Wien, Austria.

1

1 Introduction

The simplest description of a cell population, which produces a chemical signal and responds to it chemotactically, is the Keller-Segel model

∂t + ∇ · ( ∇S − ∇ ) = 0 ,

(1)

−∆S = .

(2)

The model is written in nondimensionalized form with the cell density (t, x) and the chemical concentration S(t, x) both depending on time t and position x. The ﬁrst equation is a convection-diﬀusion equation, where the drift term ∇S describes the chemotactic reaction of the cells, and the second equation is a quasistationary approximation of a reaction-diﬀusion equation for the chemical concentration. The reaction term models the production of the chemical by the cells. The model is based on the assumption that characteristic times for the dynamics of the chemical are much shorter than those for the cell dynamics. Also the chemotactic sensitivity, the cell diffusivity, the chemical diﬀusivity, and the reaction rate (coeﬃcients of ∇S, ∇ , ∆S, and, respectively, ) have been assumed constant and have been removed by an appropriate scaling.

Since its ﬁrst formulation [15] and ﬁrst mathematical investigation [13], this model (as well as variants of it) has received a considerable amount of attention in the mathematical literature. This is caused by the interesting nonlinear eﬀects it shows. In particular, it is well known that in general smooth solutions of initial(-boundary) value problems may only exist for ﬁnite time, with L∞-blow-up of the cell density at the end of the existence interval. This phenomenon strongly depends on the spatial dimension. It does not occur in one-dimensional problems, and it occurs conditionally in higher dimensional situations. For the two-dimensional whole space problem and initial data I satisfying

I ∈ L1+(IR2) ∩ L∞(IR2) ,

|x|2 I(x)dx < ∞ ,

IR2

the situation has recently been clariﬁed completely. The qualitative behaviour depends on the total mass

M=

I dx .

IR2

For M < 8π, a global bounded solution of the initial value problem with (t = 0) = I exists, which is dispersed for t → ∞. For M > 8π, blow-up

2

in ﬁnite time occurs. First results in this direction have been obtained in [12] and recently completed in [8] and [2]. Actually, even the critical case M = 8π is understood now [3]: A global solution exists in this case, which possibly becomes unbounded as t → ∞.

Blow-up scenarios have been investigated in [9], showing that at the blowup time, mass concentrates in a point. Biologically, this represents aggregation of cells, and the description of the dynamics of these aggregates and of their interaction with the non-aggregated cells is of natural interest.

This led to the study of regularized models which, in some cases, can be seen as more precise descriptions of the actual biological processes. Examples are the inclusion of volume ﬁlling eﬀects by a density dependent chemotactic sensitivity [10], [17], [18], [7], the inclusion of a ﬁnite sampling radius, which results in a regularization of the chemical concentration [11], and kinetic transport models, whose macroscopic limit is the Keller-Segel model [5]. All these models share the properties that they have global solutions and that they contain the Keller-Segel model as a formal limit.

The asymptotic behaviour of a class of regularized models with density dependent chemotactic sensitivities after blow-up has been investigated in [17], [18]. By formal asymptotics the dynamics of solutions of the regularized problem is studied under the assumption that in the limit the cell density is the sum of a smooth part and of a ﬁnite number of point aggregates, mathematically represented as Delta distributions. The result is a partial diﬀerential equation for the smooth part coupled to a system of ordinary diﬀerential equations for the dynamics of the masses and the positions of the aggregates. Well posedness of this formal limiting problem is proved locally in time. Actually, the system can only be expected to be valid on bounded time intervals between blow-up events and/or collisions of aggregates.

In this work we study the regularized model

∂t ε + ∇ · ( ε∇Sε[ ε] − ∇ ε) = 0 ,

(3)

where the Poisson equation ∆S = − is replaced by the regularized Newtonian potential solution

1 Sε[ ](x) = − 2π IR2 log(|x − y| + ε) (y)dy . (4) This regularization is similar to the ﬁnite-sampling-radius model [11] mentioned above. We consider the initial value problem

ε(t = 0) = I ∈ L1+(IR2) ∩ L∞(IR2) .

(5)

3

Our main result is a rigorous characterization of the limits of solutions as ε → 0 globally in time and for arbitrary initial mass. We use the framework developed by Poupaud in [16], which he applied to the two-dimensional incompressible Euler equations as well as to the system (3)–(4) without diffusion of cells. The limit of the cell density satisﬁes a generalized weak formulation of the Keller-Segel model (1)–(2) allowing for measure valued cell densities. Obviously, the main mathematical problem is an appropriate deﬁnition of the only nonlinear term, the convective ﬂux ∇S. Here the fact that the spatial dimension is two, is of essential importance.

The rest of this work is structured as follows: in the following section, a priori estimates for solutions of (3)–(5) are derived and the theory from [16] is outlined. In Section 3, the limit ε → 0 is carried out and the limiting problem is formulated. A strong formulation is derived under the assumption that the limiting cell density is the sum of a smooth part and a number of point aggregates. It turns out that the strong formulation is similar to the limiting model formally derived by Vela´zquez [17], [18], except one detail, showing that the limit actually depends on the type of regularization. The subject of Section 4 is the study of local density proﬁles of the regularized problem approximating point aggregates. After an appropriate rescaling, an equation for these proﬁles is rigorously derived. Finally, a free energy functional for the regularized problem is introduced and it is shown that solutions of the proﬁle equation can be obtained as minimizers.

2 A priori estimates and diagonal defect measures

Theorem 1 For every positive ε, the problem (3)–(5) has a global solution satisfying

ε(·, t) L1(IR2) = I L1(IR2) := M ,

(6)

and ε(t, ·) L∞(IR2) ≤ c 1 + ε12 , (7)

with an ε-independent constant c.

4

Proof: The existence of a local solutions and the mass conservation property (6) are well known results. As a consequence,

|∇Sε[ ε](x, t)| ≤ 1

ε(y, t)dy M ≤

2π IR2 |x − y| + ε 2πε

holds and the second a priori estimate (7) follows from Lemma 1 in [11]. Global existence is an immediate consequence.

Basic and important for what follows is the following representation of the distributional interpretation of the convective ﬂux: For ϕ ∈ C0∞(IR2),

1

(ϕ(x) − ϕ(y))(x − y)

IR2 ϕ ∇Sε[ ]dx = − 4π IR2 IR2 |x − y|(|x − y| + ε) (x) (y)dx dy (8)

holds, implying the uniform-in-ε estimate

ϕ ε∇Sε[ ε]dx ≤ M 2 |ϕ|1,∞ , (9)

IR2

4π

where |ϕ|k,∞ = maxk1+k2=k supIR2 |∂xk11∂xk22ϕ|. The form of the integral kernel in (8) suggests to introduce the family

mε(t, x) := Kε(x − y) ε(t, x) ε(t, y)dy , IR2

with Kε(x) = x⊗2 , (10) |x|(|x| + ε)

of matrix valued functions. Following [16] (to which we also refer to for some of the details omitted in the rest of this section), we consider ε(t, ·) and mε(t, ·) as time dependent measures ε(t) and mε(t).

Lemma 1 The families { ε(t)}ε>0 and {mε(t)}ε>0 are tightly bounded locally uniformly in t, and { ε(t)}ε>0 is tightly equicontinuous in t.

Proof: The proof is actually contained in the proof of Theorem 3.2 of [16] and repeated here only for completeness. We compute

d ϕ ε dx = ε(∆ϕ + ∇ϕ · ∇S [ ε])dx ,

dt IR2

IR2

ε

which, by (9), can be estimated by

d ϕ ε dx ≤ c |ϕ| .

dt IR2

2,∞

5

with c independent of ε and t. This implies equicontinuity in W 2,∞(IR2) :

ϕ ε(t, x) dx − ϕ ε(s, x) dx ≤ C(ϕ)|t − s| .

IR2

IR2

Now let ϕ ∈ Cb(IR2). Then for every δ > 0 there exists ϕδ ∈ W 2,∞(IR2)

such that ϕ − ϕδ L∞(IR2) ≤ δ. By the above inequality and by the uniform boundedness of ε, we have

ϕ ε(t, x) dx − ϕ ε(s, x) dx ≤ 2δM + C(ϕδ)|t − s| ,

IR2

IR2

implying, together with ε(t)(IR2) = M , the tight equicontinuity of ε(t). With a test function ϕR(x) = 1 − β(|x|2/R2) with β nonincreasing,

β(r) = 1 for 0 ≤ r ≤ 1/2, and β(r) = 0 for r ≥ 1, the above inequality

gives

ε(t)(IR2 \ BR) ≤ I (IR2 \ BR/2) + Rc t2 ,

which immediately implies the locally uniform tight boundedness. The result for mε is a consequence of the estimate |mε| ≤ M ε.

By the Prokhorov criterium, tight boundedness and equicontinuity of ε(t) provides compactness for ε(t) and mε(t) in the sense that there exist

nonnegative bounded time dependent measures (t) and m(t) such that, restricting to subsequences, ε(t) converges to (t) tightly and locally uniformly

in t, as ε → 0, and

t2

t2

ϕ(t, x)mε(t, x)dx dt →

ϕ(t, x)m(t, x)dx dt ,

t1 IR2

t1 IR2

for all ϕ ∈ Cb([t1, t2] × IR2). Actually, also

ϕ(x, y) ε(t, x) ε(t, y) dx dy →

ϕ(x, y) (t, x) (t, y) dx dy ,(11)

IR2 IR2

IR2 IR2

uniformly in t for all ϕ ∈ Cb(IR2 × IR2) (see [16]). However, by the discontinuity of the limiting kernel in (10), we cannot pass to the limit there, but we have to introduce the defect measure

ν(t, x) = m(t, x) − K(x − y) (t, x) (t, y)dy , IR2 6

with

K(x) = lim Kε(x) = ε→0

x|x⊗|22 for x = 0 , 0 for x = 0 .

The atomic support of the measure (t) will be denoted by Sat( (t)) := {a ∈ IR2 : (t)({a}) > 0} .

It is an at most countable set.

Lemma 2 ([16]) The defect measure ν is symmetric and nonnegative, and satisﬁes

tr(ν(t, x)) ≤

( (t)({a}))2δ(x − a) .

a∈Sat( (t))

Outline of a proof: Symmetry is obvious. For a test function ϕ ∈ Cb(IR2 × IR2), (ϕ(x, y)−ϕ(x, x))Kε(x−y) converges uniformly to the continuous func-

tion (ϕ(x, y) − ϕ(x, x))K(x − y). Therefore, by (11),

(ϕ(x, y) − ϕ(x, x))Kε(x − y) ε(t, x) ε(t, y) dx dy

IR2 IR2

→

(ϕ(x, y) − ϕ(x, x))K(x − y) (t, x) (t, y) dx dy .

IR2 IR2

By the deﬁnitions of m and ν this implies

ϕ(x, y)Kε(x − y) ε(t, x) ε(t, y) dx dy

IR2 IR2

→

ϕ(x, y)K(x − y) (t, x) (t, y) dx dy + ϕ(x, x)ν(t, x) dx .

IR2 IR2

IR2

Since Kε is nonnegative, so is the right hand side for a nonnegative test function. Choosing ϕ(x, y) = ψ(x)η(R(x − y)) ≥ 0 with an arbitrary nonnegative ψ and a nonnegative bounded η with compact support and η(0) = 1, the ﬁrst term on the right hand side tends to zero for R → ∞, proving nonnegativity of ν. The convergence is indeed a consequence of Lebesgue’s theorem of dominated convergence using the fact that K(x − y)η(R(x − y)) is bounded and converges to 0 pointwise.

7

For proving the second statement, note that tr(Kε) ≤ 1. Combined with the above this gives (again with a nonnegative test function)

ϕ(x, y) (t, x) (t, y) dx dy

IR2 IR2

≥

ϕ(x, y)tr(K(x − y)) (t, x) (t, y) dx dy + ϕ(x, x)tr(ν(t, x)) dx

IR2 IR2

IR2

=

ϕ(x, y)(1 − χD(x, y)) (t, x) (t, y) dx dy + ϕ(x, x)tr(ν(t, x)) dx ,

IR2 IR2

IR2

where χD denotes the characteristic function of the diagonal in IR2 × IR2. Since

χD(x, y) (t, x) (t, y) =

(t)({a})2 δ(x − a)δ(y − a) , (12)

a∈Sat( (t))

the desired result follows.

The limit of ε is thus characterized by the pair ( , ν) whose properties are collected in the following deﬁnition.

Deﬁnition 1 For an interval I ⊂ IR, the set of time dependent measures with diagonal defects is deﬁned by

DM+(I; IR2) = ( , ν) : (t) ∈ M+1 (IR2) ∀t ∈ I, ν ∈ M(I × IR2)2×2,

is tightly continuous with respect to t,

ν is a nonnegative, symmetric, matrix valued measure,

tr(ν(t, x)) ≤

( (t)({a}))2δ(x − a) ,

a∈Sat( (t))

where M denotes spaces of Radon measures and M+1 the subspace of nonnegative bounded measures.

3 Global measure valued solutions and a strong formulation

With the tools presented in the previous section, it is now not hard to pass to the limit in the regularized Keller-Segel model. We again follow along

8

the lines of [16]. Starting from the distributional formulation (8) for the regularized ﬂux, we observe that

(ϕ(x) − ϕ(y))(x − y) = Kε(x − y)∇ϕ(x) + Lε(ϕ)(x, y) , |x − y|(|x − y| + ε)

with

Lε(ϕ)(x, y) = (ϕ(x) − ϕ(y) − (x − y) · ∇ϕ(x))(x − y) , |x − y|(|x − y| + ε)

which converges uniformly to the continuous L0(ϕ)(x, y) for any test function ϕ ∈ Cb1(IR2). For any time interval (0, T ), we may therefore pass to the limit in

T ϕ(t, x) ε(t, x)∇Sε[ ε](t, x)dx dt = − 1 T mε(t, x)∇ϕ(t, x) dx dt

0 IR2

4π 0 IR2

1T −

ε(t, x) ε(t, y)Lε(ϕ)(t, x, y) dx dy dt .

4π 0 IR2 IR2

As a result, restricting to subsequences, ε∇Sε[ ε] converges to j[ , ν] in the sense of distributions, with the limiting ﬂux deﬁned by

T

ϕ(t, x)j[ , ν](t, x)dx dt

0 IR2

1T

=−

(ϕ(t, x) − ϕ(t, y))K(x − y) (t, x) (t, y)dx dy dt

4π 0 IR4

1T

−

ν(t, x)∇ϕ(t, x)dx dt .

(13)

4π 0 IR2

for ϕ ∈ Cb1((0, T ) × IR2) with

K(x) = |xx|2 for x = 0 , (14) 0 for x = 0 .

This actually completes the proof of our main result.

Theorem 2 For every T > 0, as ε → 0, a subsequence of solutions ε of (3)– (5) converges tightly and uniformly in time to a time dependent measure (t). There exists ν(t) such that ( , ν) ∈ DM+((0, T ); IR2) is a generalized solution of

∂t + ∇ · (j[ , ν] − ∇ ) = 0 ,

(15)

9

in the sense that the convective ﬂux j[ , ν] is given by (13)–(14) and that (15) holds in the sense of distributions. The initial condition (t = 0) = I is satisﬁed in the sense of tight continuity.

Note that, for a not charging points, ν = 0 and j[ , 0] = ∇S0[ ], implying that (15) is a generalization of the classical Keller-Segel model.

In order to derive a strong formulation of (15), we decompose the cell density as

= + ˆ, with ˆ(t, x) = Mn(t)δ(x − xn(t)) , δn(t, x) = δ(x − xn(t))

n∈N

where N ⊂ IIN, assuming is smooth and that t varies in a time interval, where the atomic support of consists of smooth paths xn(t) carrying smooth weights Mn(t). Then, by ( , ν) ∈ DM+((0, T ); IR2),

ν(t, x) = νn(t)δn(t, x) ,

n∈N

with nonnegative symmetric νn satisfying tr(νn) ≤ Mn2. The convective ﬂux can be written as

1

j[ , ν] = ∇S0[ + ˆ] + Mnδn∇S0 + Mmδm +

νn∇δn .

n

m=n

4π n

The equation (15) is then equivalent to

∂t + ∇ · ( ∇S0[ ] − ∇ ) + ∇ · ∇S0[ˆ]

+ δn(M˙ n − Mn)

n

−

Mn∇δn x˙ n − ∇S0 + Mmδm

n

m=n

+

1 ν

: ∇2δ

− M ∆δ

= 0.

n 4π n

n

nn

The terms in the ﬁrst row are in L∞ t L1x. The other three rows contain zeroth, ﬁrst and second order derivatives of δn. Therefore, all the coeﬃcients (in each row and for every n) have to vanish individually. Starting from the last

row this gives

νn = 4πMn id .

(16)

10