1. The model, a brief review of the

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

Jean Dolbeault

[email protected]

CEREMADE

CNRS & Universit´e Paris-Dauphine

http://www.ceremade.dauphine.fr/∼dolbeaul

(A JOINT WORK WITH C. SCHMEISER) La Habana, February 26, 2008

http://www.ceremade.dauphine.fr/∼dolbeaul/Preprints/Fichiers/DolbSchm-6.pdf

Outline

Write a generalized weak formulation of the Keller-Segel model allowing for measure valued cell densities

the main mathematical problem is an appropriate definition of the only nonlinear term, the convective flux (the spatial dimension is two, is of essential importance)

1. The model, a brief review of the literature

2. A priori estimates for solutions, Poupaud’s theory

3. Limiting problem, measure valued solutions, strong formulation (formal) 4. Local density profiles of the regularized problem approximating point aggregates

1. The model, a brief review of the

literature

The Keller-Segel model

Patlak-Keller-Segel model

∂_t% + ∇ · (%∇S − ∇%) = 0

−∆S = % ... a nondimensionalized form

dynamics of the chemical are much faster than those for the cell dynamics: a parabolic-elliptic system

the chemotactic sensitivity, the cell diffusivity, the chemical diffusivity, and the reaction rate are constant

... initial data

%_I ∈ L¹₊(IR²) ∩ L^∞(IR²)

Z

IR²

|x|²%_I(x)dx < ∞

The Keller-Segel model

∂_t% + ∇ · (%∇S − ∇%) = 0

−∆S = % preserves the mass

d dt

Z

IR²

% dx = 0

and has a Lyapunov functional: the free energy functional d

dt Z

IR²

% log% dx + 1 2

Z

IR²

S[%]% dx

= − Z

IR²

%

∇%

% − ∇S[%]

2

dx

S[%](x) = −_2π¹ R

IR² log|x − y]%(y)dy N.B. R

IR² S[%]% dx = − R

IR² |∇S[%]|² dx only if R

IR² % dx = 0

A standard computation with moments

d dt

Z

IR²

|x|² %(x, t) dx = Z

IR²

|x|² ∆%(x, t) dx + χ

2 π Z

IR²×IR²

2x · x − y

|x − y|² %(x, t) %(y, t) dx dy Two integrations by parts for the first term, one integration by parts for the second one and symmetrization w.r.t. x and y variables give

d dt

Z

IR²

|x|² %(x, t) dx = 4M

1 − M 8π

M = 8π is a threshold between two regimes

Finite time blow-up

Existence of smooth solutions depends on the mass M =

Z

IR²

%_I dx

M < 8π, a global bounded solution exists, vanishing occurs as t → ∞, [Jäger, Luckhaus], [Dolbeault, Perthame], [Blanchet, Dolbeault, Perthame]

M > 8π, blow-up in finite time occurs

M = 8π, a global solution exists in this case, which possibly becomes unbounded as t → ∞, [Biler, Karch, Laurençot, Nadzieja], [Blanchet, Carrillo, Masmoudi]

Aggregation: at the blow-up time, mass concentrates in a point, and then new dynamics has to be established, [Herrero, Velázquez], [Velázquez]

Regularized models

volume filling effects taken into account by a density dependent chemotactic sensitivity [Hillen, Painter], [Velázquez], [Dolak, Schmeiser],

finite sampling radius, which results in a regularization of the chemical concentration [Hillen, Painter, Schmeiser]

kinetic transport models, whose macroscopic limit is the Keller-Segel model [Chalub, Markowich, Perthame]

density dependent chemotactic sensitivities, [Velázquez]

All these models have global solutions

The Keller-Segel model appears as a formal limit

In Velázquez’ approach (formal asymptotics), the limit the cell density is the sum of a smooth part and of a finite number of point aggregates:

locally in time existence

Our regularization

Replace the Poisson equation ∆S = −% by S_ε[%](x) = − 1

2π Z

IR²

log(|x − y| + ε)%(y) dy to get the regularized model

∂_t%^ε + ∇ · (%^ε∇S_ε[%^ε] − ∇%^ε) = 0

Similar to the finite-sampling-radius model [Hillen, Painter, Schmeiser]

%^ε(t = 0) = %_I ∈ L¹₊(IR²) ∩ L^∞(IR²) .

[F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equations, Meth. Appl. Anal. 9 (2002), pp. 533–561] (two-dimensional incompressible Euler equations and Keller-Segel model without diffusion of cells)

2. A priori estimates for solu-

tions, Poupaud’s theory

A priori estimates and diagonal defect measures

Theorem 1 For every ε > 0, the regularized problem has a global solution satisfying

k%^ε(·, t)k_L¹_(IR²₎ = k%_Ik_L¹_(IR²₎ := M k%^ε(t,·)k_L^∞_(IR²₎ ≤ c

1 + 1 ε²

with an ε-independent constant c

Existence of a local solution, mass conservation are known, and by [Hillen, Painter, Schmeiser]

|∇S_ε[%^ε](x, t)| ≤ 1 2π

Z

IR²

%^ε(y, t)dy

|x − y| + ε ≤ M 2πε

The nonlinear term

For ϕ ∈ C₀^∞(IR²), Z

IR²

ϕ%∇S_ε[%]dx = − 1 4π

Z

IR²

Z

IR²

(ϕ(x) − ϕ(y))(x − y)

|x − y|(|x − y| + ε) %(x)%(y)dx dy holds, implying the uniform-in-ε estimate

Z

IR²

ϕ%^ε∇S_ε[%^ε]dx

≤ M²

4π |ϕ|_1,∞

Let

m^ε(t, x) :=

Z

IR²

K^ε(x − y)%^ε(t, x)%^ε(t, y)dy with K^ε(x) = x^⊗2

|x|(|x| + ε) of matrix valued functions. Following [Poupaud], we consider %^ε(t,·) and m^ε(t,·) as time dependent measures %^ε(t) and m^ε(t).

Tight equicontinuity

Lemma 1 [Poupaud] The families {%^ε(t)}_ε>0 and {m^ε(t)}_ε>0 are tightly bounded locally uniformly in t, and {%^ε(t)}_ε>0 is tightly equicontinuous in t.

d dt

Z

IR²

ϕ%^ε dx

≤ c |ϕ|_2,∞

with c independent of ε and t. This implies equicontinuity in W ^2,∞(IR²)⁰:

Z

IR²

ϕ%^ε(t, x) dx − Z

IR²

ϕ%^ε(s, x)dx

≤ C(ϕ)|t − s|

Now let ϕ ∈ C_b(IR²): ∀ δ > 0, ∃ϕ_δ ∈ W^2,∞(IR²) s.t. kϕ − ϕ_δk_L^∞_(IR²₎ ≤ δ

Z

IR²

ϕ%^ε(t, x) dx − Z

IR²

ϕ%^ε(s, x)dx

≤ 2δM + C(ϕ_δ)|t − s|

=⇒ tight equicontinuity of %^ε(t)

Limits

Use Prokhorov’s criterium: tight boundedness and equicontinuity of %^ε(t) provides compactness

Z

IR²

Z

IR²

ϕ(x, y)%^ε(t, x)%^ε(t, y) dx dy → Z

IR²

Z

IR²

ϕ(x, y)%(t, x)%(t, y) dx dy

Z t2

t1

Z

IR²

ϕ(t, x)m^ε(t, x) dx dt →

Z t2

t1

Z

IR²

ϕ(t, x)m(t, x)dx dt

for all ϕ ∈ C_b([t₁, t₂] × IR²) in the second case. Because of the discontinuity of the limiting kernel in K, define the defect measure

ν(t, x) = m(t, x) − Z

IR²

K(x − y)%(t, x)%(t, y) dy

with K(x) = lim_ε→0 K^ε(x) = ^x⊗2 for x 6= 0 and K(0) = 0

Atomic support

The atomic support (an at most countable set) of %(t) is S_at(%(t)) := {a ∈ IR² : %(t)({a}) > 0}

It is an at most countable set

Lemma 2 ^[Poupaud] ν is symmetric, nonnegative, and satisfies tr(ν(t, x)) ≤ X

a∈Sat(%(t))

(%(t)({a}))²δ(x − a)

The limit of %^ε is thus characterized by the pair (%, ν) M: spaces of Radon measures

M⁺₁ : subset of nonnegative bounded measures

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

DM⁺(I; IR²) = n

(%, 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)) ≤ X

a∈Sat(%(t))

(%(t)({a}))²δ(x − a)o

3. Limiting problem, measure val-

ued solutions, strong formulation

(formal)

Regularized measures

(ϕ(x) − ϕ(y))(x − y)

|x − y|(|x − y| + ε) = K^ε(x − y)∇ϕ(x) + L^ε(ϕ)(x, y) with

L^ε(ϕ)(x, y) = (ϕ(x) − ϕ(y) − (x − y) · ∇ϕ(x))(x − y)

|x − y|(|x − y| + ε)

converges uniformly to the continuous L⁰(ϕ)(x, y) for any test function ϕ ∈ C_b¹(IR²)

Z T 0

Z

IR²

ϕ(t, x)%^ε(t, x)∇S^ε[%^ε](t, x)dx dt = − 1 4π

Z T 0

Z

IR²

m^ε(t, x)∇ϕ(t, x) dx dt

− 1 4π

Z T 0

Z

IR²

Z

IR²

%^ε(t, x)%^ε(t, y)L^ε(ϕ)(t, x, y) dx dy dt

Limiting problem

Z T 0

Z

IR²

ϕ(t, x)j[%, ν](t, x)dx dt

= − 1 4π

Z T 0

Z

IR⁴

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

− 1 4π

Z T 0

Z

IR²

ν(t, x)∇ϕ(t, x)dx dt for ϕ ∈ C_b¹((0, T) × IR²)

Theorem 2 For every T > 0, %^ε converges tightly and uniformly in time to

%(t) and there exists ν(t) such that (%, ν) ∈ DM⁺((0, T); IR²) is a generalized solution of

∂_t% + ∇ · (j[%, ν] − ∇%) = 0

%(t = 0) = %_I holds in the sense of tight continuity

Strong formulation (formal)

Ansatz. % = % + ˆ%, %(t, x) =ˆ P

n∈N M_n(t)δ_n(t, x), δ_n(t, x) = δ(x − x_n(t)) (%, ν) ∈ DM⁺((0, T); IR²) =⇒ ν(t, x) = P

n∈N ν_n(t)δ_n(t, x), tr(ν_n) ≤ M_n²

j[%, ν] = %∇S₀[% + ˆ%] + X

n

M_nδ_n∇S₀



% + X

m6=n

M_mδ_m



 + 1 4π

X

n

M_nν_n∇δ_n

∂_t% + ∇ · (%∇S₀[%] − ∇%) + ∇% · ∇S₀[ˆ%]

+ X

n

δ_n( ˙M_n − %M_n)

− X

n

M_n∇δ_n

˙

x_n − ∇S₀ h

% + P

m6=n M_mδ_mi

+ X

n

1

4πν_n : ∇²δ_n − M_n∆δ_n

= 0

The last row this gives

ν_n = 4πM_n id

As a consequence of tr(ν_n) = 8πM_n ≤ M_n², point masses have to be at least 8π (there is only a finite number of them)

∂_t% + ∇ · (%∇S₀[%] − ∇%) − 1

2π∇% · X

n

M_n x − x_n

|x − x_n|² = 0 M˙ _n = %(x = x_n)M_n

˙

x_n = ∇S₀[%](x = x_n) − 1 2π

X

m6=n

M_m x_n − x_m

|x_n − x_m|² Note that _dt^d R

IR² % dx + P

n M_n

= 0

Comparison with Velázquez’ results

Formal limit of [Velázquez]: there is a factor (≤ 1) in front of the right hand side of the force term for x_n, which depends on the details of the

regularization

A local-in-time existence result can be found in [Velázquez]

In general, one has to expect blow-up events in the smooth part % and/or collisions of point aggregates in finite time. At such points in time, a restart is required with either an additional point aggregate after a blow-up event or with a smaller number of point aggregates after a collision. A rigorous theory producing global solutions by such a procedure is still missing

Long time behaviour

Assume again

ν(t, x) = 4πid X

a∈Sat(%(t))

%(t)({a})δ(x − a)

and

Z

IR²

|x|²%_Idx < ∞

With Mˆ = P

a∈Sat(%(t)) %(t)({a}) and M = M − Mˆ d

dt Z

IR²

|x|²% dx = 4M − 1 2π

Z

IR⁴

(1 − χ_D)% ⊗ % dy dx − 1 2π

Z

IR²

tr(ν)dx

= M 4 − M

2π − Mˆ 2π

!

− 1 2π

X

a6=b, a,b∈Sat(%(t))

%(t)({a})%(t)({b})

For M > 8π, the r.h.s is the sum of two nonpositive terms: the second order moment is a Lyapunov function.

The dissipation term only vanishes when M = 0, and when the atomic support of %(t) consists of only one point located at

X_I = 1 M

Z

IR²

x%_Idx

Example 1. % = 0, two aggregates:

%(t, x) = M₁δ(x − x₁(t)) + M₂δ(x − x₂(t)). The asymptotic state is reached in finite time

t = π|x₁(0) − x₂(0)|² M₁ + M₂

by collision of the aggregates

Example 2. Initial mass M(0) = R

IR² %(t = 0)dx < M_crit = sup_{q>1 q C}⁴

q of the smooth part, one aggregate. The corresponding solution may exist only on a finite time interval

∂_t% + ∇ · (%∇S₀[%] − ∇%) − ^M_2π¹∇% · _|x−x^x−x1

1|² = 0 M˙ ₁ = %(x = x₁)M₁

˙

x₁ = ∇S₀[%](x = x₁)

d dt

R

IR² %^qdx = (q − 1) R

IR² %^q+1dx − ⁴_q R

IR² |∇%^q/2|²dx

− %(x = x₁)^qM₁ Gagliardo-Nirenberg: R

IR² |u|^2(1+1/q)dx ≤ C_q R

IR² |u|^2/qdxR

IR² |∇u|²dx d

dt Z

IR²

%^qdx ≤ (q − 1)

C_qM − 4 q

Z

IR²

|∇%^q/2|²dx − %(x = x₁)^qM₁ ≤ 0

M_crit = 8π ?

4. Local density profiles of the

regularized problem approximat-

ing point aggregates

Local density profiles

For fixed t and a ∈ S_at(%(t)), let εξ = x − a and ε²%^ε = R^ε ε²∂_tR^ε + ∇_ξ · (R^ε∇_ξS₁[R^ε] − ∇_ξR^ε) = 0

R^ε is uniformly bounded, implying compactness of ∇_ξS₁[R^ε]. The L^∞-weak* limit R of R^ε (take subsequences, formal) satisfies

∇_ξ · (R∇_ξS₁[R] − ∇_ξR) = 0 Observe that

Z

IR²

R(ξ)dξ = 1 8π

Z

IR⁴

|ξ − η|

|ξ − η| + 1R(ξ)R(η)dηdξ ≤ 1 8π

Z

IR²

R(ξ)dξ 2

This shows that either R vanishes or its mass is not smaller than 8π

Free energy

F_ε[%] :=

Z

IR²

%log% − 1

2%S_ε[%]

dx

=

Z

IR²

%log% dx + 1 4π

Z

IR⁴

log(|x − y| + ε)%(x)%(y)dy dx and

d

dtF_ε[%^ε] = − Z

IR²

%^ε|∇(log %^ε − S_ε[%^ε])|²dx With an arbitrary a ∈ IR² and R(ξ) = ε²%(a + εξ) we have

F_ε[%] =

2M − M² 4π

log 1

ε + F₁[R]

Free energy

Lemma 3 Let R ∈ L¹₊(IR²) be radial, R

IR² log(1 + |x|)R(x) dx < ∞, M = R

IR² R dx 1

4π Z

IR²

log(1 + |x − y|)R(y)dy ≥ M

4π log|x| ∀ x ∈ IR² L¹_+,M := {R ∈ L¹₊(IR²) : R

IR² R dξ = M}, J_M := inf_R∈L¹

+,M F₁[R] ≥ −∞

Theorem 3 J_M = −∞ for M < 8π, and J_M > −∞ for M ≥ 8π. If M > 8π, there exists a radial nonincreasing minimizer

Logarithmic HLS inequality, [Carlen, Loss]

Riesz symmetrization inequality Variational method

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