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

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.

and has a Lyapunov functional: the free energy functional 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

8π is a threshold between two regimes
The two-dimensional Keller-Segel model after blow-up -p.6/32

Finite time blow-up
Existence of smooth solutions depends on the mass M = IR 2 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

A priori estimates and diagonal defect measures
Theorem 1 For every ε > 0, the regularized problem has a global solution satisfying Existence of a local solution, mass conservation are known, and by [Hillen, Painter, Schmeiser] The two-dimensional Keller-Segel model after blow-up -p.11/32 The nonlinear term of matrix valued functions. Following [Poupaud], we consider ε (t, ·) and m ε (t, ·) as time dependent measures ε (t) and m ε (t).
with c independent of ε and t. This implies equicontinuity in W 2,∞ (IR 2 ) :

Limits
Use Prokhorov's criterium: tight boundedness and equicontinuity of ε (t) provides compactness in the second case. Because of the discontinuity of the limiting kernel in K, define the defect measure

Atomic support
The atomic support (an at most countable set) of (t) is It is an at most countable set Lemma 2 [Poupaud] ν is symmetric, nonnegative, and satisfies The two-dimensional Keller-Segel model after blow-up -p.15/32 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 , is tightly continuous with respect to t, ν is a nonnegative, symmetric, matrix valued measure,

Regularized measures
converges uniformly to the continuous L 0 (ϕ)(x, y) for any test function The two-dimensional Keller-Segel model after blow-up -p.18/32

Limiting problem
for ϕ ∈ C 1 b ((0, T ) × IR 2 ) 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 ); IR 2 ) is a generalized solution of ∂ t + ∇ · (j[ , ν] − ∇ ) = 0 (t = 0) = I holds in the sense of tight continuity Strong formulation (formal) The last row this gives ν n = 4πM n id As a consequence of tr(ν n ) = 8πM n ≤ M 2 n , point masses have to be at least 8π (there is only a finite number of them) The two-dimensional Keller-Segel model after blow-up -p.21/32

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 a =b, a,b∈S at ( (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 . The asymptotic state is reached in finite time by collision of the aggregates The two-dimensional Keller-Segel model after blow-up -p.24/32 Example 2. Initial mass M (0) = IR 2 (t = 0)dx < M crit = sup q>1 4 q C q of the smooth part, one aggregate. The corresponding solution may exist only on a finite time interval

Local density profiles
For fixed t and a ∈ S at ( (t)), let εξ = x − a and ε 2 ε = R ε This shows that either R vanishes or its mass is not smaller than 8π With an arbitrary a ∈ IR 2 and R(ξ) = ε 2 (a + εξ) we have