6.6 The large time behavior
6.6.2 Proof of Theorem 11
As in [82], Eq. (6.1.3) can be rewritten in terms of f = (n−n1)/n1 and g = (c−c1)/c1 in the form of (6.4.1), that is
@f
@t =Lf − 1
n1r[n1fr(g c1)] .
The computation for the linearized problem established in Section 6.6.1 can be adapted to the nonlinear case and gives
d
dtQ1[f(t,·)] =−2Q2[f(t,·)] + Z
R3r(f −g c1)f n1· r(g c1)dx ,
with −2Q2[f(t,·)] −2Q1[f(t,·)] according to Theorem 23. To get an estimate on the asymptotic behaviour, we have to establish an estimate of the last term of the right hand side, which is cubic in terms of f. For this purpose, we apply H¨older’s inequality and Lemma 16 to get
✓Z
R3r(f−g c1)f n1· r(g c1)dx
◆2
Q2[f]
Z
R3
f2n1dxkr(g c1)k2L1(R2) . Using Lemma 16 and Corollary 26, we find that the right hand side can be bounded by
C(") Λ
Λ−1Q1[f]Q2[f]
-kf n1kL2−"(R2)+kf n1kL2+"(R2)
. .
From [84, Theorem 1.2] and Corollary 17 we know that limt!1kf(t,·)n1kL1(R2)= 0 and limt!1kf(t,·)n1kL1(R2)= 0 and therefore, for any given"2(0,1), there exists a continuous functiont7!δ(t, ") with limt!1δ(t, ") =
0 such that
kf(t,·)n1kL2−"(R2)+kf(t,·)n1kL2+"(R2)δ(t, "). As a consequence, we know that
d
dtQ1[f(t,·)] −2Q2[f(t,·)] +δ(t, ")p
Q1[f(t,·)]Q2[f(t,·)](δ(t, ")− 2)Q2[f(t,·)],
where the last inequality is a consequence of Theorem 23. This proves thatQ1[f(t,·)] is uniformly bounded with respect tot ast! 1: there exists a constantQ>0 such that
Q1[f(t,·)] Q 8t≥0.
Now we can give a more detailed estimate. Using again H¨older’s inequality, we find that Z 1.2] and Corollary 17 respectively. For any given "2 (0,1), there exists a continuous function (that we again denote byδ): t7!δ(t, "), with limt!1δ(t, ") = 0, such that
Fort >0 large enough, the right hand side becomes negative and we have found that d for any⌘2(0,2). Actually we know from the above estimates that
2p
Q2[f(t,·)]− δ(t, ") (Q1[f(t,·)])2 (2−")4−3" − δ(t, ") (Q1[f(t,·)])2 (2+")4+"
CHAPTER 6. ASYMPTOTIC ESTIMATES FOR THE KELLER-SEGEL MODEL 111 is positive fort >0, large enough, thus showing that
d
dtQ1[f(t,·)] −Q1[f(t,·)]h
2− δ(t, ")⇣
Q1[f(t,·)])1−"2−" +Q1[f(t,·)])2+"1 ⌘i . As a consequence, we finally get that
lim sup
t!1
e2tQ1[f(t,·)]<1,
which completes the proof of Theorem 11. ⇤
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Chapter 7
A numerical study of the linearized Keller-Segel operator in self-similar variables
This chapter is devoted to a numerical study of the Keller-Segel model in self-similar variables. We first parametrize the set of solutions in terms of the mass parameterM 2(0,8⇡) and consider the asymptotic regimes for M small or M close to 8⇡. Next we introduce the linearized operator and study its spectrum using various shooting methods: we determine its kernel, the spectrum among radial functions and use a decomposition into spherical harmonics to study the other eigenvalues. As a result, we numerically observe that the spectral gap of the linearized operator is independent of M and equal to 1, which is compatible with known results in the limiting regime corresponding toM !0+, and with the theoretical results obtained in the previous chapters. We also compute other eigenvalues, which allows to state several claims on various refined asymptotic expansions of the solutions in the large time regime.
This is a joint work with J. Dolbeault and it has been publisehd as a technical report of the CERE-MADE.
7.1 Introduction
In its simplest version, the parabolic-elliptic Keller-Segel model (also known as the Patlak-Keller-Segel model, see [124, 121])
8>
><
>>
:
@u
@t = ∆u− r ·(urv) x2R2, t >0 v=−2⇡1 log| · | ⇤u x2R2, t >0 u(0, x) =n0≥0 x2R2
(7.1.1)
describes the motion of unicellular amoebae, likedictyostelium discoideum,which move freely and diffuse.
Here u denotes their spatial density and it makes sense to consider them in a two-dimensional setting like the one of a Petri dish. Under certain circumstances, they emit a chemo-attractant and eventually start to aggregate by moving in the direction of the largest concentration of the chemo-attractant. This is modeled in the above equations by the drift termr ·(urv). The life cycle ofdictyostelium discoideum has attracted lots of attention in the community of biologists. Trying to understand the competition
117
between the diffusion and the drift is a key issue in the aggregation process, which has also motivated quite a few studies among mathematicians interested in applications of PDEs to biology. See [125] for a recent overview on the topic.
An easy computation (see [125, pages 122–124] and [122]) shows that solutions (with second moment initially finite) blow-up in finite time if the total mass is large enough (larger than 8⇡with our conventions), that is they describe an aggregate, while, for solutions with smaller masses, the diffusion dominates the large time asymptotics.
More precisely, it has been shown in [120, 118, 116] that, for initial data n0 2L1+
-R2,(1 +|x|2)dx. such that n0|logn0| 2 L1(R2) and M := R
R3n0 dx < 8⇡, there exists a solution u, in the sense of distributions, that is global in time and such that M = R
R3u(x, t) dx is preserved along the evolution.
There is no non-trivial stationary solution to (7.1.1) and any solution converges to zero locally as time gets large. In order to study the asymptotic behavior ofu, it is therefore convenient to work in self-similar variables. In the space and time scales given respectively by R(t) := p
1 + 2t and ⌧(t) := logR(t), we define the rescaled functionsnand c by
u(x, t) :=R−2n
-R−1(t)x,⌧(t).
and v(x, t) :=c
-R−1(t)x,⌧(t). .
This time-dependent rescaling is the one of the heat equation. Since the nonlinear term is invariant under this rescaling, it is also present in the rescaled system without time-dependent coefficient. This system can be written as
8>
><
>>
:
@n
@t = ∆n+r ·(n x)− r ·(nrc) x2R2, t >0 c=−2⇡1 log| · | ⇤n x2R2, t >0
n(0, x) =n0 ≥0 x2R2
(7.1.2)
and it has been shown in [116] that nand rc converge ast! 1, respectively in L1(R2) and L2(R2) to a unique stationary solution given by smooth and radially symmetric functions.
We are interested in estimating the rate of convergence towards the stationary solution in self-similar variables. After undoing the change of variables, this gives the rate of convergence towards the asymptotic profile for the solutions of (7.1.1). Existence of a stationary solution to (7.1.2) has been established in [113]
by ODE techniques, and in [123] by PDE methods. The uniqueness has been shown in [114]. In [115], it has been proved that ifM is less than some massM⇤ 2(0,8⇡), then convergence holds at an exponential rate, which is essentially governed by the linearization of System (7.1.2) around the stationary solution.
However, the estimate of the value of M⇤ was found to be significantly smaller than 8⇡. In the radially symmetric setting, V. Calvez and J.A. Carrillo have found in [117] that the rate measured with respect to Wasserstein’s distance does not depend on the mass, in the whole range (0,8⇡). To establish refined estimates a proper the functional setting for the linear operator must be characterized, this is carried out in detail in chapters 5 and 6. Here we will recoverall these results numerically and give more detailed estimates on the asymptotic behavior of the solutions.
Consider the unique stationary solution to (7.1.2), which is characterized as the solution to
−∆c=n=M e−12|x|2+c R
R3e−12|x|2+c dx , x2R2 (7.1.3) for any given mass M 2(0,8⇡). The bifurcation diagram of the solutions in terms of the parameter M will be considered in Section 7.2.
CHAPTER 7. LINEARIZATION OF THE KELLER-SEGEL MODEL AND NUMERICS 119 Next, consider f and g such that n(1 +f(x, t)) and c(x) (1 +g(x, t)) is a solution to (7.1.2). Then (f, g) solves the nonlinear problem
( @f
@t − Lf =−n1 r ·[f n(r(g c))] x2R2, t >0
−∆(c g) =f n x2R2, t >0 whereL is the linear operator defined by
Lf = 1
nr ·[nr(f −c g)]
and we know that (f n,r(g c))(t,·) has to evolve inL1(R2)⇥L2(R2), and asymptotically vanish ast! 1. To investigate the large time behavior, it is convenient to normalize the solution differently. What we actually want to investigate is the case where solutions of (7.1.2) can be written as
n(x) (1 +" f(x, t)) and c(x) (1 +" g(x, t))
in the asymptotic regime corresponding to"!0+. Formally, it is then clear that, at order", the behavior of the solution is given by @f@t = Lf. The kernel of L has been identified in chapter 6. It has also been shown that L has pure discrete spectrum and that 1 and 2 are eigenvalues. In this report, our goal is to identify the lowest eigenvalues and recover that the spectral gap is actually equal to 1, whatever the mass is in the range M⇤ 2 (0,8⇡). We will also establish the numerical value of other eigenvalues of L at the bottom of its spectrum in Section 7.4, and draw some consequences in the last section of this report: improved rates of convergence for centered initial data and faster decay rates for best matching self-similar solutions.