A kinetic Fisher-KPP equation : traveling waves and front...
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A kinetic Fisher-KPP equation : traveling waves and front acceleration CNRS UMPA Emeric Bouin* Joint work with Vincent Calvez* and Grégoire Nadin** ENS L YON INRIA * École Normale Supérieure de Lyon, UMR CNRS 5669 ’UMPA’ and INRIA project ’NUMED’, 46, allée d’Italie, F-69364 Lyon Cedex 07, France. * Université Pierre et Marie Curie - Paris 6, UMR CNRS 7598 ’LJLL’, BC187, 4 place de Jussieu, F-75252 Paris Cedex 05, France. A kinetic model for the Fisher-KPP equation We consider the following kinetic model ε 2 ∂ t f + εv ·∇ x f =(M (v )ρ − f ) Scattering +ε 2 rρ (M (v ) − f ) Monostable growth , (t, x, v ) ∈ R + × R n × V, (1) where f (t, x, v ) denotes the density of particles moving with speed v ∈ V at time t and position x. The function ρ(t, x) denotes the macroscopic density of particules: ρ(t, x)= V f (t, x, v ) dv , (t, x) ∈ R + × R n . Here V denotes a symmetric subset of R n . The Maxwellian M is symmetric and satis- fies: V M (v )dv =1 , V vM (v )dv =0 , V v 2 M (v )dv = D. The formal macroscopic limit ε → 0 is the classical Fisher-KPP equation: ∂ t ρ − D∂ xx ρ = rρ (1 − ρ) . (2) Traveling waves when V is bounded, w.l.o.g V =[−1; 1]. Definition 1. We say that a function f (t, x, v ) is a traveling front solution of speed c ∈ R + of equation (1) if it can be written f (t, x, v )= μ (ξ = x − ct, v ), where the profile μ ∈C 2 (R × V ) is nonnegative, satisfies μ (−∞, ·)= M , μ (+∞, ·)=0, and μ solves ε(v − cε)∂ ξ μ =(M (v )ν − μ)+ rε 2 ν (M (v ) − μ) , ξ ∈ R,v ∈ V. (3) where ν is the macroscopic density associated to μ, that is ν (ξ )= V μ (ξ,v ) dv . Theorem 1. Assume that ε> 0 and that Supp(M )=[−1, 1]. There exists a minimal speed c ∗ (ε) ∈ (0, 1 ε ) such that there exists a traveling wave solution of (1) of speed c for c ∈ c ∗ (ε), 1 ε . Moreover, this traveling wave is nonincreasing with respect to x. This is not a perturbative result from the Fisher-KPP equation. Finding the speed : Dispersion relation As for the Fisher-KPP equation, the rate of exponential decay in space and the speed are given by the linearized problem at the edge of the front. Yields the following spectral problem : For all λ, find c(λ) such that there exists a Maxwellian Q λ such that ∀v ∈ V, (1 + ελ (c(λ)ε − v )) Q λ (v ) = (1 + rε 2 ) V M (v )Q λ (v )dv. (4) Proposition 1. For all ε> 0, the minimal speed c ∗ (ε) is given by c ∗ (ε) = min λ>0 c(λ) where c(λ) is for all λ a solution of the following dispersion relation: (1 + rε 2 ) V M (v ) 1+ ελ(c(λ)ε − v ) dv =1 . (5) This relation is not compatible with and unbounded velocity set. We can provide estimates for the critical speed c ∗ (ε). In particular, Proposition 2. Assume that Supp(M )=[−1, 1], then c ∗ (ε) −→ ε→0 2 √ rD := c KPP . The fronts are stable in suitable Lebesgue spaces. Unbounded V : Front acceleration ( w.l.o.g. ε =1 ). In the Figures below, we plot the evolution of the value of the speed of the front, for different values of V max . We deduce that c ∗ increases with V max . Moreover, we observe that the envelop of the curves ( which represents the instanta- neous speed of the front with an unbounded V ) behaves like t 1 2 . It yields that the density ρ propagates likes x ∝ t 3 2 . 1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 "B ELL " INITIAL CONDITION . 0 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 H EAVISIDE INITIAL CONDITION . E VOLUTION OF THE SPEED OF THE FRONT FOR DIFFERENT VALUES OF THE MAXIMAL SPEED .T HE M AXWELLIAN HERE IS A G AUSSIAN : M (v )= C (V max ) exp − v 2 2 . References [1] C. Cuesta, S. Hittmeir, C. Schmeiser, Traveling waves of a kinetic transport model for the KPP-Fisher equation, preprint, (2012). [2] E. Bouin, V. Calvez, G. Nadin, A kinetic KPP equation : traveling waves and front acceleration, preprint, (2012). [3] E. Bouin, V. Calvez, A kinetic eikonal equation, Comptes Rendus Mathematique, Volume 350, Issue 5, Pages 243-248, (2012). [4] D.G. Aronson and H.F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math. 30, pp. 33-76, (1978).
Transcript of A kinetic Fisher-KPP equation : traveling waves and front...
A kinetic Fisher-KPP equation : traveling waves and frontacceleration
CNRSUMPA
Emeric Bouin*Joint work with Vincent Calvez* and Grégoire Nadin**
ENS LYON INRIA
* École Normale Supérieure de Lyon, UMR CNRS 5669 ’UMPA’ and INRIA project ’NUMED’, 46, allée d’Italie, F-69364 Lyon Cedex 07, France.* Université Pierre et Marie Curie - Paris 6, UMR CNRS 7598 ’LJLL’, BC187, 4 place de Jussieu, F-75252 Paris Cedex 05, France.
A kinetic model for the Fisher-KPP equation
We consider the followingkinetic model
ε2∂tf + εv ·∇xf = (M(v)ρ− f )︸ ︷︷ ︸
Scattering
+ε2 rρ (M(v)− f )︸ ︷︷ ︸Monostable growth
, (t, x, v) ∈ R+×Rn×V , (1)
wheref (t, x, v) denotes the density of particles moving with speedv ∈ V at timet andpositionx. The functionρ(t, x) denotes the macroscopic density of particules:
ρ(t, x) =
∫
V
f (t, x, v) dv , (t, x) ∈ R+ × Rn .
HereV denotes a symmetric subset ofRn. The MaxwellianM is symmetric and satis-
fies: ∫
V
M(v)dv = 1 ,
∫
V
vM(v)dv = 0 ,
∫
V
v2M(v)dv = D .
The formal macroscopic limitε → 0 is the classicalFisher-KPP equation:
∂tρ−D∂xxρ = rρ (1− ρ) . (2)
Traveling waves whenV is bounded, w.l.o.gV = [−1; 1].
Definition 1. We say that a functionf (t, x, v) is a traveling front solutionof speedc ∈ R
+ of equation(1) if it can be writtenf (t, x, v) = µ (ξ = x− ct, v), wheretheprofileµ ∈ C2 (R× V ) is nonnegative, satisfiesµ (−∞, ·) = M , µ (+∞, ·) = 0, andµsolves
ε(v − cε)∂ξµ = (M(v)ν − µ) + rε2ν (M(v)− µ) , ξ ∈ R, v ∈ V. (3)
whereν is the macroscopic density associated toµ, that isν (ξ) =∫
Vµ (ξ, v) dv.
Theorem 1.Assume thatε > 0 and thatSupp(M) = [−1, 1]. There exists aminimalspeedc∗(ε) ∈ (0, 1
ε) such thatthere exists a traveling wave solution of (1) of speed c
for c ∈[c∗(ε), 1
ε
]. Moreover, this traveling wave isnonincreasing with respect tox.
This isnot a perturbative result from the Fisher-KPP equation.
Finding the speed :Dispersion relation
As for the Fisher-KPP equation, the rate of exponential decay in space and the speed aregiven by the linearized problem at the edge of the front. Yields the followingspectralproblem : For allλ, find c(λ) such that there exists a MaxwellianQλ such that
∀v ∈ V, (1 + ελ (c(λ)ε− v))Qλ(v) = (1 + rε2)
∫
V
M(v)Qλ(v)dv. (4)
Proposition 1.For all ε > 0, the minimal speedc∗(ε) is given byc∗(ε) = minλ>0 c(λ)wherec(λ) is for all λ a solution of the followingdispersion relation:
(1 + rε2)
∫
V
M(v)
1 + ελ(c(λ)ε− v)dv = 1 . (5)
This relation isnot compatible with and unbounded velocity set.
We can provide estimates for the critical speedc∗(ε). In particular,
Proposition 2.Assume that Supp(M) = [−1, 1], thenc∗(ε) −→ε→0
2√rD := cKPP.
The fronts arestable in suitable Lebesgue spaces.
UnboundedV : Front acceleration ( w.l.o.g.ε = 1 ).
In the Figures below, we plot the evolution of the value of thespeed of the front, fordifferent values ofVmax. We deduce thatc∗ increases withVmax.Moreover, we observe that the envelop of the curves ( which represents theinstanta-neous speedof the front with an unboundedV ) behaves liket
1
2.
It yields thatthe densityρ propagates likesx ∝ t3
2 .
1 2 3 4 5 6 7 8 9
2
4
6
8
10
12
"BELL" INITIAL CONDITION .
0 1 2 3 4 5 6 7
1
2
3
4
5
6
7
8
HEAVISIDE INITIAL CONDITION .
EVOLUTION OF THE SPEED OF THE FRONT FOR DIFFERENT VALUES OF THEMAXIMAL SPEED . THE MAXWELLIAN HERE IS A GAUSSIAN : M(v) = C (Vmax) exp(
−v2
2
)
.
References
[1] C. Cuesta, S. Hittmeir, C. Schmeiser,Traveling waves of a kinetic transport model for the KPP-Fisher equation, preprint, (2012).
[2] E. Bouin, V. Calvez, G. Nadin,A kinetic KPP equation : traveling waves and front acceleration, preprint, (2012).
[3] E. Bouin, V. Calvez,A kinetic eikonal equation, Comptes Rendus Mathematique, Volume 350, Issue 5, Pages 243-248, (2012).
[4] D.G. Aronson and H.F. Weinberger,Multidimensional nonlinear diffusion arising in population genetics, Adv. Math. 30, pp. 33-76, (1978).
