We analyse the linear kinetic transport equation with a BGK relaxation
operator. We study the large scale hyperbolic limit $(t,x)\to (t/\eps,x/\eps)$.
We derive a new type of limiting Hamilton-Jacobi equation, which is analogous
to the classical eikonal equation derived from the heat equation with small
diffusivity. We prove well-posedness of the phase problem and convergence
towards the viscosity solution of the Hamilton-Jacobi equation. This is a
preliminary work before analysing the propagation of reaction fronts in kinetic
equations

Bouin, Emeric

Calvez, Vincent

English

arXiv

Emeric Bouin

Vincent Calvez

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 350 (2012) 243–248Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comPartial Differential EquationsA kinetic eikonal equationUne équation eikonale cinétiqueEmeric Bouin, Vincent CalvezUMR CNRS 5669 ‘UMPA’ and INRIA project ‘NUMED’, École normale supérieure de Lyon, 46, allée d’Italie, 69364 Lyon cedex 07, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 10 February 2012Accepted after revision 7 March 2012Available online 20 March 2012Presented by Pierre-Louis LionsWe analyse the linear kinetic transport equation with a BGK relaxation operator. We studythe large scale hyperbolic limit (t, x) → (t/ε, x/ε). We derive a new type of limitingHamilton–Jacobi equation, which is analogous to the classical eikonal equation derivedfrom the heat equation with small diffusivity. Interestingly, the hydrodynamic limit andthe large deviation approach do not commute. We prove well-posedness of the phaseproblem and convergence towards the viscosity solution of the Hamilton–Jacobi equation.This is a preliminary work before analyzing the propagation of reaction fronts in kineticequations.© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous analysons une équation cinétique linéaire de transport avec un opérateur derelaxation BGK. Nous étudions la limite hyperbolique de grande échelle (t, x) → (t/ε, x/ε).Nous obtenons à la limite une nouvelle équation de Hamilton–Jacobi, qui est l’analoguede l’équation eikonale classique obtenue à partir de l’équation de la chaleur avec petitediffusion. Il est alors intéressant de constater que la limite hydrodynamique ne commutepas avec l’asymptotique des grandes déviations. Nous démontrons le caractère bien poséde l’équation vérifiée par la phase, ainsi que la convergence vers une solution de viscositéde l’équation de Hamilton–Jacobi. Ceci est un travail préliminaire en vue d’analyser lapropagation de fronts de réaction pour des équations cinétiques.© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeNous considérons un modèle cinétique linéaire avec un opérateur de relaxation BGK, posé sur un ensemble de vitesses Vsymétrique et borné. On analyse le comportement de l’équation dans la limite hyperbolique de grande échelle (t, x) →( tε ,xε ),∂t fε + v · ∇x f ε = 1ε(M(v)ρε − f ε), (t, x, v) ∈R+ ×Rn × V . (1)Nous démontrons que la phase ϕε définie par la relation f ε(t, x, v) = M(v)e− ϕε(t,x,v)ε converge (localement) uniformé-ment, lorsque ε → 0, vers une fonction ϕ0(t, x) indépendante de v . De surcroît, la fonction ϕ0 est solution de viscositéE-mail addresses: emeric.bouin@ens-lyon.fr (E. Bouin), vincent.calvez@ens-lyon.fr (V. Calvez).1631-073X/$ – see front matter © 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2012.03.009244 E. Bouin, V. Calvez / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 243–248de l’équation de Hamilton–Jacobi suivante,∫VM(v)1 − ∂tϕ0(t, x) − v · ∇xϕ0(t, x) dv = 1, (t, x) ∈R+ ×Rn. (2)Cette équation peut se réécrire sous la forme canonique ∂tϕ0 + H(∇xϕ0) = 0 pour un hamiltonien effectif H(p) qui estlipschitzien et convexe. Comme dans [1], cet hamiltonien est relié à la résolution d’un problème aux valeurs propres dansl’espace des vitesses V , ce dernier s’écrivant comme suit : trouver un vecteur propre Q (v) et une valeur propre H(p) telsque(1 + H(p) − v · p)Q (v) =∫VM(v ′)Q(v ′)dv ′.La démonstration du passage à la limite de (1) vers (2) s’appuie sur une série d’estimations a priori qui démontre que ϕεappartient à l’espace de Sobolev W 1,∞(R+ ×Rn × V ), avec un contrôle uniforme en ε > 0 (Proposition 2.1 ci-dessous). Dansun deuxième temps, nous démontrons que toute fonction test ψ0(t, x) de classe C2 telle que ϕ0 − ψ0 admet un maximumlocal en (t0, x0), vérifie∫VM(v)1 − ∂tψ0(t0, x0) − v · ∇xψ0(t0, x0) dv  1.Ceci démontre que ϕ0 est une sous-solution de viscosité l’équation de Hamilton–Jacobi (2). Un raisonnement identiquemontre qu’il s’agit aussi d’une sur-solution de viscosité. La démonstration se base sur la construction d’un correcteur micro-scopique η(t, x, v) défini de façon ad-hoc par la relation∀(v, v ′) ∈ V × V , eη(t,x,v) − eη(t,x,v ′) = (v ′ − v) · ∇xψ0(t, x).1. Large scale limit and derivation of the Hamilton–Jacobi equationWe consider the following kinetic equation with BGK relaxation operator:∂t f + v · ∇x f = M(v)ρ − f , (t, x, v) ∈R+ ×Rn × V , (3)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) =∫Vf (t, x, v)dv, (t, x) ∈R+ ×Rn.Here V denotes a bounded symmetric subset of Rn . We assume that the Maxwellian M is symmetric and satisfies thefollowing moment identities:∫VM(v)dv = 1,∫VvM(v)dv = 0,∫Vv2M(v)dv = θ2.In this paper we focus on the large scale hyperbolic limit (t, x) → ( tε , xε ), ε → 0. The kinetic equation (3) reads as followsin the new scaling:∂t fε + v · ∇x f ε = 1ε(M(v)ρε − f ε), (t, x, v) ∈ R+ ×Rn × V . (4)Clearly, the velocity distribution relaxes rapidly towards the Maxwellian distribution. This motivates the introduction of thefollowing Hopf–Cole transformation:f ε(t, x, v) = M(v)e− ϕε(t,x,v)ε ,where we expect the phase ϕε to become independent of v as ε → 0. To avoid technical complications due to ill-prepareddata, we set ϕε(0, x, v) = ϕ0(x)  0 as an initial data for (4). The equation satisfied by ϕε reads∂tϕε + v · ∇xϕε =∫VM(v ′)(1 − e ϕε−ϕε ′ε)dv ′, (t, x, v) ∈ R+ ×Rn × V . (5)E. Bouin, V. Calvez / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 243–248 245Theorem 1.1. Let V ⊂ Rn be bounded and symmetric, and M ∈ L1(V ) be nonnegative and symmetric. Then ϕε converges (locally)uniformly towards ϕ0 , where ϕ0 does not depend on v. Moreover ϕ0 is the viscosity solution of the following Hamilton–Jacobi equa-tion: ∫VM(v)1 − ∂tϕ0(t, x) − v · ∇xϕ0(t, x) dv = 1, (t, x) ∈R+ ×Rn. (6)The denominator of the integrand is positive for all v ∈ V .The last observation in Theorem 1.1 is not compatible with an unbounded velocity set.As in [1,11], the Hamilton–Jacobi equation is connected with an eigenvalue problem in the velocity space V : Find aneigenvector Q (v) such that(1 − ∂tϕ0 − v · ∇xϕ0)Q (v) =∫VM(v ′)Q(v ′)dv ′.This eigenproblem can be solved explicitly, and yields formula (6).Thanks to monotonicity properties, we can boil down to the classical framework of first order Hamilton–Jacobi equations.Indeed, writing Eq. (6) as G(∂tϕ0,∇xϕ0) = 0, we observe that G is increasing with respect to the first variable. Hence theequation is equivalent to ∂tϕ0 + H(∇xϕ0) = 0, where the effective Hamiltonian H is defined through the implicit formula,∫VM(v)(1 + H(p) − v · p) dv = 1. (7)Differentiating (7) we obtain,∫VM(v)(1 + H(p) − v · p)2(∇H(p) − v)dv = 0.We deduce ‖∇H‖∞  V max. This is in accordance with the underlying kinetic equation, since ∇H can be interpreted as thegroup speed, which is bounded by the maximal speed of the particles. Differentiating (7) twice we obtain( ∫VM(v)(1 + H(p) − v · p)2 dv)D2 H(p) = 2∫VM(v)(1 + H(p) − v · p)3(∇H(p) − v) ⊗ (∇H(p) − v)dv.We deduce that the effective Hamiltonian is convex.As an example, we can compute the effective Hamiltonian H in one dimension for a constant Maxwellian M ≡ 12 onV = (−1,1). We obtain H(p) = p−tanh(p)tanh(p) . It is equivalent to θ2|p|2 for small p (θ2 = 13 ). Another example where theeffective Hamitonian is explicit is given by the Maxwellian M(v) = 12 (δ1 + δ−1), though it is not an L1 function. Thiscorresponds to a two velocities model (also known as the telegraph equation, see [10,3]). In this case we obtain the relativisticHamiltonian H(p) =√1+4p2−12 .Interestingly enough, we obtain a Hamilton–Jacobi equation which differs from the classical eikonal equation. The lattercould have been expected from the following argumentation. The formal limit of Eq. (4) at order O (ε) is the heat equationwith small diffusivity:∂tρε = εθ2xρε, (t, x) ∈R+ ×Rn.It is well known that the phase φε = −ε logρε satisfies in the limit ε → 0 the classical eikonal equation in the sense ofviscosity solutions [8,12,6,9,13,11]:∂tφ0 + θ2∣∣∇xφ0∣∣2 = 0. (8)Interestingly, the hydrodynamic limit and the large deviation approach do not commute. We only have asymptotic equiva-lence between the two approaches for small |p| as can be seen directly on (7) by Taylor expansion: H(p) ∼ θ2|p|2.In Fig. 1 we show numerical simulations of the kinetic eikonal equation (6), with a constant Maxwellian on V = (−1,1),and we compare it with the classical eikonal equation (8).We end this introduction by listing some possible extensions of Theorem 1.1 for other choices of transport and scatteringoperators. We will develop a more general framework in a future work.(i) In the case of the Vlasov–Fokker–Planck equation,∂t fε + v · ∇x f ε − ∇x V (x) · ∇v f ε = 1ε∇v ·(∇v f ε + v f ε), (t, x, v) ∈ R+ ×Rn ×Rn,we obtain simply the eikonal equation ∂tφ0 + |∇xφ0|2 = 0 in the WKB expansion f ε = M(v)e− ϕεε , where M(v) is aGaussian.246 E. Bouin, V. Calvez / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 243–248Fig. 1. Numerical simulations of the Hamilton–Jacobi equation ∂tϕ + H(∂xϕ) = 0. (Left) The kinetic eikonal equation (6) where M(v) = 12 1(−1,1) . (Right) Theclassical eikonal equation H(p) = θ2|p|2 (8). In both cases the initial condition is a parabola.(ii) It is challenging to replace the BGK operator in (4) by a convolution operator L( f ) = K ∗ f − f , where K is a probabilitykernel [2]. However in this case we are not able to solve explicitly the eigenproblem in the cell V .(iii) In a forthcoming work we will investigate the propagation of reaction fronts in kinetic equations, following [9,10,5,3].2. Proof of Theorem 1.1First let us mention that the solution ϕε remains nonnegative for all times. We proceed in two steps. First we proveuniform estimates with respect to ε > 0. It allows to extract a uniformly converging subsequence. Second we identify thelimit as the viscosity solution of Eq. (6) using the maximum principle. The second step relies on the construction of asuitable corrector η(t, x, v) [6,7].Step 1. Existence and uniform bounds.Proposition 2.1. Let V ⊂ Rn be a bounded subset. Assume M ∈ L1(V ) and ϕ0 ∈ W 1,∞(Rn). The kinetic equation (5) has a uniquesolution ϕε ∈ W 1,∞(R+ ×Rn × V ). Furthermore, the solution satisfies the following uniform estimates:0  ϕε(t, ·)  ‖ϕ0‖∞, (9)∥∥∇xϕε(t, ·)∥∥∞  ‖∇xϕ0‖∞, (10)∥∥∇vϕε(t, ·)∥∥∞  t‖∇xϕ0‖∞, (11)∥∥∂tϕε(t, ·)∥∥∞  V max‖∇xϕ0‖∞. (12)Proof. We obtain a unique solution ϕε from a fixed point method on the Duhamel formulation of (5):ϕε(t, x, v) = ϕ0(x − tv) +t∫0∫VM(v ′)(1 − e ϕε(t−s,x−sv,v)−ϕε(t−s,x−sv,v′)ε)dv ′ ds. (13)We obtain directly,∀ε > 0, 0  ϕε(t, x, v)  ϕ0(x − tv) + t.This ensures that ϕε is uniformly bounded on [0, T ] ×Rn × V . To prove the bound (9), we define ψεδ (t, x, v) = ϕε(t, x, v) −δt − δ4|x|2. For any δ > 0, ψεδ attains a maximum at point (tδ, xδ, vδ). Suppose that tδ > 0. Then, we have∂tϕε(tδ, xδ, vδ)  δ, ∇xϕε(tδ, xδ, vδ) = 2δ4xδ.As a consequence, we have at the maximum point (tδ, xδ, vδ):0 ∫M(v ′)(1 − eψεδ(tδ ,xδ ,vδ )−ψεδ (tδ ,xδ ,v′)ε)dv ′  δ + 2vδδ4xδ  δ − 2V maxδ4|xδ |. (14)VE. Bouin, V. Calvez / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 243–248 247Moreover, the maximal property of (tδ, xδ, vδ) also implies∥∥ϕε∥∥∞ − δ4|xδ |2  ϕε(tδ, xδ, vδ) − δtδ − δ4|xδ |2  ϕε(0,0, vδ)  0.We obtain a contradiction with (14) since δ−6/(2V max)  |xδ |2  δ−4‖ϕε‖∞ cannot hold for sufficiently small δ > 0. As aconsequence tδ = 0, and we have,∀(t, x, v) ∈ [0, T ] ×Rn × V , ϕε(t, x, v)  ϕ0(xδ, vδ) + δt + δ4|x|2  ‖ϕ0‖∞ + δt + δ4|x|2.Passing to the limit δ → 0, we obtain (9). To find the bound (10), we use the same ideas on the difference ϕεh (t, x, v) =ϕε(t, x + h, v) − ϕε(t, x, v). The equation for ϕεh reads as follows∂tϕεh + v · ∇xϕεh =∫VM(v ′)eϕε−ϕε ′ε(1 − eϕεh −ϕε ′hε)dv ′.Using the same argument as above with a δ-correction, we conclude that∀(t, x, v) ∈ [0, T ] ×Rn × V , ϕεh (t, x, v)  sup(x,v)∈R×V∣∣ϕ0(x + h, v) − ϕ0(x, v)∣∣.The same argument applies to −ϕεh ,∂t(−ϕεh ) + v · ∇x(−ϕεh ) = −∫VM(v ′)eϕε−ϕε ′ε(1 − e−(−ϕεh )−(−ϕε ′h )ε)dv ′,so that the r.h.s. has the right sign when −ϕεh attains a maximum. Finally,∀(t, x, v) ∈ [0, T ] ×Rn × V , ∣∣ϕεh (t, x, v)∣∣  sup(x,v)∈R×V∣∣ϕ0(x + h, v) − ϕ0(x, v)∣∣  ∥∥∇xϕ0∥∥∞|h|,from which the estimate (10) follows.To obtain regularity in the velocity variable (11), we differentiate (5) with respect to v ,(∂t + v · ∇x)(∇vϕε) = −gε(ϕε)∇vϕε − ∇xϕε,where gε(ϕε) = 1ε∫V M(v′)eϕε−ϕε ′ε dv ′  0. Multiplying by ∇vϕε|∇vϕε | , we obtain(∂t + v · ∇x)(∣∣∇vϕε∣∣) = −gε(ϕε)∣∣∇vϕε∣∣ −(∇xϕε · ∇vϕε|∇vϕε|) ‖∇xϕ0‖∞, (15)from which we deduce (11) since ∇vϕ0 = 0 by hypothesis.Finally, the bound (12) is obtained similarly as the bound on ∇xϕε (10), using the difference ϕεs (t, x, v) = ϕε(t + s, x, v)−ϕε(t, x, v). We obtain∀(t, x, v) ∈ [0, T ] ×Rn × V , ∣∣ϕεs (t, x, v)∣∣  sup(x,v)∈R×V∣∣ϕε(s, x, v) − ϕ0(x, v)∣∣.We use the Duhamel formulation (13) to estimate the last contribution:∣∣ϕε(s, x, v) − ϕ0(x, v)∣∣  ∣∣ϕ0(x − sv) − ϕ0(x)∣∣ + o(s).The estimate (12) follows. Step 2. Viscosity solution procedure.From Proposition 2.1 we deduce that the family (ϕε)ε is locally uniformly bounded in W 1,∞(R+ × Rn × V ). Then,from the Ascoli–Arzelà theorem, we can extract a locally uniformly converging subsequence. We denote by ϕ0 the limit.Furthermore, from the fact that∫V M(v′)eϕε−ϕε ′ε dv ′ is uniformly bounded on [0, T ] ×Rn × V , we deduce that ϕ0 does notdepend on v .Let ψ0 ∈ C2(R+ ×Rn) be a test function such that ϕ0 − ψ0 has a local maximum at (t0, x0). We want to show that ψ0is a subsolution of (6), yielding that ϕ0 is a viscosity subsolution [4]. The supersolution case can be performed similarly.Thereby, we define a corrective term η not depending on ε [6]: ψε = ψ0 + εη. The corrector η is defined up to an additiveconstant. We choose the renormalization∫V M(v′)e−η′ dv ′ = 1. We define η as follows∀(v, v ′) ∈ V × V , eη(t,x,v) − eη(t,x,v ′) = (v ′ − v) · ∇xψ0(t, x). (16)248 E. Bouin, V. Calvez / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 243–248The corrector η is well defined. In fact, we can choose any v0 ∈ V and define eη(t,x,v) = μ0 + (v0 − v) · ∇xψ0(t, x). There isa unique positive μ0 = eη(t,x,v0) under the condition∫V M(v′)e−η′ dv ′ = 1.The uniform convergence ensures that ϕε − ψε has a maximum at (tε, xε, vε), where (tε, xε) is close to (t0, x0). As V isa bounded set, the sequence (vε) has an accumulation point, say v∗ . We can extract a subsequence (without relabelling)such that (tε, xε, vε) → (t0, x0, v∗). We have at (tε, xε, vε):1 − ∂tψε − vε · ∇xψε = 1 − ∂tϕε − vε · ∇xϕε =∫VM(v ′)eϕε−ϕε ′ε dv ′.From the maximum property of (tε, xε, vε), the last inequality implies at this point:1 − ∂tψε(tε, xε, vε) − vε · ∇xψε(tε, xε, vε)∫VM(v ′)eη(tε,xε,vε)−η(tε,xε,v ′) dv ′.Passing to the limit, we obtain at (t0, x0):1 − ∂tψ0(t0, x0) − v∗ · ∇xψ0(t0, x0)∫VM(v ′)eη(t0,x0,v∗)−η(t0,x0,v ′) dv ′ = eη(t0,x0,v∗).From the very definition of the corrector η (16), this writes also∀v ∈ V , 1 − ∂tψ0(t0, x0) − v · ∇xψ0(t0, x0)  eη(t0,x0,v).Therefore we obtain at point (t0, x0),∫VM(v)1 − ∂tψ0(t0, x0) − v · ∇xψ0(t0, x0) dv ∫VM(v)e−η(t0,x0,v) dv = 1.We conclude that ψ0 is a subsolution of (6). References[1] G. Barles, L.C. Evans, P.E. Souganidis, Wavefront propagation for reaction–diffusion systems of PDE, Duke Math. J. 61 (3) (1990) 835–858.[2] G. Barles, S. Mirrahimi, B. Perthame, Concentration in Lotka–Volterra parabolic or integral equations: a general convergence result, Methods Appl.Anal. 16 (2009) 321–340.[3] E. Bouin, V. Calvez, G. Nadin, Hyperbolic traveling waves driven by growth, preprint, 2011.[4] M.G. Crandall, L.C. Evans, P.L. Lions, Some properties of viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc. 282 (1984) 487–502.[5] C. Cuesta, S. Hittmeir, C. Schmeiser, Traveling waves of a kinetic transport model for the KPP–Fisher equation, preprint, 2010.[6] L.C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 359–375.[7] L.C. Evans, A survey of entropy methods for partial differential equations, Bull. Amer. Math. Soc. 41 (2004) 409–438.[8] L.C. Evans, H. Ishii, A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst.H. Poincaré Anal. Non Linéaire 2 (1985) 1–20.[9] L.C. Evans, P.E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Univ. Math. J. 38 (1989) 141–172.[10] S. Fedotov, Wave front for a reaction–diffusion system and relativistic Hamilton–Jacobi dynamics, Phys. Rev. E 59 (3) (1999) 5040–5044.[11] W.H. Fleming, P.E. Souganidis, PDE-viscosity solution approach to some problems of large deviations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Sér 13 (2)(1986) 171–192.[12] M.I. Freidlin, Geometric optics approach to reaction–diffusion equations, SIAM J. Appl. Math. 46 (1986) 222–232.[13] M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, second ed., Grundlehren Math. Wiss., vol. 260, Springer-Verlag, New York,1998.

Comptes Rendus Mathematique

A kinetic eikonal equation

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We analyse the linear kinetic transport equation with a BGK relaxation operator. We study the large scale hyperbolic limit $(t,x)\to (t/\eps,x/\eps)$. We derive a new type of limiting Hamilton-Jacobi equation, which is analogous to the classical eikonal equation derived from the heat equation with small diffusivity. We prove well-posedness of the phase problem and convergence towards the viscosity solution of the Hamilton-Jacobi equation. This is a preliminary work before analysing the propagation of reaction fronts in kinetic equations

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A kinetic eikonal equationEmeric Bouin, Vincent CalvezTo cite this version:Emeric Bouin, Vincent Calvez. A kinetic eikonal equation. 2012. <hal-00668874>HAL Id: hal-00668874https://hal.archives-ouvertes.fr/hal-00668874Submitted on 10 Feb 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A kinetic eikonal equationEmeric Bouin ∗, Vincent Calvez ∗10 fe´vrier 2012AbstractWe analyse the linear kinetic transport equation with a BGK relaxation operator. We study the largescale hyperbolic limit (t, x) → (t/ε, x/ε). We derive a new type of limiting Hamilton-Jacobi equation,which is analogous to the classical eikonal equation derived from the heat equation with small diffusivity.We prove well-posedness of the phase problem and convergence towards the viscosity solution of theHamilton-Jacobi equation. This is a preliminary work before analysing the propagation of reactionfronts in kinetic equations.Re´sume´Une e´quation eikonale cine´tique. Nous analysons une e´quation cine´tique line´aire de transport avec unope´rateur de relaxation BGK. Nous e´tudions la limite hyperbolique de grande e´chelle (t, x) → (t/ε, x/ε).Nous obtenons a` la limite une nouvelle e´quation de Hamilton-Jacobi, qui est l’analogue de l’e´quationeikonale classique obtenue a` partir de l’e´quation de la chaleur avec petite diffusion. Nous de´montronsle caracte`re bien pose´ de l’e´quation ve´rifie´e par la phase, ainsi que la convergence vers une solutionde viscosite´ de l’e´quation de Hamilton-Jacobi. Ceci est un travail pre´liminaire en vue d’analyser lapropagation de fronts de re´action pour des e´quations cine´tiques.Version franc¸aise abre´ge´eNous conside´rons un mode`le cine´tique line´aire avec un ope´rateur de relaxation BGK, pose´ sur un ensemblede vitesses V syme´trique et borne´. On analyse le comportement de l’e´quation dans la limite hyperbolique degrande e´chelle (t, x)→ ( tε, xε),∂tfε + v · ∇xf ε = 1ε(M(v)ρε − f ε) , (t, x, v) ∈ R+ × Rn × V . (1)Nous de´montrons que la phase ϕε de´finie par la relation f ε(t, x, v) = M(v)e−ϕε(t,x,v)ε converge (localement)uniforme´ment, lorsque ε → 0, vers une fonction ϕ0(t, x) inde´pendante de v. De surcroˆıt, la fonction ϕ0 estsolution de viscosite´ de l’e´quation de Hamilton-Jacobi suivante,∫VM(v)1− ∂tϕ0(t, x)− v · ∇xϕ0(t, x)dv = 1 , (t, x) ∈ R+ × Rn . (2)Cette e´quation peut se re´e´crire sous la forme canonique ∂tϕ0 +H(∇xϕ0) = 0 pour un hamiltonien effectifH(p) qui est lipschitzien et convexe. En re`gle ge´ne´rale, nos travaux consistent a` homoge´ne´iser l’e´quation (1)par rapport a` la variable de vitesse. Le proble`me aux valeurs propres dans la cellule V s’e´crit comme suit :trouver un vecteur propre Q(v) et une valeur propre H(p) tels que(1 +H(p)− v · p)Q(v) =∫VM(v′)Q(v′)dv′ .∗UMR CNRS 5669 ’UMPA’ and INRIA project ’NUMED’, E´cole Normale Supe´rieure de Lyon, 46, alle´e d’Italie, F-69364Lyon Cedex 07, France. Mail : emeric.bouin@ens-lyon.fr, vincent.calvez@ens-lyon.fr1La de´monstration du passage a` la limite de (1) vers (2) s’appuie sur une se´rie d’estimations a priori quide´montre que ϕε appartient a` l’espace de Sobolev W 1,∞, avec un controˆle uniforme en ε > 0 (Proposition2.1 ci-dessous). Dans un deuxie`me temps, nous de´montrons que toute fonction test ψ0(t, x) de classe C2 telleque ϕ0 − ψ0 admet un maximum local en (t0, x0), ve´rifie∫VM(v)1− ∂tψ0(t0, x0)− v · ∇xψ0(t0, x0)dv ≤ 1 .Ceci de´montre que ϕ0 est une sous-solution de viscosite´ l’e´quation de Hamilton-Jacobi (2). Un raisonnementidentique montre qu’il s’agit aussi d’une sur-solution de viscosite´. La de´monstration se base sur la constructiond’un correcteur microscopique η(t, x, v) de´fini de fac¸on ad-hoc par la relation∀(v, v′) ∈ V × V , eη(t,x,v) − eη(t,x,v′) = (v′ − v) · ∇xψ0(t, x) .1 Large-scale limit and derivation of the Hamilton-Jacobi equa-tionWe consider the following kinetic equation with BGK relaxation operator:∂tf + v · ∇xf =M(v)ρ− f , (t, x, v) ∈ R+ × Rn × V , (3)where f(t, x, v) denotes the density of particles moving with speed v ∈ V at time t and position x. Thefunction ρ(t, x) denotes the macroscopic density of particules:ρ(t, x) =∫Vf(t, x, v) dv , (t, x) ∈ R+ × Rn .Here V denotes a bounded symmetric subset of Rn. We assume that the Maxwellian M is symmetric andsatisfies the following moment identities:∫VM(v)dv = 1 ,∫VvM(v)dv = 0 ,∫Vv2M(v)dv = θ2 .In this paper we focus on the large scale hyperbolic limit (t, x) → ( tε, xε), ε → 0. The kinetic equation (3)reads as follows in the new scaling:∂tfε + v · ∇xf ε = 1ε(M(v)ρε − f ε) , (t, x, v) ∈ R+ × Rn × V . (4)Clearly, the velocity distribution relaxes rapidly towards the Maxwellian distribution. This motivates theintroduction of the following Hopf-Cole transformation:f ε(t, x, v) =M(v)e−ϕε(t,x,v)ε .where we expect the phase ϕε to become independent of v as ε → 0. To avoid technical complications dueto ill-prepared data, we set ϕε(0, x, v) = ϕ0(x) ≥ 0 as an initial data for (4). The equation satisfied by ϕεreads:∂tϕε + v · ∇xϕε =∫VM(v′)(1− eϕε−ϕε′ε)dv′ , (t, x, v) ∈ R+ × Rn × V , (5)Theorem 1.1 Let V ⊂ Rn be bounded and symmetric, and M ∈ L1(V ) be nonnegative and symmetric.Then ϕε converges (locally) uniformly towards ϕ0, where ϕ0 does not depend on v. Moreover ϕ0 is theviscosity solution of the following Hamilton-Jacobi equation:∫VM(v)1− ∂tϕ0(t, x)− v · ∇xϕ0(t, x)dv = 1 , (t, x) ∈ R+ × Rn . (6)The denominator of the integrand is positive for all v ∈ V .2The last observation in Theorem 1.1 is not compatible with an unbounded velocity set.One can understand this as an homogenization problem in the velocity variable. Moreover, the associatedeigenproblem in the cell V writes: Find an eigenvector Q(v) such that:(1− ∂tϕ0 − v · ∇xϕ0)Q(v) =∫VM(v′)Q(v′)dv′ .This eigenproblem can be solved explicitly, and yields formula (6).Thanks to monotonicity properties, we can boil down to the classical framework of first order Hamilton-Jacobi equations. Indeed, writing equation (6) as G(∂tϕ0,∇xϕ0) = 0, we observe that G is increasing withrespect to the first variable. Hence the equation is equivalent to ∂tϕ0 +H(∇xϕ0) = 0, where the effectiveHamiltonian H is defined through the implicit formula,∫VM(v)(1 +H(p)− v · p) dv = 1 . (7)Differentiating (7) we obtain,∫VM(v)(1 +H(p)− v · p)2 (∇H(p)− v) dv = 0 .We deduce ‖∇H‖∞ ≤ Vmax. This is in accordance with the underlying kinetic equation, since ∇H can beinterpreted as the group speed, which is bounded by the maximal speed of the particles. Differentiating (7)twice we obtain(∫VM(v)(1 +H(p)− v · p)2 dv)D2H(p) = 2∫VM(v)(1 +H(p)− v · p)3 (∇H(p)− v)⊗ (∇H(p)− v) dv .We deduce that the effective Hamiltonian is convex.As an example, we can compute the effective Hamiltonian H in one dimension for a constant MaxwellianM ≡ 12 on V = (−1, 1). We obtain H(p) = p−tanh(p)tanh(p) . It is equivalent to θ2|p|2 for small p (θ2 = 13 ). Anotherexample where the effective hamitonian is explicit is given by the Maxwellian M(v) = 12 (δ1 + δ−1), thoughit is not a L1 function. This corresponds to a two velocities model (also known as the telegraph equation, see[9, 2]). In this case we obtain the relativistic hamiltonian H(p) =√1+4p2−12 .Interestingly enough, we obtain a Hamilton-Jacobi equation which differs from the classical eikonal equa-tion. The latter could have been expected from the following argumentation. The formal limit of equation(4) at order O(ε) is the heat equation with small diffusivity:∂tρε = εθ2∆xρε , (t, x) ∈ R+ × Rn .It is well-known that the phase φε = −ε log ρε satisfies in the limit ε → 0 the classical eikonal equation inthe sense of viscosity solutions [5, 10, 6, 7, 11]:∂tφ0 + θ2|∇xφ0|2 = 0 . (8)We only have asymptotic equivalence between the two approaches for small |p| as can be seen directly on(7) by Taylor expansion: H(p) ∼ θ2|p|2.In Figure 1 we show numerical simulations of the kinetic eikonal equation (6), with a constant Maxwellianon V = (−1, 1), and we compare it with the classical eikonal equation (8).We end this introduction by listing some possible extensions of Theorem 1.1 for other choices of transportand scattering operators. We will develop a more general framework in a future work.1. In the case of the Vlasov-Fokker-Planck equation,∂tfε + v · ∇xf ε −∇xV (x) · ∇vf ε = 1ε∇v · (∇vf ε + vf ε) , (t, x, v) ∈ R+ × Rn × Rn ,we obtain simply the eikonal equation ∂tφ0 + |∇xφ0|2 = 0 in the WKB expansion f ε = M(v)e−ϕεε ,where M(v) is a Gaussian.3−1 −0.5 0 0.5 1012345678910−1 −0.5 0 0.5 1012345678910Figure 1: Numerical simulations of the Hamilton-Jacobi equation ∂tϕ + H(∂xϕ) = 0. (left) The kineticeikonal equation (6) where M(v) = 121(−1,1). (right) The classical eikonal equation H(p) = θ2|p|2 (8). Inboth cases the initial condition is a parabola.2. It is challenging to replace the BGK operator in (4) by a convolution operator L(f) = K ∗f −f , whereK is a probability kernel [1]. However in this case we are not able to solve explicitly the eigenproblemin the cell V .3. In a forthcoming work we will investigate the propagation of reaction fronts in kinetic equations,following [7, 9, 4, 2].2 Proof of Theorem 1.1First let us mention that the solution ϕε remains nonnegative for all times. We proceed in two steps. Firstwe prove uniform estimates with respect to ε > 0. It allows to extract a uniformly converging subsequence.Second we identify the limit as the viscosity solution of equation (6) using the maximum principle. Thesecond step relies on the construction of a suitable corrector η(t, x, v) [6, 8].Step 1. Existence and uniform bounds.Proposition 2.1 Let V ⊂ Rn be a bounded subset. Assume M ∈ L1(V ) and ϕ0 ∈ W 1,∞ (Rn). Thekinetic equation (5) has a unique solution ϕε ∈ W 1,∞ (R+ × Rn × V ). Furthermore, the solution satisfiesthe following uniform estimates:0 ≤ ϕε(t, ·) ≤ ‖ϕ0‖∞ , (9)‖∇xϕε(t, ·)‖∞ ≤ ‖∇xϕ0‖∞ , (10)‖∇vϕε(t, ·)‖∞ ≤ t‖∇xϕ0‖∞ , (11)‖∂tϕε(t, ·)‖∞ ≤ Vmax‖∇xϕ0‖∞ . (12)Proof. We obtain a unique solution ϕε from a fixed point method on the Duhamel formulation of (5):ϕε(t, x, v) = ϕ0(x− tv) +∫ t0∫VM(v′)(1− eϕε(t−s,x−sv,v)−ϕε(t−s,x−sv,v′)ε)dv′ds, (13)4We obtain directly,∀ε > 0, 0 ≤ ϕε(t, x, v) ≤ ϕ0(x− tv) + t .This ensures that ϕε is uniformly bounded on [0, T ]×Rn×V . To prove the bound (9), we define ψεδ(t, x, v) =ϕε(t, x, v) − δt − δ4|x|2. For any δ > 0, ψεδ attains a maximum at point (tδ, xδ, vδ). Suppose that tδ > 0.Then, we have∂tϕε(tδ, xδ, vδ) ≥ δ, ∇xϕε(tδ, xδ, vδ) = 2δ4xδ.As a consequence, we have at the maximum point (tδ, xδ, vδ):0 ≥∫VM(v′)(1− eψεδ(tδ,xδ,vδ)−ψεδ(tδ ,xδ,v′)ε)dv′ ≥ δ + 2vδδ4xδ ≥ δ − 2Vmaxδ4|xδ| . (14)Moreover, the maximal property of (tδ, xδ, vδ) also implies‖ϕε‖∞ − δ4|xδ|2 ≥ ϕε(tδ, xδ, vδ)− δtδ − δ4|xδ|2 ≥ ϕε(0, 0, vδ) ≥ 0 .We obtain a contradiction with (14) since δ−6/(2Vmax) ≤ |xδ|2 ≤ δ−4‖ϕε‖∞ cannot hold for sufficientlysmall δ > 0. As a consequence tδ = 0, and we have,∀(t, x, v) ∈ [0, T ]× Rn × V, ϕε(t, x, v) ≤ ϕ0(xδ, vδ) + δt+ δ4|x|2 ≤ ‖ϕ0‖∞ + δt+ δ4|x|2.Passing to the limit δ → 0, we obtain (9). To find the bound (10), we use the same ideas on the differenceϕεh(t, x, v) = ϕε(t, x+ h, v)− ϕε(t, x, v). The equation for ϕεh reads as follows,∂tϕεh + v · ∇xϕεh =∫VM(v′)eϕε−ϕε′ε(1− eϕεh−ϕε′hε)dv′.Using the same argument as above with a δ−correction, we conclude that∀(t, x, v) ∈ [0, T ]× Rn × V, ϕεh(t, x, v) ≤ sup(x,v)∈R×V∣∣ϕ0(x+ h, v)− ϕ0(x, v)∣∣The same argument applies to −ϕεh,∂t (−ϕεh) + v · ∇x (−ϕεh) = −∫VM(v′)eϕε−ϕε′ε(1− e− (−ϕεh)−(−ϕε′h )ε)dv′.so that the r.h.s has the right sign when −ϕεh attains a maximum. Finally,∀(t, x, v) ∈ [0, T ]× Rn × V, |ϕεh(t, x, v)| ≤ sup(x,v)∈R×V∣∣ϕ0(x+ h, v)− ϕ0(x, v)∣∣ ≤ ∥∥∇xϕ0∥∥∞ |h|.from which the estimate (10) follows.To obtain regularity in the velocity variable (11), we differentiate (5) with respect to v,(∂t + v · ∇x) (∇vϕε) = −gε(ϕε)∇vϕε −∇xϕε,where gε(ϕε) = 1ε∫VM(v′)eϕε−ϕε′ε dv′ ≥ 0. Multiplying by ∇vϕε|∇vϕε| , we obtain(∂t + v · ∇x) (|∇vϕε|) = −gε(ϕε)|∇vϕε| −(∇xϕε · ∇vϕε|∇vϕε|)(15)≤ ‖∇xϕ0‖∞ .from which we deduce (11) since ∇vϕ0 = 0 by hypothesis.Finally, the bound (12) is obtained similarly as the bound on ∇xϕε (10), using the difference ϕεs(t, x, v) =ϕε(t+ s, x, v)− ϕε(t, x, v). We obtain∀(t, x, v) ∈ [0, T ]× Rn × V, |ϕεs(t, x, v)| ≤ sup(x,v)∈R×V∣∣ϕε(s, x, v)− ϕ0(x, v)∣∣ .5We use the Duhamel formulation (13) to estimate the last contribution:∣∣ϕε(s, x, v)− ϕ0(x, v)∣∣ ≤ |ϕ0(x− sv)− ϕ0(x)|+ o(s) .The estimate (12) follows. Step 2. Viscosity solution procedure.From Proposition 2.1 we deduce that the familly (ϕε)ε is locally uniformly bounded inW1,∞ (R+ × Rn × V ).Then, from the Ascoli-Arzela` theorem, we can extract a locally uniformly converging subsequence. We denoteby ϕ0 the limit. Furthermore, from the fact that∫VM(v′)eϕε−ϕε′ε dv′ is uniformly bounded on [0, T ]×Rn×V ,we deduce that ϕ0 does not depend on v.Let ψ0 ∈ C2 (R+ × Rn) be a test function such that ϕ0 − ψ0 has a local maximum at(t0, x0). We wantto show that ψ0 is a subsolution of (6), yielding that ϕ0 is a viscosity subsolution [3]. The supersolutioncase can be performed similarly. Thereby, we define a corrective term η not depending on ε: ψε = ψ0 + εη.The corrector η is defined up to an additive constant. We choose the renormalization∫VM(v′)e−η′dv′ = 1.We define η as follows,∀(v, v′) ∈ V × V , eη(t,x,v) − eη(t,x,v′) = (v′ − v) · ∇xψ0(t, x) . (16)The corrector η is well defined. In fact, we can choose any v0 ∈ V and define eη(t,x,v) = µ0 + (v0 − v) ·∇xψ0(t, x). There is a unique positive µ0 = eη(t,x,v0) under the condition∫VM(v′)e−η′dv′ = 1.The uniform convergence ensures that ϕε − ψε has a maximum at (tε, xε, vε), where (tε, xε) is close to(t0, x0). As V is a bounded set, the sequence (vε) has an accumulation point, say v∗. We can extract asubsequence (without relabelling) such that (tε, xε, vε)→ (t0, x0, v∗). We have at (tε, xε, vε):1− ∂tψε − vε · ∇xψε = 1− ∂tϕε − vε · ∇xϕε =∫VM(v′)eϕε−ϕε′ε dv′ .From the maximum property of (tε, xε, vε), the last inequality implies at this point :1− ∂tψε(tε, xε, vε)− vε · ∇xψε(tε, xε, vε) ≥∫VM(v′)eη(tε,xε,vε)−η(tε,xε,v′)dv′.Passing to the limit, we obtain at (t0, x0):1− ∂tψ0(t0, x0)− v∗ · ∇xψ0(t0, x0) ≥∫VM(v′)eη(t0,x0,v∗)−η(t0,x0,v′)dv′ = eη(t0,x0,v∗) .From the very definition of the corrector η (16), this writes also:∀v ∈ V , 1− ∂tψ0(t0, x0)− v · ∇xψ0(t0, x0) ≥ eη(t0,x0,v) .Therefore we obtain at point (t0, x0),∫VM(v)1− ∂tψ0(t0, x0)− v · ∇xψ0(t0, x0)dv ≤∫VM(v)e−η(t0,x0,v)dv = 1 .We conclude that ψ0 is a subsolution of (6).References[1] G. Barles, S. Mirrahimi and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equa-tions: a general convergence result, Methods Appl. Anal. 16, 321–340 (2009).[2] E. Bouin, V. Calvez and G. Nadin, Hyperbolic traveling waves driven by growth, preprint (2011).6[3] M.G. Crandall, L.C. Evans and P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobiequations, Trans. Amer. Math. Soc. 282, 487–502 (1984).[4] C. Cuesta, S. Hittmeir and C. Schmeiser, Traveling waves of a kinetic transport model for the KPP-Fisher equation, preprint (2010).[5] L.C. Evans and H. Ishii, A PDE approach to some asymptotic problems concerning random differentialequations with small noise intensities, Ann. Inst. H. Poincare´ Anal. Non Line´aire 2, 1–20 (1985).[6] L.C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy.Soc. Edinburgh Sect. A 111, 359–375 (1989).[7] L.C. Evans and P.E. Souganidis, A PDE approach to geometric optics for certain semilinear parabolicequations, Indiana Univ. Math. J. 38, 141–172 (1989).[8] L.C. Evans, A survey of entropy methods for partial differential equations, Bull. Amer. Math. Soc. 41,409–438 (2004).[9] S. Fedotov, Wave front for a reaction-diffusion system and relativistic Hamilton-Jacobi dynamics, Phys.Rev. E 59(3) 5040–5044 (1999).[10] M.I. Freidlin, Geometric optics approach to reaction-diffusion equations, SIAM J. Appl. Math. 46, 222–232 (1986).[11] M.I. Freidlin and A.D. Wentzell, Random perturbations of dynamical systems. Second edition.Grundlehren der Mathematischen Wissenschaften 260. Springer-Verlag, New York, 1998.7

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HAL Id: hal-00668874https://hal.archives-ouvertes.fr/hal-00668874Preprint submitted on 10 Feb 2012HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.A kinetic eikonal equationEmeric Bouin, Vincent CalvezTo cite this version:Emeric Bouin, Vincent Calvez. A kinetic eikonal equation. 2012. ￿hal-00668874￿A kinetic eikonal equationEmeric Bouin ∗, Vincent Calvez ∗10 février 2012AbstractWe analyse the linear kinetic transport equation with a BGK relaxation operator. We study the largescale hyperbolic limit (t, x) → (t/ε, x/ε). We derive a new type of limiting Hamilton-Jacobi equation,which is analogous to the classical eikonal equation derived from the heat equation with small diffusivity.We prove well-posedness of the phase problem and convergence towards the viscosity solution of theHamilton-Jacobi equation. This is a preliminary work before analysing the propagation of reactionfronts in kinetic equations.RésuméUne équation eikonale cinétique. Nous analysons une équation cinétique linéaire de transport avec unopérateur de relaxation BGK. Nous étudions la limite hyperbolique de grande échelle (t, x) → (t/ε, x/ε).Nous obtenons à la limite une nouvelle équation de Hamilton-Jacobi, qui est l’analogue de l’équationeikonale classique obtenue à partir de l’équation de la chaleur avec petite diffusion. Nous démontronsle caractère bien posé de l’équation vérifiée par la phase, ainsi que la convergence vers une solutionde viscosité de l’équation de Hamilton-Jacobi. Ceci est un travail préliminaire en vue d’analyser lapropagation de fronts de réaction pour des équations cinétiques.Version française abrégéeNous considérons un modèle cinétique linéaire avec un opérateur de relaxation BGK, posé sur un ensemblede vitesses V symétrique et borné. On analyse le comportement de l’équation dans la limite hyperbolique degrande échelle (t, x) →(tε, xε),∂tfε + v · ∇xf ε =1ε(M(v)ρε − f ε) , (t, x, v) ∈ R+ × Rn × V . (1)Nous démontrons que la phase ϕε définie par la relation f ε(t, x, v) = M(v)e−ϕε(t,x,v)ε converge (localement)uniformément, lorsque ε → 0, vers une fonction ϕ0(t, x) indépendante de v. De surcrôıt, la fonction ϕ0 estsolution de viscosité de l’équation de Hamilton-Jacobi suivante,∫VM(v)1− ∂tϕ0(t, x)− v · ∇xϕ0(t, x)dv = 1 , (t, x) ∈ R+ × Rn . (2)Cette équation peut se réécrire sous la forme canonique ∂tϕ0 +H(∇xϕ0) = 0 pour un hamiltonien effectifH(p) qui est lipschitzien et convexe. En règle générale, nos travaux consistent à homogénéiser l’équation (1)par rapport à la variable de vitesse. Le problème aux valeurs propres dans la cellule V s’écrit comme suit :trouver un vecteur propre Q(v) et une valeur propre H(p) tels que(1 +H(p)− v · p)Q(v) =∫VM(v′)Q(v′)dv′ .∗UMR CNRS 5669 ’UMPA’ and INRIA project ’NUMED’, École Normale Supérieure de Lyon, 46, allée d’Italie, F-69364Lyon Cedex 07, France. Mail : emeric.bouin@ens-lyon.fr, vincent.calvez@ens-lyon.fr1La démonstration du passage à la limite de (1) vers (2) s’appuie sur une série d’estimations a priori quidémontre que ϕε appartient à l’espace de Sobolev W 1,∞, avec un contrôle uniforme en ε > 0 (Proposition2.1 ci-dessous). Dans un deuxième temps, nous démontrons que toute fonction test ψ0(t, x) de classe C2 telleque ϕ0 − ψ0 admet un maximum local en(t0, x0), vérifie∫VM(v)1− ∂tψ0(t0, x0)− v · ∇xψ0(t0, x0)dv ≤ 1 .Ceci démontre que ϕ0 est une sous-solution de viscosité l’équation de Hamilton-Jacobi (2). Un raisonnementidentique montre qu’il s’agit aussi d’une sur-solution de viscosité. La démonstration se base sur la constructiond’un correcteur microscopique η(t, x, v) défini de façon ad-hoc par la relation∀(v, v′) ∈ V × V , eη(t,x,v) − eη(t,x,v′) = (v′ − v) · ∇xψ0(t, x) .1 Large-scale limit and derivation of the Hamilton-Jacobi equa-tionWe consider the following kinetic equation with BGK relaxation operator:∂tf + v · ∇xf =M(v)ρ− f , (t, x, v) ∈ R+ × Rn × V , (3)where f(t, x, v) denotes the density of particles moving with speed v ∈ V at time t and position x. Thefunction ρ(t, x) denotes the macroscopic density of particules:ρ(t, x) =∫Vf(t, x, v) dv , (t, x) ∈ R+ × Rn .Here V denotes a bounded symmetric subset of Rn. We assume that the Maxwellian M is symmetric andsatisfies the following moment identities:∫VM(v)dv = 1 ,∫VvM(v)dv = 0 ,∫Vv2M(v)dv = θ2 .In this paper we focus on the large scale hyperbolic limit (t, x) →(tε, xε), ε → 0. The kinetic equation (3)reads as follows in the new scaling:∂tfε + v · ∇xf ε =1ε(M(v)ρε − f ε) , (t, x, v) ∈ R+ × Rn × V . (4)Clearly, the velocity distribution relaxes rapidly towards the Maxwellian distribution. This motivates theintroduction of the following Hopf-Cole transformation:f ε(t, x, v) =M(v)e−ϕε(t,x,v)ε .where we expect the phase ϕε to become independent of v as ε → 0. To avoid technical complications dueto ill-prepared data, we set ϕε(0, x, v) = ϕ0(x) ≥ 0 as an initial data for (4). The equation satisfied by ϕεreads:∂tϕε + v · ∇xϕε =∫VM(v′)(1− eϕε−ϕε′ε)dv′ , (t, x, v) ∈ R+ × Rn × V , (5)Theorem 1.1 Let V ⊂ Rn be bounded and symmetric, and M ∈ L1(V ) be nonnegative and symmetric.Then ϕε converges (locally) uniformly towards ϕ0, where ϕ0 does not depend on v. Moreover ϕ0 is theviscosity solution of the following Hamilton-Jacobi equation:∫VM(v)1− ∂tϕ0(t, x)− v · ∇xϕ0(t, x)dv = 1 , (t, x) ∈ R+ × Rn . (6)The denominator of the integrand is positive for all v ∈ V .2The last observation in Theorem 1.1 is not compatible with an unbounded velocity set.One can understand this as an homogenization problem in the velocity variable. Moreover, the associatedeigenproblem in the cell V writes: Find an eigenvector Q(v) such that:(1− ∂tϕ0 − v · ∇xϕ0)Q(v) =∫VM(v′)Q(v′)dv′ .This eigenproblem can be solved explicitly, and yields formula (6).Thanks to monotonicity properties, we can boil down to the classical framework of first order Hamilton-Jacobi equations. Indeed, writing equation (6) as G(∂tϕ0,∇xϕ0) = 0, we observe that G is increasing withrespect to the first variable. Hence the equation is equivalent to ∂tϕ0 +H(∇xϕ0) = 0, where the effectiveHamiltonian H is defined through the implicit formula,∫VM(v)(1 +H(p)− v · p) dv = 1 . (7)Differentiating (7) we obtain,∫VM(v)(1 +H(p)− v · p)2 (∇H(p)− v) dv = 0 .We deduce ‖∇H‖∞ ≤ Vmax. This is in accordance with the underlying kinetic equation, since ∇H can beinterpreted as the group speed, which is bounded by the maximal speed of the particles. Differentiating (7)twice we obtain(∫VM(v)(1 +H(p)− v · p)2 dv)D2H(p) = 2∫VM(v)(1 +H(p)− v · p)3 (∇H(p)− v)⊗ (∇H(p)− v) dv .We deduce that the effective Hamiltonian is convex.As an example, we can compute the effective Hamiltonian H in one dimension for a constant MaxwellianM ≡ 12 on V = (−1, 1). We obtain H(p) =p−tanh(p)tanh(p) . It is equivalent to θ2|p|2 for small p (θ2 = 13 ). Anotherexample where the effective hamitonian is explicit is given by the Maxwellian M(v) = 12 (δ1 + δ−1), thoughit is not a L1 function. This corresponds to a two velocities model (also known as the telegraph equation, see[9, 2]). In this case we obtain the relativistic hamiltonian H(p) =√1+4p2−12 .Interestingly enough, we obtain a Hamilton-Jacobi equation which differs from the classical eikonal equa-tion. The latter could have been expected from the following argumentation. The formal limit of equation(4) at order O(ε) is the heat equation with small diffusivity:∂tρε = εθ2∆xρε , (t, x) ∈ R+ × Rn .It is well-known that the phase φε = −ε log ρε satisfies in the limit ε → 0 the classical eikonal equation inthe sense of viscosity solutions [5, 10, 6, 7, 11]:∂tφ0 + θ2|∇xφ0|2 = 0 . (8)We only have asymptotic equivalence between the two approaches for small |p| as can be seen directly on(7) by Taylor expansion: H(p) ∼ θ2|p|2.In Figure 1 we show numerical simulations of the kinetic eikonal equation (6), with a constant Maxwellianon V = (−1, 1), and we compare it with the classical eikonal equation (8).We end this introduction by listing some possible extensions of Theorem 1.1 for other choices of transportand scattering operators. We will develop a more general framework in a future work.1. In the case of the Vlasov-Fokker-Planck equation,∂tfε + v · ∇xf ε −∇xV (x) · ∇vf ε =1ε∇v · (∇vf ε + vf ε) , (t, x, v) ∈ R+ × Rn × Rn ,we obtain simply the eikonal equation ∂tφ0 + |∇xφ0|2 = 0 in the WKB expansion f ε = M(v)e−ϕεε ,where M(v) is a Gaussian.3−1 −0.5 0 0.5 1012345678910−1 −0.5 0 0.5 1012345678910Figure 1: Numerical simulations of the Hamilton-Jacobi equation ∂tϕ + H(∂xϕ) = 0. (left) The kineticeikonal equation (6) where M(v) = 121(−1,1). (right) The classical eikonal equation H(p) = θ2|p|2 (8). Inboth cases the initial condition is a parabola.2. It is challenging to replace the BGK operator in (4) by a convolution operator L(f) = K ∗f −f , whereK is a probability kernel [1]. However in this case we are not able to solve explicitly the eigenproblemin the cell V .3. In a forthcoming work we will investigate the propagation of reaction fronts in kinetic equations,following [7, 9, 4, 2].2 Proof of Theorem 1.1First let us mention that the solution ϕε remains nonnegative for all times. We proceed in two steps. Firstwe prove uniform estimates with respect to ε > 0. It allows to extract a uniformly converging subsequence.Second we identify the limit as the viscosity solution of equation (6) using the maximum principle. Thesecond step relies on the construction of a suitable corrector η(t, x, v) [6, 8].Step 1. Existence and uniform bounds.Proposition 2.1 Let V ⊂ Rn be a bounded subset. Assume M ∈ L1(V ) and ϕ0 ∈ W 1,∞ (Rn). Thekinetic equation (5) has a unique solution ϕε ∈ W 1,∞ (R+ × Rn × V ). Furthermore, the solution satisfiesthe following uniform estimates:0 ≤ ϕε(t, ·) ≤ ‖ϕ0‖∞ , (9)‖∇xϕε(t, ·)‖∞ ≤ ‖∇xϕ0‖∞ , (10)‖∇vϕε(t, ·)‖∞ ≤ t‖∇xϕ0‖∞ , (11)‖∂tϕε(t, ·)‖∞ ≤ Vmax‖∇xϕ0‖∞ . (12)Proof. We obtain a unique solution ϕε from a fixed point method on the Duhamel formulation of (5):ϕε(t, x, v) = ϕ0(x− tv) +∫ t0∫VM(v′)(1− eϕε(t−s,x−sv,v)−ϕε(t−s,x−sv,v′)ε)dv′ds, (13)4We obtain directly,∀ε > 0, 0 ≤ ϕε(t, x, v) ≤ ϕ0(x− tv) + t .This ensures that ϕε is uniformly bounded on [0, T ]×Rn×V . To prove the bound (9), we define ψεδ(t, x, v) =ϕε(t, x, v) − δt − δ4|x|2. For any δ > 0, ψεδ attains a maximum at point (tδ, xδ, vδ). Suppose that tδ > 0.Then, we have∂tϕε(tδ, xδ, vδ) ≥ δ, ∇xϕε(tδ, xδ, vδ) = 2δ4xδ.As a consequence, we have at the maximum point (tδ, xδ, vδ):0 ≥∫VM(v′)(1− eψεδ(tδ,xδ,vδ)−ψεδ(tδ ,xδ,v′)ε)dv′ ≥ δ + 2vδδ4xδ ≥ δ − 2Vmaxδ4|xδ| . (14)Moreover, the maximal property of (tδ, xδ, vδ) also implies‖ϕε‖∞ − δ4|xδ|2 ≥ ϕε(tδ, xδ, vδ)− δtδ − δ4|xδ|2 ≥ ϕε(0, 0, vδ) ≥ 0 .We obtain a contradiction with (14) since δ−6/(2Vmax) ≤ |xδ|2 ≤ δ−4‖ϕε‖∞ cannot hold for sufficientlysmall δ > 0. As a consequence tδ = 0, and we have,∀(t, x, v) ∈ [0, T ]× Rn × V, ϕε(t, x, v) ≤ ϕ0(xδ, vδ) + δt+ δ4|x|2 ≤ ‖ϕ0‖∞ + δt+ δ4|x|2.Passing to the limit δ → 0, we obtain (9). To find the bound (10), we use the same ideas on the differenceϕεh(t, x, v) = ϕε(t, x+ h, v)− ϕε(t, x, v). The equation for ϕεh reads as follows,∂tϕεh + v · ∇xϕεh =∫VM(v′)eϕε−ϕε′ε(1− eϕεh−ϕε′hε)dv′.Using the same argument as above with a δ−correction, we conclude that∀(t, x, v) ∈ [0, T ]× Rn × V, ϕεh(t, x, v) ≤ sup(x,v)∈R×V∣∣ϕ0(x+ h, v)− ϕ0(x, v)∣∣The same argument applies to −ϕεh,∂t (−ϕεh) + v · ∇x (−ϕεh) = −∫VM(v′)eϕε−ϕε′ε(1− e−(−ϕεh)−(−ϕε′h )ε)dv′.so that the r.h.s has the right sign when −ϕεh attains a maximum. Finally,∀(t, x, v) ∈ [0, T ]× Rn × V, |ϕεh(t, x, v)| ≤ sup(x,v)∈R×V∣∣ϕ0(x+ h, v)− ϕ0(x, v)∣∣ ≤∥∥∇xϕ0∥∥∞|h|.from which the estimate (10) follows.To obtain regularity in the velocity variable (11), we differentiate (5) with respect to v,(∂t + v · ∇x) (∇vϕε) = −gε(ϕε)∇vϕε −∇xϕε,where gε(ϕε) = 1ε∫VM(v′)eϕε−ϕε′ε dv′ ≥ 0. Multiplying by ∇vϕε|∇vϕε| , we obtain(∂t + v · ∇x) (|∇vϕε|) = −gε(ϕε)|∇vϕε| −(∇xϕε ·∇vϕε|∇vϕε|)(15)≤ ‖∇xϕ0‖∞ .from which we deduce (11) since ∇vϕ0 = 0 by hypothesis.Finally, the bound (12) is obtained similarly as the bound on ∇xϕε (10), using the difference ϕεs(t, x, v) =ϕε(t+ s, x, v)− ϕε(t, x, v). We obtain∀(t, x, v) ∈ [0, T ]× Rn × V, |ϕεs(t, x, v)| ≤ sup(x,v)∈R×V∣∣ϕε(s, x, v)− ϕ0(x, v)∣∣ .5We use the Duhamel formulation (13) to estimate the last contribution:∣∣ϕε(s, x, v)− ϕ0(x, v)∣∣ ≤ |ϕ0(x− sv)− ϕ0(x)|+ o(s) .The estimate (12) follows. Step 2. Viscosity solution procedure.From Proposition 2.1 we deduce that the familly (ϕε)ε is locally uniformly bounded inW1,∞ (R+ × Rn × V ).Then, from the Ascoli-Arzelà theorem, we can extract a locally uniformly converging subsequence. We denoteby ϕ0 the limit. Furthermore, from the fact that∫VM(v′)eϕε−ϕε′ε dv′ is uniformly bounded on [0, T ]×Rn×V ,we deduce that ϕ0 does not depend on v.Let ψ0 ∈ C2 (R+ × Rn) be a test function such that ϕ0 − ψ0 has a local maximum at(t0, x0). We wantto show that ψ0 is a subsolution of (6), yielding that ϕ0 is a viscosity subsolution [3]. The supersolutioncase can be performed similarly. Thereby, we define a corrective term η not depending on ε: ψε = ψ0 + εη.The corrector η is defined up to an additive constant. We choose the renormalization∫VM(v′)e−η′dv′ = 1.We define η as follows,∀(v, v′) ∈ V × V , eη(t,x,v) − eη(t,x,v′) = (v′ − v) · ∇xψ0(t, x) . (16)The corrector η is well defined. In fact, we can choose any v0 ∈ V and define eη(t,x,v) = µ0 + (v0 − v) ·∇xψ0(t, x). There is a unique positive µ0 = eη(t,x,v0) under the condition∫VM(v′)e−η′dv′ = 1.The uniform convergence ensures that ϕε − ψε has a maximum at (tε, xε, vε), where (tε, xε) is close to(t0, x0). As V is a bounded set, the sequence (vε) has an accumulation point, say v∗. We can extract asubsequence (without relabelling) such that (tε, xε, vε) → (t0, x0, v∗). We have at (tε, xε, vε):1− ∂tψε − vε · ∇xψε = 1− ∂tϕε − vε · ∇xϕε =∫VM(v′)eϕε−ϕε′ε dv′ .From the maximum property of (tε, xε, vε), the last inequality implies at this point :1− ∂tψε(tε, xε, vε)− vε · ∇xψε(tε, xε, vε) ≥∫VM(v′)eη(tε,xε,vε)−η(tε,xε,v′)dv′.Passing to the limit, we obtain at (t0, x0):1− ∂tψ0(t0, x0)− v∗ · ∇xψ0(t0, x0) ≥∫VM(v′)eη(t0,x0,v∗)−η(t0,x0,v′)dv′ = eη(t0,x0,v∗) .From the very definition of the corrector η (16), this writes also:∀v ∈ V , 1− ∂tψ0(t0, x0)− v · ∇xψ0(t0, x0) ≥ eη(t0,x0,v) .Therefore we obtain at point (t0, x0),∫VM(v)1− ∂tψ0(t0, x0)− v · ∇xψ0(t0, x0)dv ≤∫VM(v)e−η(t0,x0,v)dv = 1 .We conclude that ψ0 is a subsolution of (6).References[1] G. Barles, S. Mirrahimi and B. Perthame, Concentration in Lotka-Volterra parabolic or integral equa-tions: a general convergence result, Methods Appl. Anal. 16, 321–340 (2009).[2] E. Bouin, V. Calvez and G. Nadin, Hyperbolic traveling waves driven by growth, preprint (2011).6[3] M.G. Crandall, L.C. Evans and P.L. 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A kinetic eikonal equation

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