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Averaging

Lemmas and Dispersion for kinetic equations

Pierre-Emmanuel Jabin email: jabin@unice.fr

LaboratoireJ-ADieudonn´e

Universite´deNice

Parc Valrose, 06108 Nice Cedex 02

Estimates

Abstract.Averaging lemmas consist in a regularizing eﬀect on the average of the solution to a linear kinetic equation. Some of the main results are reviewed and their proofs presented in as self contained a way as possible. The use of kinetic formulations for the well posedness of scalar conservation laws is eventually explained as an example of application. Key words.Regularizing eﬀects, averaging lemmas, dispersion estimates, conservation laws. Mathematics Subject Classiﬁcation35B65, 82C40, 47G10.

Introduction

Kinetic equations are a particular case of transport equation in the phase space,i.e.on functionsf(x, v) of physicalandvelocity variables like

∂tf+v∙ rxf=g, t≥0 v, x,∈Rd.

As a solution to a hyperbolic equation, the solution cannot be more regular than the initial data or the right hand-side. However a speciﬁc feature of kinetic equations is that the averages in velocity, like ρ(t, xZRd ) =f(t, x, v)φ(v)dv,

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φ∈Cc∞(Rd),

are indeed more regular. This phenomenon is called velocity averaging. It was ﬁrst observed in [24] and then in [23] in aL2 ﬁnalframework. The Lpestimate was obtained in [17] (and slighty reﬁned in [3] to get a Sobolev space instead of Besov). The case of a full derivativeg=rx∙hwas treated in [45] and although it is in many ways a limit case, it is important for some applications as it can replace compensated compactness arguments. In addition to these works, this course presents and sometimes reformu-lates some of the results of [6], [17], [22], [31], [32], [36], [37], [45]. There are of course many other interesting contributions investigating averaging lemmas that are not quoted or only brieﬂy mentioned through the text.

1 Kinetic equations: Basic tools

1.1 A short introduction to kinetic equations

For a more complete introduction to kinetic equations and the basic theory, we refer to [6] or [21]. Many proofs are omitted here but are generally well known and not diﬃcult.

1.1.1 The basic equations

During most of this course, we will deal with the simplest equations

∂tf+α(v)∙ rxf=g(t, x, v), t∈R+, x∈Rd, v∈ω,(1.1) whereωis oftenRd(but might only be a subdomain); or with the stationary version α(v)∙ rxf=g(x, v), t∈R+, x∈O, v∈ω,(1.2) whereOis an open, regular subset ofRdandωis usually rather the sphere Sd−1 transport coeﬃcient. Theαwill always be regular, typically Lipschitz although here bounded would be enough. Of course (1.1) is really a subcase of (1.2) in dimensiond+ 1 and with α0(v) = (1, α(v)),O=R∗+×Rd,ω=ω. Neither (1.1) nor (1.2) have a unique solution as there are many solutions to ∂tf+α(v)∙ rxf= 0,

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for instance. Indeed for (1.1) an initial data must be provided f(t= 0, x, v) =f0(x, v),

(1.3)

and for (1.2) the incoming value offon the boundary must be speciﬁed f(x, v) =fin(x, v), x∈∂O, α(v)∙ν(x)≤0,(1.4)

whereν(x) is the outward normal toOatx. It is then possible to have existence and uniqueness in the space of dis-tributions Theorem 1.1Letf0∈ D0(Rd×ω)andg∈Ll1oc(R+,D0(Rd×ω)). Then there is a unique solution inLl1oc(R+,D0(Rd×ω))to(1.1)with(1.3)in the sense of distribution given by Zt−s, x−α(v)s, v)ds.(1.5) f(t, x, v) =f0(x−α(v)t, v) +g(t 0

Note that iffsolves (1.1) then for anyφ∈Cc∞(Rd×ω) ddtZf(t, x, v)φ(x, v)∈Lloc(R+), 1 Rd×ω sofhas a trace att= 0 in the weak sense and (1.3) perfectly makes sense. Proof.It is easy to check Ifthat (1.5) indeed gives a solution.fis another solution then deﬁne f¯=f−f0(x−α(v)t, v)−Z0tg(t−s, x−α(v)s, v)ds. Remark that ¯ ¯ ∂tf+α(v)∙ rxf= 0, ¯ ¯ and hence∂t(f(t, x+α(v)t, v) = 0 so thatf= 0. An equivalent result may be proved for (1.2) with the condition that the support of the singular part (inx) of the distributiongdoes not extend to the boundary∂O.

On the other hand, the modiﬁed equation, which we will frequently use, α(v)∙ rxf+f=g, x∈Rd, v∈ω,(1.6) is well posed in the wholeRdwithout the need for any boundary condition

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Theorem 1.2Letg∈ S0(Rd×ω), there exists a uniquef solution to(1.6). It is given by

) =Z0∞)t, v)e−tdt. f( gx, v(x−α(v

1.1.2 Liouville equation

The equation reads

inS0(Rd×ω)

(1.7)

∂tf+α(v)∙ rxf+F(t, x, v)∙ rvf= 0, t≥0, x∈Rd, v∈Rd,(1.8) whereF many applications, like the Vlasov-Maxwell Inis a given force ﬁeld. system 1.2,Fis in fact computed from the solutionf. Eq. (1.8) describes the dynamics of particles submitted to the forceF and as such is connected to the solution of the ODE

d X(x,vdtt,s,=)α(V(t, s, x, v))Vd,(,s,tv,xtd)=F(t, X, V), X(s, s, x, v) =x, V(s, s, x, v) =v,

(1.9)

which represents the trajectory of a particle starting with position and ve-locity (x, v) at timet=s. The ODE (1.9) is well posed for instance if loc α(v)∈W1,∞(Rd),1F+|∈x|Wl+1oc,|∞v(|)R+×R2d),(1.10) |α|+|F| ≤C(t) (,

thanks to Cauchy-Lipschitz Theorem. Weaker assumptions are however enough,Wl1,oc1and bounded divergence in [16] or evenBVlocin [1], but will not be required here. Under (1.10), (1.8) is also well posed

Theorem 1.3Assume(1.10)andrv∙F∈L∞(R+×R2d), for any mea-sure valued initial dataf0∈M1(R2d), there exists a uniquefincluded in L∞([0, T], M1(Rd))solution to(1.8)in the sense of distribution and satis-fying(1.3). It is given by f(t, x, v) =f0(X(0, t, x, v), V(0, t, x, v)).

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IfFandαare regular enough (C∞), the same theorem holds iff0is only a distribution. This theorem implies many properties onf, for example Proposition 1.1(i)f≥0if and only iff0≥0. (ii)Iff0∈L∞(R2d)thenf∈L∞(R+×R2d)and kf(t, ., .)kL∞(R2d)≤ kf0kL∞(R2d) (iii)Iff0∈Lp(R2d)thenf∈L∞([0, T], Lp(R2d))and kf(t, ., .)kLp(R2d)≤ kf0kLp(R2d)etkrvFk∞/p . From the point of view of averaging lemma, Eq. (1.8) does not have a particularly interesting structure. Indeed most of the time, the acceleration termF∙rvfwill be considered as a right hand side with no particular relation tof enough this is generally optimal.. Surprisingly

1.1.3 A simple case: local equilibrium

Let us consider (1.2) in the special case where

f(x, v) =ρ(x)M(v).

This might seem like an over simpliﬁcation but it will nevertheless provide many examples of optimality later on. For the moment we will be satisﬁed with a few remarks. We have M(v)α(v)∙ rxρ(x) =g. Let us hence writeg=M(v)h(x, v). Assuming thathis a regular function (L1∩L∞for example), this provides some regularity forρbut not necessarily in term of Sobolev spaces. Notice ﬁrst that some assumption is needed onα if there exists. Indeed a directionξ∈Sd−1such thatα(v) is colinear toξorαkξfor enoughv |{v∈Rd|α(v)kξ}| 6= 0,

and ifM(no matter how regular) then it is onlyis supported in this set possible to deduce from (1.2) that

ξ∙ rxρ∈L∞.

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Nothing can be said a priori about the derivatives in the other directions. Even ifα(v) is not concentrated along some directions likeα(v) =ξ, some assumption is needed onM. If not,Mitself may be concentrated along one directionξin which case the same phenomenon occurs. This shows the two features of all the averaging results that will be proved: Some assumption is needed on|{v∈Rd|α(v)kξ}|and the more regular in velocityfis, the more regularρis. In fact the regularity provided by averaging lemmas (in terms of Sobolev spaces) is in many situations not the optimal way of describing the regularity of solutions to (1.2) (see [10], [12] and [52] for example in the case of scalar conservation laws).

1.2 An application: The Vlasov-Maxwell system

The Vlasov-Maxwell system describes the evolution of charged particles and it reads

∂tf+v(p)∙rxf+(E(t, x)+v(p)×B(t, x))∙rpf= 0, t≥0, x, p∈Rd.(1.11) The ﬁeldsEandBare the electric and magnetic ﬁelds and are solutions to Maxwell equations

∂∂ttBE−+lcruclurBE=0=−,,jivdidBE==ρ,( .12) 1 v 0, whereρandjare the density and current of charged particles and therefore computed fromf ρ(t, x) =ZRdf(t, x, p)dp, j(t, x) =Zdv(p)f(t, x, p)dp.(1.13) R

Initial data are required for the system f(t= 0, x, p) =f0(x, p), E(t= 0, x) =E0(x), B(t= 0, x) =B0(x). (1.14) Finally the variableprepresents the impulsion of the particles. the classical In case (velocities of the particles much lower than the light speed), it is simply the velocity and v(p) =p.

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In the relativistic case, the velocity is related to the impulsion through v(p + (1) =|pp|2)1/2 .

For simplicity all physical constants were taken equal to 1. Globally in time and in dimension 3 and more, only the existence of solutions in the sense of distributions is known (and thus no uniqueness). This was proved in [15] and it is one of the ﬁrst examples of application of averaging lemmas. As usual one considers a sequence of classical solutionsfε,Eε,Bεto a regularized system. The form of this system is essentially unimportant as long as it has the same a priori estimates as (1.11)-(1.12). For (1.11) and from the analysis in 1.1.2, one ﬁrst has kfε(t, ., .)kLp(R2d)≤ kfε0kLp(R2d),∀t≥0,∀p∈[1,∞].(1.15) The only other available a priori estimate is the conservation of energy ZR2dE(p)fε(t, x, p)dx dp+ZRd(|Eε(t, x)|2+|Bε(t, x)|2)dx≤(1.16) Z2dE(p)fε0(x, p)dx dp+ZRd(|Eε0(x)|2+|Bε0(x)|2)dx. R

This relation is an inequality instead of an equality as the regularized system typically dissipates a bit. The termE(p) is equal to the usual kinetic energy |p|2in the classical case and to (1 +|p|2)1/2in the relativistic case. Therefore assuming that f0≥0, f0∈L1∩L∞(R2d),Z2dE(p)f0dx dp <∞, E0, B0∈L2(Rd), R (1.17) then the same bounds are uniformly true inεforfε(t, ., .),Eε(t, .) andB(t, .). On the other hand, we obviously have that ZRdρε(t, x) =ZR2dfε(t, x, p) =ZR2dfε0.(1.18)

In the relativistic case ZRd|jε(t, x)|dx≤ZR2d|fε(t, x, p)|= 7

ZR2dfε0,

(1.19)