About Lanford's theorem in the half-space with specular reflection
AAbout Lanford’s theorem in the half-space with specular reflection
Th´eophile Dolmaire ∗ February 11, 2021
Abstract
The present article proposes a rigorous derivation of the Boltzmann equation in the half-space. Weshow an analog of the Lanford’s theorem in this domain, with specular reflection boundary condition,stating the convergence in the low density limit of the first marginal of the density function of a system of N hard spheres towards the solution of the Boltzmann equation associated to the initial data correspondingto the initial state of the one-particle-density function.The original contributions of this work consist in two main points: the rigorous definition of the collisionoperator and of the functional space in which the BBGKY hierarchy is solved in a strong sense; andthe adaptation to the case of the half-space of the control of the recollisions performed by Gallagher,Saint-Raymond and Texier, which is a crucial step to obtain the Lanford’s theorem. Contents N hard spheres: the BBGKY hierarchy . . . . . . . . . 5 ‹ X N,ε, (cid:101) β, (cid:101) µ and ‹ X , (cid:101) β, (cid:101) µ B.1 Stability of the spaces ‹ X N,ε, (cid:101) β, (cid:101) µ α and ‹ X , (cid:101) β, (cid:101) µ α under the action of the operators E N,ε and E . 43B.2 The contracting property of the operators E N,ε and E . . . . . . . . . . . . . . . . . . . . . . 44 ∗ Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland, e-mail: [email protected] a r X i v : . [ m a t h . A P ] F e b Introduction
In 1872, starting from an atomistic description of matter, Boltzmann obtained an evolution equation for thedensity of the particles describing a rarefied gas, marking a milestone in kinetic theory (see [4]). In particular,he managed to grasp the irreversible trend of fluids of tending to equilibirum states. Although this is one ofthe strengths of his model, it was also at the origin of important debates: how could a reversible, microscopicdescription of a fluid generate irreversible behaviours on a macroscopic scale? Is this model trustable andrigorously obtained? The first rigorous derivation of the Boltzmann equation was obtained in 1973 by Lanford (see his pioneeringwork [13]), for a non trivial (albeit small) time interval, and was completed over time by Cercignani, Illnerand Pulvirenti ([8]), Gerasimenko and Petrina ([7]) and more recently by Pulvirenti, Saffirio and Simonellafor short-range potentials ([16]), and by Gallagher, Saint-Raymond and Texier for the hard sphere model in[10]. This derivation may be presented with the following statement, which is not completely formalized atthis step for the sake of simplicity.
Theorem 1 (Lanford’s theorem) . We consider a system of N particles of radius ε interacting via the hardsphere model, or via a radial, singular at and repulsive potential Φ ε supported in B (0 , ε ) which enables theparametrization of the scattering of particles by their deflection angle.Let f : R d → R + be a continuous density of probability such that, for some β > , µ ∈ R , we have: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f exp (cid:0) β | v | (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ( R dx × R dv ) ≤ exp( − µ ) . Let us assume that the N particles are initially identically distributed according to f , and “independent”.Then, there exists T > (depending only on β and µ ) such that, in the Boltzmann-Grad limit N → + ∞ , N ε d − = 1 , the distribution function of the particles converges to the solution of the Boltzmann equation: ∂ t f + v · ∇ x f = (cid:90) S d − × R d (cid:2) f ( v (cid:48) ) f ( v (cid:48)∗ ) − f ( v ) f ( v ∗ ) (cid:3) b ( v − v ∗ , ω ) d v ∗ d ω with initial data f , and with the cross-section b ( w, ω ) = [ ω · w ] + for the hard sphere interactions, or witha bounded cross-section depending implicitly on the potential Φ ε in the case of the interactions through arepulsive potential. So far, the previous theorem was stated only for domains without any boundary (namely, R d or T d ). Inthis work, we address the question of the rigorous derivation of the Boltzmann equation in the half-space,prescribing the specular reflection as boundary condition. We will start from the hard shere model, to obtainthe analog of Theorem 1 in our setting, which is Theorem 4, stated page 16. To the best of our knowledge,it is the first rigorous derivation of the Boltzmann equation in a domain with a boundary. Let d be an integer strictly larger than 1, N be a positive integer, and ε be a positive real number. Oneconsiders a system of N spherical particles of mass 1 and of radius ε/
2, evolving inside a domain of theEuclidean space R d , namely the half-space { x ∈ R d / x · e > } , where e denotes the first vector of thecanonical basis. The complement of the domain will be called an obstacle, denotes as Ω. The boundary { x ∈ R d / x · e = 0 } of the domain in which the particles evolve (which is of course also the boundary ofthe obstacle) will be called the wall . The notations for the state of the system of particles.
The position, respectively the velocity, attime t of the i -th particle (for 1 ≤ i ≤ N ) will be denoted x i ( t ), respectively v i ( t ). One will assume that the See for instance the criticisms from Loschmidt ([14]) and Zermelo ([21]). { x ∈ R d / x · e = 0 } , so that, if for all time t onecollects all the positions and velocities of the particles of the system to create the vector Z N defined as Z N ( t ) = (cid:0) x ( t ) , . . . , x N ( t ) , v ( t ) , . . . , v N ( t ) (cid:1) (such a vector is called a configuration of the system), then this vector Z N ( t ) has to lie in (cid:8) ( x , . . . , x N , v , . . . , v N ) ∈ R dN / ∀ ≤ i ≤ N, x i · e ≥ ε/ , and ∀ ≤ j < k ≤ N, | x j − x k | ≥ ε (cid:9) . (1)This part, defined by the expression (1), will be called the phase space for N hard spheres of radius ε/
2, andit will be denoted D εN .Sometimes, it will be useful to designate the position or the velocity of a certain particle i of a configuration Z N = ( x , v , . . . , x N , v N ). One will then use the notations Z X,iN = x i for the position of the particle i ,and Z V,iN = v i for its velocity. The collection of all the positions ( x , . . . , x N ) of the configuration Z N willbe denoted Z XN = ( x , . . . , x N ), and the collection of all the velocities ( v , . . . , v N ) of Z N will be denoted Z VN = ( v , . . . , v N ). The dynamics inside the phase space.
Inside this phase space D εN , one will prescribe the most simpledynamics: the particles will travel in straight lines, conserving a constant velocity. In other words, far fromthe boundary they are subject to the free flow. Interaction with the boundary of the domain: the specular reflection.
When a particle reachesthe boundary, that is when there exists a time t b and an integer 1 ≤ i ≤ N such that x i ( t b ) · e = ε/
2, thevelocity of the particle has to be changed in order to keep this particle inside the domain { x · e ≥ ε/ } for times larger than t b . The law of reflection chosen here will be the specular reflection, also known as theSnell-Descartes law, which takes here a very simple form, namely: v i ( t + b ) = v i ( t − b ) − (cid:0) v i ( t − b ) · e (cid:1) e . (2)Figure 1: A representation of the specular reflection against an obstacle Ω. Interaction between the particles: the hard sphere model.
The particles are assumed to be spherical,with a non zero diameter ε . In addition to the dynamics prescribed by the free flow and the bouncings againstthe wall, one has to impose another change of velocity when two particles are about to overlap. For twoparticles at x and x that are about to overlap (that is such that x = x + εω with ω ∈ S d − ) with respectivepre-collisional velocities v and v ∗ (that is such that ( x − x ) · ( v ∗ − v ) = ω · ( v ∗ − v ) < ß v (cid:48) = v − ( v − v ∗ ) · ωω,v (cid:48)∗ = v ∗ + ( v − v ∗ ) · ωω (3)(that is one has ( x − x ) · ( v (cid:48)∗ − v (cid:48) ) = ω · ( v (cid:48)∗ − v (cid:48) ) > | v (cid:48) | / | v (cid:48)∗ | / | v | / | v ∗ | / v (cid:48) + v (cid:48)∗ = v + v ∗ . For ω ∈ S d − fixed, the mappingthat associates to the pair of velocities ( v, v ∗ ) the new pair( v (cid:48) , v (cid:48)∗ ) = (cid:0) v − ( v − v ∗ ) · ωω, v ∗ + ( v − v ∗ ) · ωω (cid:1) , scattering mapping , is an involution and sends the pre-collisional velocities ( v, v ∗ ) (suchthat ω · ( v ∗ − v ) <
0) onto the post-collisional velocities ( v (cid:48) , v (cid:48)∗ ) (such that ω · ( v (cid:48)∗ − v (cid:48) ) > . Let us focus here on an important problemconcerning this transport: for a given system of particles, let us call an event a time such that a particlebounces against the obstacle or collide with another particle. Depending on the initial configuration, twoevents involving the same particle can occure at the same time, leading to an ill-defined dynamics. Here arethe possible cases.We first deal with the case of a bouncing against the obstacle: thanks to the convexity of the obstacle (in factany obstacle with a bounded curvature provides the same property up to assume that the size of the particlesis small enough), none of the initial configurations could lead to a situation in which a particle touches atthe same time the obstacle at two or more different points. However, some initial configurations could leadto a situation in which a particle bounces against the obstacle, and collides with at least another particle atthe same time.One should also consider an initial configuration leading to a situation in which a particle collides with twoother particles (or more) at the same time. There is no other kind of simultaneous events involving the sameparticle when the obstacle is the half-space, but in those cases, two velocities or more are assigned to the sameparticle at the time of the considered events, we call pathological a trajectory for which the dynamics becomesill-defined due to this phenomenon at some point. Let us then study the initial configurations leading to apathological trajectory. Proposition 1 (Rigorous definition of the hard sphere dynamics almost everywhere, globally in time) . Let N be an integer larger than and < ε ≤ be a positive number. Then the two following assertions hold. • The set of initial configurations Z N in the phase space D εN for N hard spheres of radius ε/ leading toa pathological trajectory during the time interval R + is of measure zero. • For every initial configuration Z N in the phase space D εN , one considers the subset E ( Z N ) of R + composed of all the times of the events of this dynamics in the largest time interval where it is well-defined. Then for any initial configuration Z N outside a subset of the phase space D εN of measure zero, E ( Z N ) is a discrete set. The proof of this result, originally published in [1] , is presented in a modern way in [10] for the casewithout obstacle. The case of the half-space is presented in detail in [9] but for the sake of completeness, thereader may find a shortened proof in appendix, page 41. Remark 1.
This result shows that the hard sphere dynamics is globally defined on time, for almost everyinitial configurations of particles, and the accumulation of events cannot happen except for a subset of initialdata of zero measure. See in particular Appendix 4.A “More About Hard-Sphere Dynamics”. This reference deals with the most general setting possible, taking into account a wide variety of obstacles. See Proposition 4.1.1 page 28. efinition 1 (Hard sphere transport) . For any positive integer N and any positive real number ε , we definethe hard sphere transport of N particles of radius ε/ as the map: ( t, Z N ) (cid:55)→ T N,εt ( Z N ) , defined for almost every Z N ∈ D εN (according to Proposition 1) and any time t ∈ R , where T N,εt ( Z N ) is theconfiguration starting from Z N and obtained after following the hard sphere dynamics for a time t . N hard spheres: the BBGKY hierarchy In statistical physics the central object turns out to be the density function of the system of particles, whichrepresents the probability, along time, of finding the system of N particles in a given state. The relevantinformation will be obtained as moments of this function (which corresponds to a ”mean information”).Another very important object in the following, derived from the density function, will be the family itsmarginals, that is the integral with respect to some of the variables of the density. In particular, the firstmarginal, which is obtained as the integral with respect to all of the variables except the position and thevelocity of the first particle of the system, represents the mean behaviour of a single particle of the system.One will recall in this section the key observation due to Bogolyubov, Born, Green, Kirkwood and Yvon (seerespectively [3], [5], [12] and [20]), proving that it is possible to link together those marginals. This link, beingknown as the BBGKY hierarchy , provides a crucial family of equations, deeply linked with the Boltzmannequation (see [11], [6], [8] and [10]), and which will be the central object of study of the present work.
The distribution function of a system of N hard spheres. We denote f N ( t, Z N ) = f N ( t, x , v , . . . , x N , v N ) , the density function of the system of N hard spheres. In other words, at time t , and for A ⊂ D εN ⊂ R dx × R dv × · · · × R dx N × R dv N measurable, the quantity (cid:90) A f N ( t, x , v , . . . , x N , v N ) d x d v . . . d x N d v N represents the probability of finding the system in a configuration belonging to the subpart A of the phasespace.We also introduce the following boundary conditions, which incode the hard spheres dynamics, according tothe introductive Sections 2.1 and 2.1: Definition 2 (Boundary condition for the hard sphere dynamics) . Let s be a positive integer and ε be apositive number. One defines the boundary condition for the hard sphere dynamics of s particles of radius ε/ as the map defined on the boundary D εs of the phase space into itself as χ εs : ß ∂ D εs → ∂ D εs ,Z s (cid:55)→ χ εs ( Z s ) , with (cid:0) χ εs ( Z s ) (cid:1) X = Z Xs (the map does not act on the positions of the configurations) and such that: • if for some ≤ i < j ≤ s , one has | x i − x j | = ε and | x k − x l | > ε for all ( k, l ) (cid:54) = ( i, j ) (a single collisionhappens, between the two particles i and j ), while in addition x k · e > ε/ for all ≤ k ≤ s , one defines: (cid:0) χ εs ( Z s ) (cid:1) V,i = v i − (cid:0) /ε (cid:1)(cid:0) ( v i − v j ) · ( x i − x j ) (cid:1) ( x i − x j ) , (cid:0) χ εs ( Z s ) (cid:1) V,j = v j + (cid:0) /ε (cid:1)(cid:0) ( v i − v j ) · ( x i − x j ) (cid:1) ( x i − x j ) , (cid:0) χ εs ( Z s ) (cid:1) V,k = v k for all ≤ k ≤ s, k (cid:54) = i, k (cid:54) = j. • if for some ≤ i ≤ s , one has x i · e = ε/ and x j · e > ε/ for all j (cid:54) = i (a single particle bounces againstthe wall), while in addition | x k − x l | > ε for all ≤ k < l ≤ s , one defines: ® (cid:0) χ εs ( Z s ) (cid:1) V,i = v i − v i · e e , (cid:0) χ εs ( Z s ) (cid:1) V,j = v j for all ≤ j ≤ s, j (cid:54) = i. f N is a solution of the Liouville equation : ∀ t ≥ , ∀ Z N ∈ D εN , ∂ t f N ( t, Z N ) + N (cid:88) i =1 v i · ∇ x i f N ( t, Z N ) = 0 , (4)with boundary conditions: ∀ t ≥ , ∀ Z N ∈ ’ ∂ D εN in = (cid:0) (cid:83) ≤ i
2: this is also an incoming configuration), then χ εs sends theincoming configurations of B in i,j and C in i onto the outgoing configurations of B out i,j and C out i respectively, andconversely, where B out i,j = (cid:110) | x i − x j | = ε, | x k − x l | > ε for all ( k, l ) (cid:54) = ( i, j ) , x k · e > ε/ k and ( x i − x j ) · ( v i − v j ) < (cid:111) and C out i = (cid:110) x i · e = ε/ , x j · e > ε/ j (cid:54) = i, | x k − x l | > ε for all ( k, l ) and v i · e < (cid:111) . From the Liouville equation to the BBGKY hierarchy.
Following the computation (which is now aclassic) that can be found originally in [11], or in [8] and [10] for a more modern presentation, we can derivean equation verified by the marginals of the distribution function of the hard sphere system. Concerning theparticularities appearing due to the presence of the wall, the reader may refer to [9].We can show that the marginals f ( s ) N of the distribution function solve the following equation on R + × D εs : ∂ t f ( s ) N + s (cid:88) i =1 v i · ∇ x i f ( s ) N = C N,εs,s +1 f ( s +1) N , (6)6here C N,εs,s +1 , called the s-th collision operator , denotes C N,εs,s +1 f ( s +1) N ( t, Z s ) = s (cid:88) i =1 ( N − s ) ε d − (cid:90) S d − × R d ω · ( v s +1 − v i ) f ( s +1) N ( t, Z s , x i + εω, v s +1 ) d ω d v s +1 . (7)This generic equation (for 1 ≤ s ≤ N −
1) constitutes the so-called
BBGKY hierarchy . Nevertheless, we willnot use this version of the BBGKY hierarchy, that has to be considered with the analog of the boundaryconditions (5), namely: ∀ ≤ s ≤ N, ∀ t ≥ , ∀ Z s ∈ ‘ ∂ D εs in , f ( s ) N ( t, Z s ) = f ( s ) N ( t, χ εs ( Z s )) . (8)We will use instead an integrated with respect to time version, which is on the one hand more self-contained(since it contains the boundary conditions), and which will be also more convenient to deal with the fixedpoint argument. This version, equivalent up to assume enough regularity of the solutions, writes f ( s ) N ( t, Z s )) = (cid:0) T s,εt f ( s ) N (0 , · ) (cid:1) ( Z s ) + (cid:90) t T s,εt − u C N,εs,s +1 f ( s +1) N ( u, Z s ) d u, (9)where T s,εt denotes the backwards hard sphere flow, defined using the transport introduced in Definition 1: (cid:0) T s,εt f (cid:1) ( Z s ) = f (cid:0) T s,ε − t ( Z s ) (cid:1) . The formal limit of the BBGKY hierarchy when ε → . One presents now briefly the bridge builtby Grad in [11] between the BBGKY hierarchy and the Boltzmann equation, which is now a famous step inthe derivation. One can refer to [6], [8] or again [10] for more details.The first step consists in a change of variable in the pre-collisional configurations in the collision term ofthe BBGKY hierarchy, that is one will rewrite the integral in order to integrate only over pre-collisionalconfigurations. One writes: C N,εs,s +1 f ( s +1) N ( t, Z s ) = s (cid:88) i =1 ( N − s ) ε d − (cid:90) S d − × R d ω · ( v s +1 − v i ) f ( s +1) N ( t, Z s , x i + εω, v s +1 ) d ω d v s +1 = s (cid:88) i =1 ( N − s ) ε d − (cid:104) (cid:90) ω · ( v s +1 − v i ) < + (cid:90) ω · ( v s +1 − v i ) > (cid:105) ω · ( v s +1 − v i ) × f ( s +1) N ( t, Z s , x i + εω, v s +1 ) d ω d v s +1 = s (cid:88) i =1 ( N − s ) ε d − (cid:90) ω · ( v s +1 − v i ) > (cid:2) ω · ( v s +1 − v i ) (cid:3) + × (cid:16) f ( s +1) N ( t, Z s , x i + εω, v s +1 ) − f ( s +1) N ( t, Z s , x i − εω, v s +1 ) (cid:17) d ω d v s +1 , where the last line is obtained after performing the change of variables ω → − ω in the first term constitutedof the pre-collisional velocities.Now we can use the boundary conditions verified by the marginal f ( s +1) N in order to remove the post-collisionalarguments in the integrand, replacing f ( s +1) N ( t, Z s , x i + εω, v s +1 ) by f ( s +1) N ( t, x , x , . . . , x i , v (cid:48) i , . . . , x i + εω, v (cid:48) s +1 ).Finally, taking now formally the limit ε →
0, and up to assume that
N ε d − →
1, we find the limiting collisionoperator: C s,s +1 f ( s +1) ( t, Z s ) = s (cid:88) i =1 (cid:90) S d − × R d (cid:2) ω · ( v s +1 − v i ) (cid:3) + × (cid:16) f ( s +1) ( t, x , v , . . . , x i , v (cid:48) i , . . . , x i , v (cid:48) s +1 ) − f ( s +1) ( t, Z s , x i , v s +1 ) (cid:17) d ω d v s +1 , where [ x ] + denotes the nonnegative part of x ∈ R , that is [ x ] + = x if x ≥
0, and [ x ] + = 0 if x < oltzmann hierarchy , which writes ∂ t f ( s ) + s (cid:88) i =1 v i · ∇ x i f ( s ) = C s,s +1 f ( s +1) . (10)As for the BBGKY hierarchy, the integrated version of the Boltzmann hierarchy, which follows, will be themost useful thereafter: f ( s ) ( t, Z s )) = (cid:0) T s, t f ( s ) (0 , · ) (cid:1) ( Z s ) + (cid:90) t T s, t − u C s,s +1 f ( s +1) ( u, Z s ) d u, (11)where T s, t denotes the backwards free flow with the specular boundary conditions, defined using the associ-ated free transport, that is (cid:0) T s, t f (cid:1) ( Z s ) = f (cid:0) T s, − t ( Z s ) (cid:1) . The Boltzmann equation as the first equation of the Boltzmann hierarchy for tensorized func-tions.
The Boltzmann hierarchy is called this way because it is deeply linked with the Boltzmann equation.To be more accurate, if the second unknown f (2) of the sequence of solutions ( f ( s ) ) s ≥ of the Boltzmann hier-archy is the tensorization of the first unknown f (1) , that is if f (2) ( t, x , v , x , v ) = f (1) ( t, x , v ) f (1) ( t, x , v ),then f (1) solves the Boltzmann equation.Conversely, if f is a solution of the Boltzmann equation, then the sequence of its tensorizations ( f ( s ) ) s ≥ =( f ⊗ s ) s ≥ provides a solution of the Boltzmann hierarchy. This remark, together with the formal derivation ofthe Boltzmann hierarchy from the BBGKY hierarchy suggests then an interesting plan to obtain a derivationof the Boltzmann equation itself.The assertion that the quantity N ε d − stays constant when N goes to infinity is called the Boltzmann-Grad limit, introduced by Grad in his pioneering work [11] casting for the first time the bridge described betweenthe BBGKY hierarchy and the Boltzmann equation. Physically, it means that the mean free path of a par-ticle remains constant. It also implies that the volume
N ε d occupied by the particles is going to zero as thenumber of the particles increases, hence one usually calls this condition the low density limit .Concerning now the different steps providing formally the Boltzmann hierarchy, it is quite clear that theirorder crucially matters: if we had performed the limit ε → before using the boundary condition f ( s +1) ( . . . , v (cid:48) i , . . . , v (cid:48) s +1 ) = f ( s +1) ( . . . , v i , . . . , v s +1 ) , then the collision term would have been simply 0, that is we would have recovered the free transport in thelimit. In addition, we decided to remove the post-collisional arguments in the integrand: this can be formallymotivated by the fact that the equation involves a backwards in time transport, so when two particles collide,it is important to give the pre-collisional velocities associated to a post-collisional pair in order to be able toreconstruct the path of the particles backwards. If we had removed the pre-collisional arguments instead, wewould have recover the opposite of the collision term, and then the backwards in time Boltzmann equation.Finally, and even if it was already mentionned several times, all those manipulations and the hierarchiesobtained are only formal so far. A first important challenge is to give a rigorous sense to those objects, whichis the purpose of the following section. The free transport with specular reflexion preserves the continuity, and then the Boltzmann hierarchy makessense for continuous functions. But the hard sphere transport is only defined almost everywhere, so we haveto deal with another set of functions. Let us then here study the BBGKY hierarchy, that has to make sensefor Lebesgue functions.The general formula (9), using the collision operator described by (7), is based on an integration on a manifoldwith a positive codimension in the phase space. Indeed, if we focus only on the collision term (7) (and forgetfor the moment about the integration in time), we see that this collision term is obtained by integrating overthe variables ω ∈ S d − and v s +1 ∈ R d . Since the trace of a Lebesgue function is not well defined in general,8ne will need an additional result to give a sense to this term.The problem is for the first time mentionned (and addressed) in [10], we will here only sketch the main stepsof the solution. However, the presentation of the rigorous definition of this term in [10] is quite fast, one mayrefer to [9] for a more detailed proof, which leads to the following result. For the sake of completeness, theproof is sketched below. Theorem 2 (Definition of the collision operator of the BBGKY hierarchy for functions of C (cid:0) [0 , T ] , L ∞ ( D εs +1 ) (cid:1) decaying sufficiently fast at infinity in the velocity variables) . Let s be a positive integer, ε and T be twopositive numbers.Let in addition g s +1 : [0 , T ] × R + → R + be a function verifying: • ( t, x ) (cid:55)→ g s +1 ( t, x ) is measurable and almost everywhere positive, • for all x ∈ R + , the function t (cid:55)→ g s +1 ( t, x ) is increasing, • for all t ∈ [0 , T ] and almost every ( v , . . . , v s ) ∈ R ds , the function v s +1 (cid:55)→ (cid:12)(cid:12) V s +1 (cid:12)(cid:12) g s +1 (cid:0) t, (cid:12)(cid:12) V s +1 (cid:12)(cid:12)(cid:1) is integrable on R d , • for all t ∈ [0 , T ] , the function ( v , . . . , v s ) (cid:55)→ (cid:90) R d (cid:12)(cid:12) V s +1 (cid:12)(cid:12) g s +1 (cid:0) t, (cid:12)(cid:12) V s +1 (cid:12)(cid:12)(cid:1) d v s +1 is bounded almost everywhere,and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R d | V s +1 |≥ R (cid:12)(cid:12) V s +1 (cid:12)(cid:12) g s +1 (cid:0) t, (cid:12)(cid:12) V s +1 (cid:12)(cid:12)(cid:1) d v s +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ([0 ,T ] ,L ∞ ( D εs +1 )) converges to zero as R goes to infinity.Then, for every integer ≤ i ≤ s , and for any function h ( s +1) ∈ C (cid:0) [0 , T ] , L ∞ ( D εs +1 ) (cid:1) such that there exists λ ∈ R + such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ( s +1) ( t, Z s +1 ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ([0 ,T ] ,L ∞ ( D εs +1 )) ≤ λ (cid:12)(cid:12)(cid:12)(cid:12) g s +1 (cid:0) t, (cid:12)(cid:12) V s +1 (cid:12)(cid:12)(cid:1)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ([0 ,T ] ,L ∞ ( D εs +1 )) , the function C εs,s +1 , ± ,i T s +1 ,εt h ( s +1) is a well defined element of L ∞ (cid:0) [0 , T ] × D εs +1 (cid:1) , and one has almost ev-erywhere on [0 , T ] × D εs : (cid:12)(cid:12)(cid:12) C εs,s +1 , ± ,i T s +1 ,εt h ( s +1) ( t, Z s ) (cid:12)(cid:12)(cid:12) ≤ λε d − (cid:12)(cid:12) S d − (cid:12)(cid:12) (cid:90) R d (cid:0) | v i | + | v s +1 | (cid:1) g s +1 (cid:0) t, (cid:12)(cid:12) V s +1 (cid:12)(cid:12)(cid:1) d v s +1 . Sketch of proof of Theorem 2.
Let us assume, in order to simplify the presentation, that the function f ( s +1) N on which the collision term (7) is acting does not depend on time, and let focus only on the second term ofthe collision operator, which does not involve scattering.The first ingredient is the Fubini theorem: for a function f : X × Y → R which is integrable for the productmeasure d x ⊗ d y on X × Y , one knows that y (cid:55)→ (cid:90) X f ( x, y ) d x is defined almost everywhere and is integrable with respect to the measure d y on Y . In other words, thistheorem can be seen as a way to define traces in some particular cases.A function f ( s +1) N defined on the phase space D εs +1 (with s + 1 particles), which is composed with the map( Z s , ω, v s +1 ) (cid:55)→ ( Z s , x i + εω, v s +1 ), defined on R ds × S d − × R d and taking its values in R d ( s +1) , andintegrated with respect to the variables ω and v s +1 , depends in the end on the variable Z s . So in orderto apply the Fubini theorem in this case, one should integrate again with respect to this last variable Z s .However it is not enough, since if f ( s +1) N is a measurable function defined on D εs +1 , the integration variables Z s , ω and v s +1 cover only a manifold of codimension 1.The second important idea is then to compose f ( s +1) N with the hard sphere transport for s + 1 particles,depending of course on the configuration of the system of those s + 1 hard spheres, but also on an additionalparameter: the time u . This will play the role of the missing variable in the integration.9n the end, this insertion of a transport operator inside the collision term will lead us to consider a ”shifted”in time version of the BBGKY hierarchy, called the conjugated BBGKY hierarchy , on which the regularityresults can be stated : if one denotes h ( s ) N ( Z s +1 ) = f ( s ) N (cid:0) T s +1 ,εu ( Z s +1 ) (cid:1) , that is h ( s +1) N = T s +1 ,ε − u f ( s +1) N , then (formally) the s -th equation of the BBGKY hierarchy holds if and only if h ( s +1) N ( t, · ) = f ( s +1) N (0 , · ) + (cid:90) t T s,ε − u C N,εs,s +1 T s +1 ,εu h ( s +1) N ( u, · ) d u. (12)In the end, the collision operator will not be defined alone, but composed with the hard sphere transport.That is why we will talk about the transport-collision operator of the BBGKY hierarchy in the sequel, andnot about the collision operator only.The very last step, which is of technical order, but which leads to a lot of work, is a series of restrictionsthat have to be relaxed one by one (see [9]). To be more explicit, we start by decomposing the collisionoperator into elementary terms (each one concerning a single particle 1 ≤ i ≤ s chosen to collide, andeach one being either in a pre- or in a post-collisional configuration according to the adjunction parameters( ω, v s +1 )), that is we write C N,εs,s +1 = ( N − s ) s (cid:88) i =1 (cid:0) C εs,s +1 , + ,i − C εs,s +1 , − ,i (cid:1) where C εs,s +1 , ± ,i h ( s +1) = ε d − (cid:90) S d − ω × R dvs +1 (cid:0) ω · ( v s +1 − v i (cid:1) ± h ( s +1) ( Z s , x i + εω, v s +1 ) d ω d v s +1 (here again, this last term is only formally introduced for h ( s +1) being a L p function), and we introduce threecut-off parameters δ , R and R restricting the domain of integration such that • | x j − x k | > ε + √ δR for all 1 ≤ j < k ≤ s + 1 with ( j, k ) (cid:54) = ( i, s + 1), • x l · e > ε/ δR for all 1 ≤ l ≤ s + 1, • | X s +1 | = | ( x , x , . . . , x i , . . . , x i + εω ) | ≤ R , • | V s +1 | = | ( v , . . . , v s , v s +1 ) | ≤ R .On this restricted domain of integration, that will be denoted D εs +1 ( δ, R , R ), if the pair of particles( x i , v i ) and ( x i + εω, v s +1 ) is in a pre-collisional configuration, for δ small enough (depending on R and R )the backwards hard sphere transport coincides with the free transport for small times, that is for all 0 ≤ t ≤ δ one has T s +1 ,ε − t ( Z s , x i + εω, v s +1 ) = ( X s − tV s , V s , x i + εω − tv s +1 , v s +1 ) , and since the function S − s +1 = ß D εs × [0 , δ ] × S d − × R d → R d ( s +1) ( Z s , t, ω, v s +1 ) (cid:55)→ ( X s − tV s , x i + εω − tv s +1 , v s +1 )is such that its Jacobian determinant has an absolute value equal to ε d − (cid:12)(cid:12) ω · ( v s +1 − v i ) (cid:12)(cid:12) , In [10], this hierarchy is said to be “more regular”, but it is actually the only one on which one can properly work.
10e can at last define the pre-collisional elementary terms of the (truncated) transport-collision operator C εs,s +1 , − ,i ( δ, R , R ) T s +1 ,ε using the formula (cid:90) S − s +1 ( D εs +1 ( δ,R ,R )) h ( s +1) ( Z s +1 ) d Z s +1 = (cid:90) δ (cid:90) D εs (cid:90) S d − × R d D εs +1 ( δ,R ,R ) ε d − (cid:0) ω · ( v s +1 − v i ) (cid:1) − × h ( s +1) (cid:0) S − s +1 ( Z s , x i + εω, v s +1 ) (cid:1) d ω d v s +1 d Z s d t = (cid:90) δ (cid:90) D εs C εs,s +1 , − ,i ( δ, R , R ) T s +1 ,ε h ( s +1) ( t, Z s ) d Z s d t, for h ( s +1) a L ∞ function of the phase space D εs +1 .One sees that there is a restriction on the time interval, which can be relaxed (that is one can define thetruncated transport-collision operator on any time interval [0 , T ]) thanks to the conservation of the L ∞ normby the hard sphere transport and a decomposition of any time interval into sub-intervals of length ≤ δ . Thepost-collisional terms C εs,s +1 , + ,i are defined in the same way, replacing only the mapping S − s +1 by S + s +1 , whichis defined as the scattering mapping with S − s +1 .Finally, we relax first the cut-off in the time variable (the parameter δ ). We can show that the se-quence (cid:0) C εs,s +1 , ± ,i ( δ, R , R ) T s +1 ,εt h ( s +1) (cid:1) δ converges strongly in L as δ → C εs,s +1 , ± ,i ( R , R ) T s +1 ,εt h ( s +1) , which is also L ∞ , and the convergence holds also in the weak sense in L ∞ .The cut-off in the position variable (the parameter R ) can then be relaxed: (cid:0) C εs,s +1 , ± ,i ( R , R ) T s +1 ,εt h ( s +1) (cid:1) R converges almost everywhere as R → + ∞ towards a limit, denoted C εs,s +1 , ± ,i ( R ) T s +1 ,εt h ( s +1) , which is a L ∞ function (with a supremum which depends on R ).To counter-balance the growth in R of the L ∞ norm of C εs,s +1 , ± ,i ( R ) T s +1 ,εt h ( s +1) , one has to impose adecrease in the velocity variable for h ( s +1) . The condition, explicited in Theorem 2 below, is quite strong,but we can notice that among the few functions verifying this condition can be found the gaussians. With thiscondition, the sequence (cid:0) C εs,s +1 , ± ,i ( R ) T s +1 ,εt h ( s +1) (cid:1) R is a Cauchy sequence in L ∞ , and then it is convergingas R → + ∞ .This long process provides the rigourous definition of the transport-collision operator for the BBGKY hier-archy, up to consider it acting on the set of functions described in the theorem. This concludes the sketch ofproof of Theorem 2. To the best of our knowledge, the previous theorem provides the most general setting in which the collisionoperator of the BBGKY hierarchy makes sense when it acts on functions (and not on distributions). However,it does not answer the question on the functional setting in which the BBGKY hierarchy is rigourouslydefined: a solution of the BBGKY hierarchy is a family of functions (cid:0) f ( s ) N (cid:1) s such that (formally) f ( s ) N = T s,εt f ( s ) N, + (cid:82) t T s,εt − u C N,εs,s +1 f ( s +1) N d u , for all 1 ≤ s ≤ N −
1. In other words, we need to define families offunctional spaces, that are on the one hand consistent with the action of the collision operator, and on theother hand consistent with the low density limit, in order to be able to compare the solutions of the twohierarchies in the end.Let us then introduce first the relevant functional spaces containing individually the elements f ( s ) N and f ( s ) of the solutions of the two hierarchies, and then complete the introduction of the functional setting with thespaces containing the whole sequences ( f ( s ) N ) ≤ s ≤ N and ( f ( s ) ) s ≥ . Definition of the spaces X ε,s,β and X ,s,β , the functions of the phase space of s particles boundedby a gaussian in the velocity variables. One starts with the definition of the first kind of functionalspace, in which each marginal will lie. The main difference between the spaces for the BBGKY and theBoltzmann hierarchies, except of course the domain of definition, is the continuity of the functions.11 efinition 3 (Norms | · | ε,s,β and | · | ,s,β , spaces X ε,s,β and X ,s,β ) . Let ε and β > be two strictly positivenumbers and s be a positive integer. For any function h ( s ) belonging to L ∞ (cid:0) D εs (cid:1) , one defines: | h ( s ) | ε,s,β = sup ess Z s ∈D εs (cid:34)(cid:12)(cid:12) h ( s ) ( Z s ) (cid:12)(cid:12) exp (cid:32) β s (cid:88) i =1 | v i | (cid:33)(cid:35) , and the space X ε,s,β as the space of the functions of L ∞ (cid:0) D εs (cid:1) with a finite | · | ε,s,β norm, that is: X ε,s,β = (cid:110) h ( s ) ∈ L ∞ ( D εs ) / | h ( s ) | ε,s,β < + ∞ (cid:111) . For any function f ( s ) belonging to C (cid:0)(cid:0) Ω c × R d (cid:1) s (cid:1) , one defines: | f ( s ) | ,s,β = sup Z s ∈ (Ω c × R d ) s (cid:34)(cid:12)(cid:12) f ( s ) ( Z s ) (cid:12)(cid:12) exp (cid:32) β s (cid:88) i =1 | v i | (cid:33)(cid:35) , and the space X ,s,β as the space of the continuous functions vanishing at infinity defined on (cid:0) Ω c × R d (cid:1) s witha finite | · | ,s,β norm, that is: X ,s,β = (cid:8) f ( s ) ∈ C (cid:0)(cid:0) Ω c × R d (cid:1) s (cid:1) / | f ( s ) | ,s,β < + ∞ (cid:9) , and satisfying the following boundary condition f ( s ) ( Z s ) = f ( s ) ( χ s ( Z s )) for all Z s belonging to the boundaryof (cid:0) Ω c × R d (cid:1) s , that is such that there exists at least an integer ≤ i ≤ s such that x i · e = 0 and v i · e > . Definition of the spaces X ε,β,µ α and X ,β,µ α , the sequence of functions of X · ,s,β with an exponen-tial weight with respect to the number of particles. Now that we introduced the functional spacesin which each of the marginals will lie, let us introduce a structure on the sequence of such spaces, with, inaddition to a real parameter µ , which is the activity of the solution from a physical point of view, anotherparameter α , strictly positive, and which will be taken equal to 1 or 2 in the sequel. The choice of thisparameter α plays a role in the definition of the continuity in time introduced in the final spaces, introducedbelow. Definition 4 (Norms ||·||
N,ε,β,µ α and ||·|| ,β,µ α , spaces X N,ε,β,µ α and X ,β,µ α ) . Let N be a positive integer.Let ε an β be two strictly positive numbers, µ be a real number and α > be a strictly positive number. Forany finite sequence H N = (cid:0) h ( s ) N (cid:1) ≤ s ≤ N of functions h ( s ) N of X ε,s,β , one defines : || H N || N,ε,β,µ α = max ≤ s ≤ N (cid:16) | h ( s ) N | ε,s,β exp( s α µ ) (cid:17) , and the space X N,ε,β,µ α as the space of the finite sequences H N = (cid:0) h ( s ) N (cid:1) ≤ s ≤ N such that for every ≤ s ≤ N , h ( s ) N belongs to X ε,s,β , and such that the sequence (cid:0) h ( s ) N (cid:1) ≤ s ≤ N has a finite ||·|| N,ε,β,µ α norm, that is : X N,ε,β,µ α = (cid:110) H N = (cid:0) h ( s ) N (cid:1) ≤ s ≤ N ∈ (cid:0) X ε,s,β (cid:1) ≤ s ≤ N / || H N || N,ε,β,µ α < + ∞ (cid:111) . Similarly, for any infinite sequence F = (cid:0) f ( s ) (cid:1) s ≥ of functions f ( s ) of X ,s,β , one defines : || F || ,β,µ α = sup s ≥ (cid:16) | f ( s ) | ,s,β exp( s α µ ) (cid:17) , and the space X ,β,µ α as the space of the infinite sequences (cid:0) f ( s ) (cid:1) s ≥ such that for every s ≥ , f ( s ) belongsto X ,s,β , and such that the sequence (cid:0) f ( s ) (cid:1) s ≥ has a finite ||·|| ,β,µ α norm, that is : X ,β,µ α = (cid:110) F = (cid:0) f ( s ) (cid:1) s ≥ ∈ (cid:0) X ,s,β (cid:1) s ≥ / || F || ,β,µ α < + ∞ (cid:111) . X ε,s,β for different parameters β on the one hand, and on the other hand between the spaces X N,ε,β,µ α for different parameters β and µ .This will be useful to define functional spaces that are stable under the action of the collision operators. Proposition 2 (Embeddings of the spaces X ε,s,β , X ,s,β , X N,ε,β,µ α and X ,β,µ α ) . Let s be a positive integerand ε be a strictly positive number. • For any β ≤ β (cid:48) , one has X ε,s,β (cid:48) ⊂ X ε,s,β and X ,s,β (cid:48) ⊂ X ,s,β , and if h ( s ) belongs to X ε,s,β (cid:48) (respectively f ( s ) belongs to X ,s,β (cid:48) ), one has | h ( s ) | ε,s,β ≤ | h ( s ) | ε,s,β (cid:48) (respectively | f ( s ) | ,s,β ≤ | f ( s ) | ,s,β (cid:48) ). • For any β ≤ β (cid:48) , any µ ≤ µ (cid:48) and any α > , one has X N,ε,β (cid:48) ,µ (cid:48) α ⊂ X N,ε,β,µ α and X ,β (cid:48) ,µ (cid:48) α ⊂ X ,β,µ α , andif (cid:0) h ( s ) N (cid:1) ≤ s ≤ N belongs to X N,ε,β (cid:48) ,µ (cid:48) α (respectively (cid:0) f ( s ) (cid:1) s ≥ belongs to X ,β (cid:48) ,µ (cid:48) α ), one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) h ( s ) N (cid:1) ≤ s ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε,β,µ α ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) h ( s ) N (cid:1) ≤ s ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε,β (cid:48) ,µ (cid:48) α (respectively (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) f ( s ) (cid:1) s ≥ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,β,µ α ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) f ( s ) (cid:1) s ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,β (cid:48) ,µ (cid:48) α ) . Definition of the spaces ‹ X ε, (cid:101) β, (cid:101) µ α and ‹ X , (cid:101) β, (cid:101) µ α , the functions of sequences belonging to X · , (cid:101) β ( t ) , (cid:101) µ ( t ) α at time t . It will be important in the sequel to enable a loss of regularity of the marginals when time grows(translated into a growth of the parameters β and µ , for the embeddings of Proposition 2 hold). One willthen define spaces of time-dependent functions.From this point, there are mainly two possibilities to define the relevant spaces of time-dependent functionstaking their values in X · , (cid:101) β ( t ) , (cid:101) µ ( t ) , depending on the value of α : this value has to be balanced with theregularity with respect to time, in order to have stable spaces under the action of the collision operator.The motivation of introducting such a parameter is the choice of the continuity in time introduced in [10],which is uniform in the number of particles s . With such a strong continuity in time property, it is possibleto show (see [9]) that the weight α = 1 is too weak, and has to be replaced by α >
1. On the other hand,this value would lead in practice to consider weird initial data, that are not meeting the expected physicalproperties for marginals of a distribution function. As a consequence, in the sequel we will focus on thepractical choice α = 1, up to relax the continuity in time property. Only in this section, we will give theproper definition of the spaces ‹ X · , (cid:101) β, (cid:101) µ α , for α = 1 or 2 such that the collision operator is stable on thosespaces. The case of α = 2 , and uniform continuity in time in the parameter s . We follow here the definitiongiven in the erratum version of the article [10] , in the sense that we require a uniformly s continuity in time. Definition 5 (Norms |||·|||
N,ε, ˜ β, ˜ µ and |||·||| , ˜ β, ˜ µ , spaces ‹ X N,ε, ˜ β, ˜ µ and ‹ X , ˜ β, ˜ µ ) . Let N be a positive integer.Let ε be a strictly positive number. For any T > , any strictly positive, non increasing function ˜ β , anynon increasing function ˜ µ , both defined on [0 , T ] , and any function ‹ H N : [0 , T ] → (cid:83) t ∈ [0 ,T ] X N,ε, (cid:101) β ( t ) , (cid:101) µ ( t ) , t (cid:55)→ ‹ H N ( t ) = Ä h ( s ) N ( t ) ä ≤ s ≤ N such that ‹ H N ( t ) ∈ X N,ε, (cid:101) β ( t ) , (cid:101) µ ( t ) for all t ∈ [0 , T ] , we define (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‹ H N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε, ˜ β, ˜ µ = sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‹ H N ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε, ˜ β ( t ) , ˜ µ ( t ) , and we define the space ‹ X N,ε, ˜ β, ˜ µ as the space of such functions ‹ H N with a finite |||·||| N,ε, (cid:101) β, (cid:101) µ norm, andverifying the left continuity in time hypothesis: ∀ t ∈ ]0 , T ] , lim u → t − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‹ H N ( t ) − ‹ H N ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε, (cid:101) β ( t ) , (cid:101) µ ( t ) = 0 . (13) Similarly, for any
T > , any strictly positive, non increasing function ˜ β and any non increasing function ˜ µ , both defined on [0 , T ] , and any function ‹ F : [0 , T ] → (cid:83) t ∈ [0 ,T ] X N,ε, (cid:101) β ( t ) , (cid:101) µ ( t ) , t (cid:55)→ ‹ F ( t ) = (cid:0) f ( s ) ( t ) (cid:1) s ≥ suchthat ‹ F ( t ) ∈ X , (cid:101) β ( t ) , (cid:101) µ ( t ) for all t ∈ [0 , T ] , we define (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‹ F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ˜ β, ˜ µ = sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‹ F ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ˜ β ( t ) , ˜ µ ( t ) , See the last Definition 5.2.4 of Section 5.2 “Functional spaces and statement of the results”. nd we define the space ‹ X , ˜ β, ˜ µ as the space of such functions ‹ F with a finite |||·||| , (cid:101) β, (cid:101) µ norm, and verifyingthe left continuity in time hypothesis: ∀ t ∈ ]0 , T ] , lim u → t − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‹ F ( t ) − ‹ F ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:101) β ( t ) , (cid:101) µ ( t ) = 0 . (14) Remark 2.
To be meaningful, the continuity conditions (13) and (14) use Proposition 2, together with thecrucial fact that the functions (cid:101) β and (cid:101) µ are assumed to be non increasing. The case of α = 1 , and continuity in time for every integer s . Let us now introduce the space thatwill be the most useful for the rest of this work. We choose α = 1, and require a less restrictive condition ofcontinuity in time than for the case α = 2: instead of having a continuity condition in the ||·|| · , (cid:101) β ( t ) , (cid:101) µ ( t ) norm,we will require, for any value of the parameter s , a continuity condition in the | · | · ,s, (cid:101) β ( t ) norm. Definition 6 (Norms |||·|||
N,ε, ˜ β, ˜ µ and |||·||| , ˜ β, ˜ µ , spaces ‹ X N,ε, ˜ β, ˜ µ and ‹ X , ˜ β, ˜ µ ) . Let N be a positive integer, ε be a strictly positive number. For any T > , any strictly positive, non increasing function ˜ β , any nonincreasing function ˜ µ , both defined on [0 , T ] , and any function ‹ H N : [0 , T ] → (cid:83) t ∈ [0 ,T ] X N,ε, (cid:101) β ( t ) , (cid:101) µ ( t ) , t (cid:55)→ ‹ H N ( t ) = Ä h ( s ) N ( t ) ä ≤ s ≤ N such that ‹ H N ( t ) ∈ X N,ε, (cid:101) β ( t ) , (cid:101) µ ( t ) for all t ∈ [0 , T ] , we define (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‹ H N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε, ˜ β, ˜ µ = sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‹ H N ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε, ˜ β ( t ) , ˜ µ ( t ) , and we define the space ‹ X N,ε, ˜ β, ˜ µ as the space of such functions ‹ H N with a finite |||·||| N,ε, (cid:101) β, (cid:101) µ norm, andverifying the left continuity in time hypothesis: ∀ t ∈ ]0 , T ] , ∀ ≤ s ≤ N, lim u → t − (cid:12)(cid:12) h ( s ) N ( t ) − h ( s ) N ( u ) (cid:12)(cid:12) ε,s, (cid:101) β ( t ) = 0 . (15) Similarly, for any
T > , any strictly positive, non increasing function ˜ β and any non increasing function ˜ µ both defined on [0 , T ] , and any function ‹ F : [0 , T ] → (cid:83) t ∈ [0 ,T ] X N,ε, (cid:101) β ( t ) , (cid:101) µ ( t ) , t (cid:55)→ ‹ F ( t ) = (cid:0) f ( s ) ( t ) (cid:1) s ≥ suchthat ‹ F ( t ) ∈ X , (cid:101) β ( t ) , (cid:101) µ ( t ) for all t ∈ [0 , T ] , we define (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‹ F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ˜ β, ˜ µ = sup ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‹ F ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ˜ β ( t ) , ˜ µ ( t ) , and we define the space ‹ X , ˜ β, ˜ µ as the space of such functions ‹ F with a finite |||·||| , (cid:101) β, (cid:101) µ norm, and verifyingthe left continuity in time hypothesis: ∀ t ∈ ]0 , T ] , ∀ ≤ s ≤ N, lim u → t − (cid:12)(cid:12) h ( s ) N ( t ) − h ( s ) N ( u ) (cid:12)(cid:12) ,s, (cid:101) β ( t ) = 0 . (16)The spaces ‹ X N,ε, (cid:101) β, (cid:101) µ α and ‹ X , (cid:101) β, (cid:101) µ α defined in the previous section satisfy the following regularity property: Proposition 3 (Banach space structure of the spaces ‹ X N,ε, (cid:101) β, (cid:101) µ α and ‹ X , (cid:101) β, (cid:101) µ α ) . Let N be a positive integer.Let ε be a strictly positive number. For any T > , any strictly positive, non increasing function ˜ β and anynon increasing function ˜ µ , both defined on [0 , T ] , and for α = 1 or , the spaces ‹ X N,ε, (cid:101) β, (cid:101) µ α and ‹ X , (cid:101) β, (cid:101) µ areBanach spaces. Since the norm |||·||| · , ˜ β, ˜ µ is defined using suprema, the proof of the previous proposition is very close tothe Riesz-Fischer theorem, establishing the completeness of the L ∞ spaces. See [9] for a proof of Proposition 3.From this point, the parameter α will always be taken equal to 1, and will be omitted in the notationsin what follows. 14 .3 Existence and uniqueness of the solutions of the hierarchies in the spaces ‹ X N,ε, (cid:101) β, (cid:101) µ and ‹ X , (cid:101) β, (cid:101) µ We are now able to state a result of existence and uniqueness for the solutions of the two hierarchies. Thisresult will use the Banach space structure of the spaces ‹ X N,ε, (cid:101) β, (cid:101) µ and ‹ X , (cid:101) β, (cid:101) µ , and the rewriting of the genericequations of the hierarchies as a fixed point problem. Definition 7 (BBGKY operator, Boltzmann operator) . For a sequence of initial data ( f ( s ) N, ) ≤ s ≤ N ∈ X N,ε, (cid:101) β (0) , (cid:101) µ (0) , we introduce the BBGKY operator , acting on the sequences ( h ( s ) N ) ≤ s ≤ N of ‹ X N,ε, (cid:101) β, (cid:101) µ , denotedas E N,ε (cid:0) ( f ( s ) N, ) ≤ s ≤ N , · (cid:1) , and defined as E N,ε (cid:0) ( f ( s ) N, ) ≤ s ≤ N , ( h ( s ) N ) ≤ s ≤ N (cid:1) = (cid:16) E ( s ) N,ε (cid:0) ( f ( s ) N, ) ≤ s ≤ N , ( h ( s ) N ) ≤ s ≤ N (cid:1)(cid:17) ≤ s ≤ N where E ( s ) N,ε (cid:0) ( f ( s ) N, ) s , ( h ( s ) N ) s (cid:1) ( t, · ) = f ( s ) N, ( · ) + (cid:90) t T s,ε − u C N,εs,s +1 T s +1 ,εu h ( s +1) N ( u, · ) d u, and this, for all ≤ s ≤ N − and all t (the case s = N is just given by E ( N ) N,ε (cid:0) ( f ( s ) N, ) s , ( h ( s ) N ) s (cid:1) ( t, · ) = f ( N ) N, ( · ) ).The same kind of operator can be introduced as well for the Boltzmann hierarchy: for a sequence of initialdata ( f ( s )0 ) s ≥ ∈ X , (cid:101) β (0) , (cid:101) µ (0) , we introduce the Boltzmann operator , acting on the sequences ( f ( s ) ) s ≥ of ‹ X , (cid:101) β, (cid:101) µ , denoted as E (cid:0) ( f ( s )0 ) s ≥ , · (cid:1) , and defined as E (cid:0) ( f ( s )0 ) s ≥ , ( f ( s ) ) s ≥ (cid:1) = (cid:16) E ( s )0 (cid:0) ( f ( s )0 ) s ≥ , ( f ( s ) ) s ≥ (cid:1)(cid:17) s ≥ where E ( s )0 (cid:0) ( f ( s )0 ) s , ( f ( s ) ) s (cid:1) ( t, · ) = T s, t f ( s )0 ( · ) + (cid:90) t T s, t − u C s,s +1 f ( s +1) ( u, · ) d u, for all s ≥ and all t . We have then the following reformulation: ( h ( s ) N ) ≤ s ≤ N is a solution of the conjugated BBGKY hierarchy(12) associated to the initial data ( f ( s ) N, ) ≤ s ≤ N if and only if ( h ( s ) N ) ≤ s ≤ N = E N,ε (cid:0) ( f ( s ) N, ) ≤ s ≤ N , ( h ( s ) N ) ≤ s ≤ N (cid:1) (and of course the same rewritting holds also for the Boltzmann hierarchy).Thanks to this reformulation into a fixed point problem, we can now obtain the following theorem. Theorem 3 (Joint local in time existence and uniqueness of solutions to the BBGKY and Boltzmannhierarchies) . Let β be a strictly positive real number and µ a real number. There exist a time T > , a strictlypositive decreasing function ˜ β and a decreasing function ˜ µ defined on [0 , T ] such that ˜ β (0) = β , ˜ µ (0) = µ and such that for any positive integer N and any strictly positive number ε > verifying the Boltzmann-Gradlimit N ε d − = 1 , any pair of sequences of initial data F N, = (cid:0) f ( s ) N, (cid:1) ≤ s ≤ N and F = (cid:0) f ( s )0 (cid:1) s ≥ belongingrespectively to X N,ε,β ,µ and X ,β ,µ give rise respectively to a unique solution H N = t (cid:55)→ (cid:0) h ( s ) N ( t, · ) (cid:1) ≤ s ≤ N in ‹ X N,ε, ˜ β, ˜ µ to the BBGKY hierarchy with initial datum F N, and F = t (cid:55)→ (cid:0) f ( s ) ( t, · ) (cid:1) s ≥ in ‹ X , ˜ β, ˜ µ to theBoltzmann hierarchy with initial datum F , that is there exists a unique pair of elements H N and F belongingrespectively to the spaces ‹ X N,ε, (cid:101) β, (cid:101) µ and ‹ X , (cid:101) β, (cid:101) µ such that, for every t ∈ [0 , T ] : H N ( t ) = E N,ε (cid:0) F N, , H N (cid:1) ( t ) , and F ( t ) = E (cid:0) F , F (cid:1) ( t ) . Moreover, the decreasing functions (cid:101) β = (cid:101) β λ and (cid:101) µ = (cid:101) µ λ are affine, given by the expressions (cid:101) β λ = [0 , T ] → R ∗ + , t (cid:55)→ (cid:101) β λ ( t ) = β − λt and (cid:101) µ λ = [0 , T ] → R , t (cid:55)→ (cid:101) µ λ ( t ) = µ − λt , for λ depending only on β and µ , and for T depending only on β , µ and λ . emark 3. Here it is important to notice that, since in the end the goal is to obtain a convergence result ofthe solutions of the BBGKY hierarchy towards the solution of the Boltzmann hierarchy, the time of existenceof solutions has to be the same for the two hierarchies (this comes from the fact that the upper bound in thecontrol (53) in Section B.2 is exactly the same for E N,ε (cid:0) ( f ( s ) N, ) ≤ s ≤ N , · (cid:1) and E (cid:0) ( f ( s )0 ) s ≥ , · (cid:1) ), and it hasalso to be same for all N concerning the BBGKY hierarchy (this comes from the fact that the dependency on N and ε for C ( d, N, ε ) in (53) is exactly through the term N ε d − , as one may see by tracking the constantsalong the sketch of proof that was presented, which explains why we have to work in the Boltzmann-Gradlimit N ε d − = 1 ). The proof of Theorem 3, which is presented in [10], and of course in [9] with much details, relies on acrucial contracting inequality due to Nishida [15], Uchiyama [18] and Ukai [19]. A shortened version is alsopresented at the end of this work, in appendix B: first we adress the problem of the stability of the spaces ‹ X · , (cid:101) β, (cid:101) µ by the operators of Definition 7, and second we study the norm of those operators. In particular, it ispossible to show that, up to choose wisely the weights (cid:101) β and (cid:101) µ , they are contracting mappings. Nevertheless,the choice of those weights comes with a (serious) time restriction on the validity of the existence of thesolutions. This section will be devoted to prove the main result of this work, which can be presented as follows.
Theorem 4 (Lanford’s theorem: convergence of the BBGKY hierarchy towards the Boltzmann hierarchy) . Let β > and µ be two real numbers. Then there exists a time T > such that the following holds:let F = (cid:0) f ( s )0 (cid:1) s ≥ be a sequence of initial data of the Boltzmann hierarchy belonging to X ,β ,µ , and for anypositive integer N , let F N, = (cid:0) f ( s ) N, (cid:1) ≤ s ≤ N be a sequence of initial data of the BBGKY hierarchy belongingto X N,ε,β ,µ . We assume that for any s ∈ N ∗ , f ( s ) N, converges locally uniformly towards f ( s )0 on the phasespace of s particles, with in addition sup N ≥ || F N, || N,ε,β ,µ < + ∞ .Then, in the Boltzmann-Grad limit N → + ∞ , N ε d − = 1 , if one denotes F = (cid:0) f ( s ) (cid:1) s ≥ the solution on [0 , T ] of the Boltzmann hierarchy with initial data F , and F N the solution on [0 , T ] of the BBGKY hierarchywith initial data F N, , one has that, for any positive integer s , the locally uniform convergence on the domainof local uniform convergence Ω s (see Definition 11 page 34 below), uniformly on [0 , T ] , of f ( s ) N towards f ( s ) . Remark 4.
This result is the analog of the Lanford’s theorem (see [13]) when the particles evolve in thehalf-space, in its qualitative version. A modern proof of the original theorem, stated for domains withoutboundary ( R d or T d ), can be found in [10]. In this reference, the authors were able to perform an importantbreakthrough by achieving the most detailed proof of Lanford’s result, with in addition an explicit rate ofconvergence.At this step, the presence of the obstacle does not essentially change the statement of [10]. However, we willsee along the proof that this obstacle complicates the argument, and although we can refine Theorem 4 andprovide a quantitative convergence (as in [10]), stated at the very end of this work in Theorem 6 page 40, therate of convergence is less sharp when there is an obstacle.We note the significant fact that the theorems presented here provide a convergence in a strong sense, implyingin particular the one obtained in [10] (which was the convergence in the sense of the observables, that is auniform convergence in the time and the position variables, but only in the sense of the distributions for thevelocity variable). The counterpart is the domain on which this convergence holds: we are forced to considercompact sets in the phase space that are in particular not crossing the wall, nor containing grazing velocities.The convergence between the hierarchies implies two important results: first, for s = 1 , we obtain a rigorousderivation of the Boltzmann equation from finite systems of hard spheres, and second, since the result holdsfor any integer s , when f ( s ) is tensorized we also recover the propagation of chaos, since the s -th marginalof the hard sphere system converges towards a chaotic distribution function. .1 An explicit formula for the solutions of the hierarchies Now that the problem of existence and uniqueness has been addressed, let us see how the solutions of thehierarchies can be rewritten explicitely in terms of the initial data, and iterations of the integrated in time(transport)-collision-transport operator.To describe this result, which can be seen as an analog of the Duhamel formula, we introduces the fol-lowing notations: the integrated in time transport-collision-transport operator of the BBGKY hierarchy t (cid:55)→ (cid:90) t T s,ε − u C N,εs,s +1 T s +1 ,εu f ( s +1) N, d u will be denoted as t (cid:55)→ (cid:0) I N,εs f ( s ) N, (cid:1) ( t, · ), while the k -th iterate of thisoperator, that is t (cid:55)→ (cid:90) t T s,ε − t C N,εs,s +1 T s +1 ,εt (cid:90) t T s +1 ,ε − t C N,εs +1 ,s +2 T s +2 ,εt . . . (cid:90) t k − T s + k − ,ε − t k C N,εs + k − ,s + k T s + k,εt k f ( s + k ) N, d t k . . . d t d t ,which can be denoted as t (cid:55)→ (cid:16) I N,εs ◦ I
N,εs +1 ◦ · · · ◦ I N,εs + k − f ( s + k ) N, (cid:17) ( t, · ), thanks to the new notations, will bein fact denoted as t (cid:55)→ (cid:0) I N,εs,s + k − f ( s + k ) N, (cid:1) ( t, · ) (the second subscript index describes the number of iterations).Similarly, the integrated in time collision-transport operator of the Boltzmann hierarchy t (cid:55)→ (cid:90) t T s, t − u C s,s +1 f ( s +1) ( u, · ) d u will be denoted as t (cid:55)→ (cid:0) I s f ( s ) (cid:1) ( t, · ).The k -th iterate of this operator, that is: t (cid:55)→ (cid:90) t T s, t − t C N,εs,s +1 (cid:90) t T s, t − t C s +1 ,s +2 . . . (cid:90) t k − T s + k − , t k − − t k C s + k − ,s + k f ( s + k ) ( t k , · ) d t k . . . d t d t will be denoted as t (cid:55)→ (cid:0) I s,s + k − f ( s + k ) (cid:1) ( t, · ). Proposition 4 (Iterated Duhamel formula for the solution of the hierarchies) . Let N be a positive integerand ε be a strictly positive number. In the Boltzmann-Grad limit N ε d − = 1 , for any strictly positive number β , any real number µ , and for any sequence of initial data F N, = (cid:0) f ( s ) N, (cid:1) ≤ s ≤ N belonging to the space X N,ε,β ,µ , the unique solution of the integrated form of the conjugated BBGKY hierarchy with initial datum F N, on the time interval [0 , T ] ( T given by Theorem 2) is given by H N = t (cid:55)→ (cid:16) f ( s ) N, + N − s (cid:88) k =1 s ≤ N − k (cid:0) I N,εs,s + k − f ( s + k ) N, (cid:1) ( t, · ) (cid:17) ≤ s ≤ N . (17) Similarly for any sequence of initial data F = (cid:0) f ( s )0 (cid:1) s ≥ belonging to the space X ,β ,µ , the unique solutionof the integrated form of the Boltzmann hierarchy with initial datum F on the same time interval [0 , T ] isgiven by F = t (cid:55)→ (cid:16) T s, t f ( s )0 ( · ) + + ∞ (cid:88) k =1 (cid:16) I s,s + k − (cid:0) u (cid:55)→ T s + k, u f ( s + k )0 (cid:1)(cid:17) ( t, · ) (cid:17) s ≥ . (18) Sketch of proof of Proposition 4.
The proof of Proposition 4 comes from the fact that the two series in (17)and (18) (which are well defined as limits of Cauchy sequences in the two respective Banach spaces ‹ X N,ε, (cid:101) β, (cid:101) µ and ‹ X , (cid:101) β, (cid:101) µ , thanks to the contracting property of the integrated in time collision operators) are fixed points,respectively of the BBGKY and the Boltzmann operators, which concludes the proof of Proposition 4 thanksto the uniqueness of the solutions for the two hierarchies. See [9] for more details. The Duhamel formula will guide us to a geometrical interpretation, suggesting a somehow natural proof forthe convergence. Let us discuss here the formula (18), for the Boltzmann hierarchy, in order to avoid thequestions about the meaning of the collision operator for the BBGKY hierarchy.First, the s -th marginal is the sum of the terms I s,s + k − (cid:0) T s + k, u f ( s + k )0 (cid:1) , for all k ≥
1, where k represents the17umber of iterations of the integrated in time collision-transport operator.Now, each collision operator C s,s +1 being a sum of the 2 s terms C s,s +1 , ± ,i (for 1 ≤ i ≤ s and ± = + or − ),each term I s,s + k − can be decomposed as a sum of 2 k s ( s + 1) . . . ( s + k −
1) terms. Each of those termscorresponds to, first, choosing a first particle j among s particles, and a configuration ± , being either pre-collisional ( ± = − ) or post-collisional ( ± = +), then choosing a second particle j among s + 1 particles,and a configuration ± , and so on, k times. If we write explicitely such a term, for example the one with k = 2 (two iterations of the integrated in time collision-transport operator), ± = − and ± = +, we obtain: − (cid:90) t (cid:90) S d − ω × R dvs +1 (cid:104) ω · (cid:0) v s +1 − v s, j ( t ) (cid:1)(cid:105) − (cid:90) t (cid:90) S d − ω × R dvs +2 ï ω · (cid:0) v s +2 − v ,j s, ( t ) (cid:1) ò + f ( s +2)0 (cid:0) Z s, (0) (cid:1) d ω d v s +2 d t d ω d v s +1 d t , where v ,j s, ( t ) = (cid:0) T s, t − t ( Z s ) (cid:1) V,j ,v ,j s, ( t ) = (cid:16) T s +1 , t − t (cid:0) T s, t − t ( Z s ) , (cid:0) T s, t − t ( Z s ) (cid:1) X,j , v s +1 (cid:1)(cid:17) V,j ,Z s, (0) = T s +2 , − t (cid:32)(cid:32) T s +1 , t − t (cid:16) T s, t − t ( Z s ) , (cid:0) T s, t − t ( Z s ) (cid:1) X,j , v s +1 (cid:17) , (cid:16) T s +1 , t − t (cid:16) T s, t − t ( Z s ) , (cid:0) T s, t − t ( Z s ) (cid:1) X,j , v s +1 (cid:17)(cid:17) X,j , v s +2 (cid:33) (cid:48) j ,s +2 (cid:33) . Considering the expressions from the left to the right, one sees that the iterations of the operators are com-plicating more and more the arguments below the integrals.Let us investigate the structure of those arguments: in the expression of this term of the solution of theBoltzmann hierarchy at time t , the first argument v ,j s, ( t ) corresponds to the velocity of the j -th particleof the system starting from a configuration Z s , and after following the backward free flow with boundarycondition for a time t − t (where t is the integration variable of the first integrated in time transport-collisionoperator). Then, a particle is added to this system of s particles at time t − t , next to the j -th particle,with v s +1 as its initial velocity. Then, this new system with s + 1 particles follows again the backward freeflow for a time t − t . Here we select the velocity of the j -th particles of this new configuration: this isthe argument v ,j s, ( t ). Finally, to obtain the last argument Z s, (0), we restart from the last configurationwhich was described, picks the j -th particle and add just next to it (at time ( t − t ) + ( t − t ) its positionis (cid:16) T s +1 , t − t (cid:16) T s, t − t ( Z s ) , (cid:0) T s, t − t ( Z s ) (cid:1) X,j , v s +1 (cid:17)(cid:17) X,j ) another particle, with an initial velocity v s +2 . Since thistime ± was chosen to be +, the particle which is added starts in a post-collisional configuration, so oneapplies in addition the scattering operator, and in the end this new system of s + 2 particles undergoes theaction of the backward free transport for a time t : this is the final argument Z s, (0).This process, complicated at first glance, can be pictured as in Figure 3 below. Such process, mixing trans-ports and adjunctions of particles, builds what is usually called in the literature pseudo-trajectories (pseudobecause the number of particles of the system changes along time).Formally, the same decomposition, and the same process to describe the elementary terms can be done forthe BBGKY hierarchy as well. Of course, there is an important difference: here the particles have a non-zeroradius, and the backward free transport is replaced by the backward hard sphere transport, allowing possibleinteractions between the particles.Let us introduce here some notations. A pseudo-trajectory is entirely determined by its initial configuration Z s , its number of adjunctions k , and finally by the choice of its adjunction parameters (the time of adjunction,the particle chosen for it, and the angular parameter and the velocity of the particle added). We introducethen the sets T k = (cid:8) ( t , . . . , t k ) ∈ [0 , t ] k / t < · · · < t k (cid:9) , J sk = (cid:8) ( j , . . . , j k ) ∈ N k / ∀ i, ≤ j i ≤ s + i − (cid:9) and A k = (cid:8)(cid:0) ( ω , v s +1 ) , . . . , ( ω k , v s + k ) (cid:1)(cid:9) = (cid:0) S d − × R d (cid:1) k . Note that the difference between an adjunction ina pre- or post-collisional configuration lies in the sign of ω i · (cid:0) v s + i − v · ,i − s,j i ( t i ) (cid:1) (where v · ,i − s,j i ( t i ) is the velocityof the particle j i undergoing the adjunction, at the time t i of this adjunction), this sign being ± i .18igure 3: Construction of a pseudo-trajectory for the BBGKY hierarchy.We call then a pseudo-trajectory the collection of all the configurations, along time, of the system start-ing from Z s , and undergoing the adjunctions described by T k ∈ T k , J k ∈ J sk and A k ∈ A k , and denoted Z ( Z s , T k , J k , A k ) (or Z ε for the BBGKY hierarchy), or more simply Z . To specify that we consider thepseudo-trajectory at a fixed time τ , we denote its configuration Z ( τ ). A pseudo-trajectory is then a time-dependent function taking its values in the configurations, with an increasing number of particles along time.The position of a generic particle j at time τ of this pseudo-trajectory will be denoted x ,j s,i ( τ ) (or x ε,j forthe BBGKY hierarchy), where the subscripts s and i represent respectively the initial number of particles,and the number of adjunctions performed before τ . For a velocity, we simply replace x by v .The main idea of the proof of the convergence can be then described easily: one expects that for particlesof small radius, the process described above produces pseudo-trajectories for the BBGKY hierarchy that areuniformly close to the pseudo-trajectories for the Boltzmann hierarchy, as pictured in Figure 4. Then, ifFigure 4: Comparison of the pseudo-trajectories of the two hierarchies.the pseudo-trajectories can be compared, a continuity argument will show that the integrands in the expres-sion of the Duhamel formula will converge, and a dominated convergence argument would conclude the proof.19evertheless, the uniform comparison of the pseudo-trajectories is not always possible, for initial config-urations of the system may lead to drastically diverging pseudo-trajectories. This is the case when twoparticles, evolving according to the hard sphere transport, collide one with another. Since this behaviourcannot happen for the corresponding Boltzmann pseudo-trajectory (that is, with the same initial configu-ration, the same choice of particles for the adjunctions, and the same choice for the pre- or post-collisionalsettings), a radical difference between the positions and the velocities may suddenly appear, as it is picturedin Figure 5. This is a well-known obstruction in Lanford’s proof, called recollision (see [8]). Those recollisionswere studied in much details in [10] in the case of the Euclidean space (without any obstacle in the domain).Figure 5: Case of a recollision in the dynamics of the hard spheres, and divergence from the dynamics ofparticles of radius zero, following the free flow.In the case of our work, the presence of the wall may also produce divergence between the pseudo-trajectories. If a particle hits the wall, starting from a given point, with a given velocity, the point of impactwill depend on the radius of the particle, as well as the trajectory of the particle after the bouncing. Thisvariation can be uniformly controlled in terms of the radii of the particles though. A much more seriousproblem comes from the divergence between the times of bouncing of two particles of different radii: betweenthe two bouncings, one has already bounced and has already a reflected velocity, while the other particle hasstill its pre-bouncing velocity, leading to an important difference. Those phenomena are pictured in Figure6 below.Figure 6: The two phenomena of divergence of the pseudo-trajectories appearing during a bouncing againstthe obstacle. 20he task will be now to prepare the solutions for those geometrical comparisons: some cut-offs willsimplify the proof, while we will have to control the size of the configurations leading to pathological pseudo-trajectories preventing the comparison. As in [10] , we start with considering only pseudo-trajectories with a finite number of adjunctions, thatis obtained with a finite number of iterations of the collision operators. In addition, we will remove thepseudo-trajectories with particles travelling too fast: we will perfom a cut-off in large velocities. Finally,an important tool to simplify the study of the geometry of the pseudo-trajectories will be to prevent theadjunctions to be to close in time: the time difference between two adjunctions will be bounded from below.All those simplifications have a cost, that is described in the three lemmas below. Lemma 1 (Cut-off in high number of collisions) . Let β be a strictly positive number and µ be a realnumber. For any positive integer n , any positive integer N and any strictly positive number ε > verifyingthe Boltzmann-Grad limit N ε d − = 1 , and any couple of sequences of initial data F N, = (cid:0) f ( s ) N, (cid:1) ≤ s ≤ N and F = (cid:0) f ( s )0 (cid:1) s ≥ belonging respectively to X N,ε,β ,µ and X ,β ,µ , the respective unique solutions H N ∈ ‹ X N,ε, (cid:101) β λ , (cid:101) µ λ to the BBGKY hierarchy with initial datum F N, and F ∈ ‹ X , (cid:101) β λ , (cid:101) µ λ to the Boltzmann hierarchywith inital datum F (where (cid:101) β λ , (cid:101) µ λ , H N and F are given by Theorem 3) satisfy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H N − (cid:16) f ( s ) N, + n (cid:88) k =1 s ≤ N − k (cid:0) I N,εs,s + k − f ( s + k ) N, (cid:1)(cid:17) ≤ s ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε, (cid:101) β λ , (cid:101) µ λ ≤ (cid:16) (cid:17) n || F N, || N,ε,β ,µ , (19) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F − (cid:16) t (cid:55)→ T s, t f ( s )0 + n (cid:88) k =1 (cid:0) I s,s + k − (cid:0) u (cid:55)→ T s + k, u f ( s + k )0 (cid:1)(cid:1) ( t, · ) (cid:17) s ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:101) β λ , (cid:101) µ λ ≤ (cid:16) (cid:17) n || F || ,β ,µ . (20) Proof of Lemma 1.
The proof is a simple consequence of the contracting property of the collision operatorson the time interval [0 , T ] given by Theorem 3. The proof is presented in details in [9].In the following, we will denote: H nN = t (cid:55)→ (cid:16) f ( s ) N, ( · ) + n (cid:88) k =1 s ≤ N − k (cid:0) I N,εs,s + k − f ( s + k ) N, (cid:1) ( t, · ) (cid:17) ≤ s ≤ N . and F n = t (cid:55)→ (cid:16) T s, t f ( s )0 + n (cid:88) k =1 (cid:0) I s,s + k − (cid:0) u (cid:55)→ T s + k, u f ( s + k )0 (cid:1)(cid:1) ( t, · ) (cid:17) s ≥ . In addition we introduce for any parameter
R > H n,RN = t (cid:55)→ (cid:16) f ( s ) N, ( · ) | V s |≤ R + n (cid:88) k =1 s ≤ N − k (cid:0) I N,εs,s + k − (cid:0) f ( s + k ) N, | V s + k |≤ R (cid:1)(cid:1) ( t, · ) (cid:17) ≤ s ≤ N and F n,R = t (cid:55)→ (cid:16) T s, t f ( s )0 ( · ) | V s |≤ R + n (cid:88) k =1 (cid:0) I N,εs,s + k − (cid:0) u (cid:55)→ T s + k, u f ( s + k ) N, | V s + k |≤ R (cid:1)(cid:1) ( t, · ) (cid:17) s ≥ . Lemma 2 (Cut-off in large energy configurations) . Let β be a strictly positive number and µ be a realnumber. There exists an affine, strictly positive, decreasing function (cid:101) β (cid:48) < (cid:101) β defined on [0 , T ] (where (cid:101) β = (cid:101) β λ is given by Theorem 3) and two constants C ( d, β , µ ) and C ( d, β , µ ) , depending only on the dimension d See Chapter 7. nd on the numbers β and µ , such that for any positive integer n , any strictly positive number R , any positiveinteger N and any strictly positive number ε > verifying the Boltzmann-Grad limit N ε d − = 1 , and anypair of sequences of initial data F N, = (cid:0) f ( s ) N, (cid:1) ≤ s ≤ N and F = (cid:0) f ( s )0 (cid:1) s ≥ belonging respectively to X N,ε,β ,µ and X ,β ,µ , the truncated in high number of collisions solutions H nN ∈ ‹ X N,ε, (cid:101) β λ , (cid:101) µ λ and F n ∈ ‹ X , (cid:101) β λ , (cid:101) µ λ satisfy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H nN − H n,RN (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε, (cid:101) β (cid:48) , (cid:101) µ λ ≤ C exp (cid:0) − C R (cid:1) || F N, || N,ε, (cid:101) β (cid:48) (0) ,µ , (21) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n − F n,R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:101) β (cid:48) , (cid:101) µ λ ≤ C exp (cid:0) − C R (cid:1) || F || , (cid:101) β (cid:48) (0) ,µ . (22)We introduce here the notations for the integrated in time collision operators, truncated in small timedifference between the collisions , that is we consider t ≥ ( k − δ (cid:90) t ( k − δ T s,ε − t C N,εs,s +1 T s +1 ,εt (cid:16) t ≥ ( k − δ (cid:90) t − δ ( k − δ T s +1 ,ε − t C N,εs +1 ,s +2 T s +2 ,εt . . . (cid:16) t k − ≥ δ (cid:90) t k − − δ T s + k − ,ε − t k C N,εs + k − ,s + k T s + k,εt k f ( s + k ) N, ( t k , · ) d t k (cid:17) . . . (cid:17) d t , so that we have t j − − t j ≥ δ for all 2 ≤ j ≤ k . It is indeed a cut-off in small differences between theadjunctions, since they are performed at the times t j .Those truncated in small differences between the adjunctions, iterated collision operators will be denoted I N,ε,δs,s + k − and I ,δs,s + k − respectively for the BBGKY hierarchy and the Boltzmann hierarchy. We introducethen H n,R,δN = t (cid:55)→ (cid:16) f ( s ) N, ( · ) | V s |≤ R + n (cid:88) k =1 s ≤ N − k (cid:16) I N,ε,δs,s + k − (cid:0) f ( s + k ) N, | V s + k |≤ R (cid:1)(cid:17) ( t, · ) (cid:17) ≤ s ≤ N and F n,R,δ = t (cid:55)→ (cid:16) T s, t f ( s )0 ( · ) | V s |≤ R + n (cid:88) k =1 (cid:16) I N,ε,δs,s + k − (cid:0) u (cid:55)→ T s + k, u f ( s + k ) N, | V s + k |≤ R (cid:1)(cid:17) ( t, · ) (cid:17) s ≥ . Lemma 3 (Cut-off in small time difference between the collisions) . Let β be a strictly positive numberand µ be a real number. There exists a constant C ( d, β , µ ) , depending only on the dimension d and onthe numbers β and µ , such that for any positive integer n , any strictly positive numbers R and δ , anypositive integer N and any strictly positive number ε > verifying the Boltzmann-Grad limit N ε d − = 1 ,and any couple of sequences of initial data F N, = (cid:0) f ( s ) N, (cid:1) ≤ s ≤ N and F = (cid:0) f ( s )0 (cid:1) s ≥ belonging respectively to X N,ε,β ,µ and X ,β ,µ , the respective truncated in high number of collisions and in large energy solutions H n,RN ∈ ‹ X N,ε, (cid:101) β, (cid:101) µ and F ∈ ‹ X , (cid:101) β, (cid:101) µ verify, for all integer ≤ s ≤ N and time t ∈ [0 , T ] : (cid:12)(cid:12)(cid:12)(cid:0) H n,RN (cid:1) ( s ) ( t, · ) − (cid:0) H n,R,δN (cid:1) ( s ) ( t, · ) (cid:12)(cid:12)(cid:12) ε,s, (cid:101) β ( t ) ≤ C || F N, || N,ε,β ,µ √ sn / √ δ, (23) and (cid:12)(cid:12)(cid:12)(cid:0) F n,R (cid:1) ( s ) ( t, · ) − (cid:0) F n,R,δ (cid:1) ( s ) ( t, · ) (cid:12)(cid:12)(cid:12) ,s, (cid:101) β ( t ) ≤ C || F || ,β ,µ √ sn / √ δ. (24)The proofs of Lemmas 2 and 3 are not specific to the case of the half-plane, and use only the propertiesof the collision operators. See [10] or [9], where the details are presented. We will control here the configurations leading to recollisions. To do so, we will introduce the concept of good configurations , that is, the initial data for the system of particles such that all the particles remain at22 certain fixed distance one from another, for all time. The goal is to show that, except for a few adjunctionparameters, the good configurations are stable under adjunctions, following [10] . Here the main difference,due of course to the presence of the wall, is the requirement of an additional condition on the particleundergoing the adjunction: it has to be far enough from the obstacle. The reason of this restriction is easyto understand, since if a particle k + 1 is added next to a particle k which is close to the boundary of thedomain, there are many possible velocities to choose for the new particle such that a recollision will happenbetween the particles k and k + 1, after a bouncing of the new particle k + 1 (see Figure 7 below).Figure 7: The possible directions (in red and blue) of the velocity of the new particle k + 1 leading to arecollision between t and t with the particle k undergoing the adjunction (the particle k undergoes thecollision at time t , and travels then until t on the figure), depending on the distance between this particleand the wall. Definition 8 (Good configuration) . Let ε and c be two strictly positive numbers and k be a positive integer.One defines the set of good configurations for k hard spheres separated by at least c , respectively the set ofgood configurations for k particles following the free flow separated by at least c , as the subset of D εk of theconfigurations Z s such that (cid:12)(cid:12)(cid:12) Ä T k,ε − τ ( Z k ) ä X,i − Ä T k,ε − τ ( Z k ) ä X,j (cid:12)(cid:12)(cid:12) > c, respectively as the subset of (cid:0) Ω c × R d (cid:1) k of the configuration Z s such that (cid:12)(cid:12)(cid:12) Ä T k, − τ ( Z k ) ä X,i − Ä T k, − τ ( Z k ) ä X,j (cid:12)(cid:12)(cid:12) > c, for all τ > and ≤ i (cid:54) = j ≤ k .Those sets will be denoted respectively G εk ( c ) and G k ( c ) . Let us introduce first an accurate definition of the stability under the adjunctions. We will also use thefollowing notations for the orthogonal symmetries: S ( x ) will denote x − x · e ) e and S ε ( x ) will denote x − x · e ) e + εe . Definition 9 (Stability by adjunction of the good configurations) . Let k be a positive integer, and R, δ, ε , a , ε be five strictly positive numbers. For Z k ∈ G k ( ε ) , we define B k ( R, δ, ε, a, ε )( Z k ) ⊂ S d − × B (0 , R ) as the complement of the set of the elements ( ω, v ) of S d − × B (0 , R ) such that, for all Z k ∈ G εk ( ε ) with, forall ≤ i ≤ k , we have (cid:12)(cid:12) x i − x i (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:0) Z k (cid:1) X,i − (cid:0) Z k (cid:1) X,i (cid:12)(cid:12)(cid:12) ≤ a and v k = v k , that is (cid:0) Z k (cid:1) V,k = (cid:0) Z k (cid:1) V,k , and forall ≤ i ≤ k − , v i = v i or v i = S ( v i ) , that is ( Z k ) V,i = (cid:0) Z k (cid:1) V,i or ( Z k ) V,i = S (cid:0) ( Z k ) V,i (cid:1) , then • if ω · ( v − v k ) < : See Proposition 12.1.1 page 94, their key geometrical result allowing to compare the pseudo-trajectories. the configuration ( Z k , x k + εω, v ) does not lead to a further recollision, that is (cid:0) Z k , x k + εω, v (cid:1) ∈ G εk +1 ( ε ) , – the configuration (cid:0) Z k , x k , v (cid:1) is a good configuration (for the free-flow) separated by at least ε after a time δ , that is T k +1 , − δ (cid:0) Z k , x k , v (cid:1) ∈ G k +1 ( ε ) , • if ω · ( v − v k ) > : – the configuration (cid:0) Z s , x k + εω, v (cid:1) (cid:48) k,k +1 does not lead to a further recollision, that is (cid:0) Z s , x k + εω, v (cid:1) (cid:48) k,k +1 ∈G εk +1 ( ε ) , – the configuration (cid:0) Z k , x k , v (cid:1) is a good configuration (for the free flow) separated by at least ε after a time δ , that is T k +1 , − δ (cid:16)(cid:0) Z k , x k , v (cid:1) (cid:48) k,k +1 (cid:17) ∈ G k +1 ( ε ) . Remark 5.
Here, we wants to find a small upper bound on the size of the set (cid:12)(cid:12) B k (cid:0) R, δ, ε, a, ε (cid:1)(cid:0) Z k (cid:1)(cid:12)(cid:12) , toshow that there are a lot of ways to add a particle to a system of k particles in a good configuration, whichlead to a new system of k + 1 particles in a good configuration.There are two important degrees of freedom introduced in Definition 9: one authorizes a possible smalldifference between the positions of the configurations Z k ∈ (cid:0) Ω c × R d (cid:1) k and the positions of the vector Z k ∈ D εk ,and the velocities of those two configurations may differ by a symmetry (that is v k = v k or S ( v k ) ).The necessity of those degrees of freedom is actually clear: differences between the pseudo-trajectories of theBBGKY and the Boltzmann hierarchies will appear, due to the radius of the particles and the interactionwith the obstacle (see Figure 6). Theorem 5 (Control of the size of the good configurations by adjunction of particles) . There exists a strictlypositive constant C ( d ) depending only on the dimension d such that for any positive integer k , and for all R, δ, ε, a, ε , ρ, η strictly positive real numbers such that ε ≤ a, √ a ≤ ε , ε ≤ ηδ, a ≤ ρ , R ≥ and ε /δ ≤ , and for all Z k ∈ G k ( ε ) such that x k · e = (cid:0) Z k (cid:1) X,k · e ≥ ρ, (25) there exists a measurable subset (cid:102) B k ( R, δ, ε, a, ε , ρ, η )( Z k ) ⊂ S d − × B (0 , R ) such that : B k ( R, δ, ε, a, ε )( Z k ) ⊂ (cid:102) B k ( R, δ, ε, a, ε , ρ, η )( Z k ) and (cid:91) Z k ∈ G k ( ε ) ,x k · e ≥ ρ { Z k } × (cid:102) B k ( R, δ, ε, a, ε , ρ, η )( Z k ) is measurable. Moreover, one has: (cid:12)(cid:12)(cid:12)(cid:102) B k ( R, δ, ε, a, ε , ρ, η )( Z k ) (cid:12)(cid:12)(cid:12) ≤ C ( d ) Å η d + R d (cid:16) aρ (cid:17) d − + kR d − (cid:16) aε (cid:17) d − / + kR d +1 / (cid:16) ε δ (cid:17) d − / ã . (26)The proof of such a theorem lies on two cornerstones, as in [10]: on the one hand, a “Shooting Lemma”,proving that for a particle k which starts from a ball, with a given velocity, there is only a small amount ofvelocities, lying in a cylinder with a small radius, for another particle k + 1 starting from another ball suchthat the particles k and k + 1 will collide (or, more generally, will be close at some time). We also have toshow that the scattering operator maps small cylinders into small subsets of the adjunction parameters, inthe case when the new particle is added in a post-collisional configuration.Let us introduce here the notation for the cylinders. For two vectors v, w ∈ R d and a positive real number ρ ,we set: K ( v, w, ρ ) = (cid:110) x ∈ R d / ∀ u ∈ S d − such that u · w = 0 , ( x − v ) · u ≤ ρ (cid:111) . Lemma 4 (Shooting Lemma with fixed axes) . Let R , δ , ε , a and ε be five strictly positive numbers, suchthat ε ≤ a, √ a ≤ ε . We consider two points x , x ∈ (cid:8) x ∈ R d / x · e > (cid:9) such that | x − x | ≥ ε , and v ∈ B (0 , R ) .Then for all x ∈ B ( x , a ) , x ∈ B ( x , a ) , and v ∈ B (0 , R ) : . if for some δ > v / ∈ K ( v , x − x , ε /δ ) ∪ K ( S ( v ) , S ( x ) − x , ε /δ ) , for all τ ≥ δ , we have: (cid:12)(cid:12)(cid:12)(cid:16) T , − τ (cid:0) Z (cid:1)(cid:17) X, − (cid:16) T , − τ (cid:0) Z (cid:1)(cid:17) X, (cid:12)(cid:12)(cid:12) > ε ,2. if in addition v / ∈ K ( v , x − x , Ra/ε ) ∪ K ( S ( v ) , S ( x ) − x , Ra/ε ) , for all τ > , we have: (cid:12)(cid:12)(cid:12) Ä T ,ε − τ ( Z ) ä X, − Ä T ,ε − τ ( Z ) ä X, (cid:12)(cid:12)(cid:12) > ε .Proof of Lemma 4. Let us start with the second point, and assume that the two particles (of radius ε ) collide,and we consider the smallest time τ such that it happens. Then by definition, the particles follow the freeflow, with boundary conditions, before τ . From this point, there are two possibilities for the expression ofthe position of the two particles, depending on wether they have already bounced against the obstacle, ornot, before τ . If none of the particles have bounced, the situation is already studied in [10] . The conclusionin that case is that v has to lie in the cylinder K ( v , x − x , Ra/ε ), as soon as ε ≥ √ a .It remains then three cases to study, specific to the presence of the wall. Before the collision at τ : • when only the first particle has bounced against the wall, one has (cid:0) T ,ε − τ ( Z ) (cid:1) X, = S ε ( x − τ v ) and (cid:0) T ,ε − τ ( Z ) (cid:1) X, = x − τ v . We have (cid:12)(cid:12)(cid:0) T ,ε − τ ( Z ) (cid:1) X, − (cid:0) T ,ε − τ ( Z ) (cid:1) X, (cid:12)(cid:12) = (cid:12)(cid:12)(cid:0) S ε ( x ) − x (cid:1) − τ (cid:0) S ( v ) − v (cid:1)(cid:12)(cid:12) , which is equal to ε by definition of τ . We now get rid of the vectors x and x using the fact that S ε ( u ) − S ε ( v ) = S ( u ) − S ( v ) and that S is linear, writing: S ε ( x ) − x = S ( x − x ) + S ε ( x ) − x + ( x − x ) , which provides, since | x i − x i | ≤ a (for i = 1 ,
2) and ε ≤ a , that (cid:12)(cid:12) S ε ( x ) − x − τ (cid:0) S ( v ) − v (cid:1)(cid:12)(cid:12) ≤ a. (27)Repeating the proof of [10], we find that S ( v ) − v belongs to the cone C of vertex 0 (in R d ), based on theball centered on S ε ( x ) − x and of radius 3 a . Now, since (cid:12)(cid:12) S ε ( x ) − x (cid:12)(cid:12) ≥ | x − x | ≥ ε (the first inequalityis an easy consequence of the fact that, if we denote x θ the element of the segment (cid:2) x , S ε ( x ) (cid:3) , one has onthe one hand (cid:12)(cid:12) S ε ( x ) − x θ (cid:12)(cid:12) = | x − x θ | , and on the other hand (cid:12)(cid:12) S ε ( x ) − x (cid:12)(cid:12) = (cid:12)(cid:12) S ε ( x ) − x θ (cid:12)(cid:12) + | x θ − x | ),if ε ≥ √ a , the cylinder K (0 , S ε ( x ) − x , Ra/ε ) contains the intersection of the cone C with the ball B (0 , R ), so this cylinder contains v − S ( v ).However, this result is not entirely satisfactory, since the axis of the cylinder depends on ε (which would causetrouble in the preparation of the pseudo-trajectories, see Proposition 5 below). To eliminate this dependency,we simply write S ε ( x ) = S ( x ) + εe , so that we replace (27) by (cid:12)(cid:12) S ( x ) − x − τ (cid:0) S ( v ) − v (cid:1)(cid:12)(cid:12) ≤ a + ε ≤ a, leading to the conclusion that v ∈ K ( S ( v ) , S ( x ) − x , Ra/ε ). • When only the second particle has bounced, we obtain here the condition (cid:12)(cid:12)(cid:0) x −S ( x ) (cid:1) − τ (cid:0) v −S ( v ) (cid:1)(cid:12)(cid:12) = ε . We use now the fact that S is a linear, involutive isometry, so that the condition can be rewritten as (cid:12)(cid:12) S (cid:0) x −S ε ( x ) (cid:1) − τ (cid:0) S ( v ) − v (cid:1)(cid:12)(cid:12) = ε . Taking care again of removing the dependency on ε in S ( x −S ε ( x ))and replacing x i by x i (for i = 1 , (cid:12)(cid:12) S ( x − S ( x )) − τ (cid:0) S ( v ) − v (cid:1)(cid:12)(cid:12) ≤ a after writing S (cid:0) x − S ε ( x ) (cid:1) = S (cid:0) x − S ( x ) − εe (cid:1) = S (cid:0) x − S ( x ) (cid:1) − ε S ( e ). We deduce again that v has to lie in K (cid:0) S ( v ) , S ( x − S ( x )) , Ra/ε (cid:1) . See Lemma 12.2.1 page 96. And finally when the two particles have bounced, we have in that case (cid:0) T ,ε − τ ( Z ) (cid:1) X,i = S ε ( x i − τ v i ) for i = 1 ,
2, so that, using again the identity S ε ( u ) − S ε ( v ) = S ( u ) − S ( v ), we obtain that (cid:12)(cid:12)(cid:0) T ,ε − τ ( Z ) (cid:1) X, − (cid:0) T ,ε − τ ( Z ) (cid:1) X, (cid:12)(cid:12) = (cid:12)(cid:12) S (cid:0) ( x − τ v ) − ( x − τ v ) (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) ( x − x ) − τ ( v − v ) (cid:12)(cid:12) = ε, which turns out to be exactly the condition obtained without any bouncing before τ , corresponding to thecase studied in [10].For the first point of the lemma, we notice that it is the free flow which is involved, so there is no concernabout collisions between the particles to have. Let us start to say that the same four sub-cases as for theprevious point have to be considered, and the one without bouncing is already adressed in [10]. Let us presentthe proof when only the first particle has bounced against the wall before τ , a time such that: (cid:12)(cid:12)(cid:0) T , − τ ( Z ) (cid:1) X, − (cid:0) T , − τ ( Z ) (cid:1) X, (cid:12)(cid:12) ≤ ε . In that case, the condition writes explicitely (cid:12)(cid:12) S ( x ) − x − τ (cid:0) S ( v ) − v (cid:1)(cid:12)(cid:12) ≤ ε . Then, for all unit vector n orthogonal to S ( x ) − x , we have thanks to the Cauchy-Schwarz inequality that τ (cid:12)(cid:12) n · (cid:0) S ( v ) − v (cid:1)(cid:12)(cid:12) ≤ ε ,which means that S ( v ) − v belongs to K (0 , S ( x ) − x , ε /δ ), hence the conclusion. The other cases areobtained in the same way. This concludes the proof of Lemma 4.The second lemma studies the effect of the scattering of the cylinders, already stated and proved in [10]: Lemma 5 (Scattering Lemma for cylinders) . There exists a strictly positive constant C ( d ) depending onlyon the dimension d such that for all strictly positive numbers ρ and R , all vectors ( y, w ) ∈ R d × B (0 , R ) , and v ∈ B (0 , R ) , one has : |N ∗ ( w, y, ρ )( v ) | ≤ C R d +1 / ρ d − / with N ∗ ( w, y, ρ )( v ) = (cid:110) ( ω, v ) ∈ S d − × B (0 , R ) / ( v − v ) · ω > , v (cid:48) ∈ K ( w, y, ρ ) or v (cid:48) ∈ K ( w, y, ρ ) (cid:111) . Proof of Theorem 5.
There are two cases that have to be considered separately: whether the particles k and k + 1 are in a pre-collisional configuration or not. • Let us start with the pre-collisional case. It implies that the velocities of the pair of particles ( k, k + 1) arenot modified by the scattering.Considering now the recollisions (that is, if Z εk +1 belongs to G εk +1 ( ε ) or not, with Z εk +1 = ( Z k , x k + εω, v )), theycannot happen between the particles i and j with 1 ≤ i < j ≤ k by hypothesis. If now i < k and j = k + 1,a simple application of the Shooting Lemma 4 provides 4( k −
1) cylinders (for the velocity v ) of respectivevolume C ( d )(4 R ) (cid:0) Ra/ε (cid:1) d − to exclude in order to prevent recollision. The most interesting case is thenwhen i = k and j = k + 1. In that case, except if the velocities of the two particles are the same, no recollisioncan occur if none or both of the two particles have bounced against the wall. If only one of the two particleshas already bounced against the wall at τ , a time such that (cid:12)(cid:12)(cid:0) T k +1 ,ε − τ ( Z εk +1 ) (cid:1) X,k − (cid:0) T k +1 ,ε − τ ( Z εk +1 ) (cid:1) X,k +1 (cid:12)(cid:12) = ε ,this condition can be rewritten as (cid:12)(cid:12) − x k · e e − εω + εe − τ (cid:0) S ( v k ) − v (cid:1)(cid:12)(cid:12) = ε. (28)To obtain a control on v − S ( v k ) which does not depend on the positions of the particles nor on the angularparameter ω , we write: ε ≥ (cid:12)(cid:12) x k · e e − τ (cid:0) v − S ( v k ) (cid:1)(cid:12)(cid:12) − (cid:12)(cid:12) x k · e e − x k · e e (cid:12)(cid:12) − ε | ω | − ε | e |≥ (cid:12)(cid:12) x k · e e − τ (cid:0) v − S ( v k ) (cid:1)(cid:12)(cid:12) − a − ε, so that (cid:12)(cid:12) x k · e e − τ (cid:0) v − S ( v k ) (cid:1)(cid:12)(cid:12) ≤ a . Following the same proof as for Lemma 4, we deduce that v − S ( v k ) belongs to the cone of vertex 0 and based on the ball B (2 x k · e e , a ). We see here why we needto assume that the particle k is far enough from the boundary: if (cid:12)(cid:12) x k · e e (cid:12)(cid:12) ≤ a , the condition on thecone is empty, since we would describe the whole space with a cone centered on 0 and based on a ball which26ontains 0. So here, since ρ ≤ x k · e and 5 a/ √ ≤ ρ , we can deduce that the condition (28) implies that v ∈ K (cid:0) S ( v k ) , x k · e e , Ra/ρ (cid:1) . The measure of this cylinder, intersected with B (0 , R ), is then controlledby C ( d )(4 R ) (cid:0) Ra/ρ (cid:1) d − .Let us consider now the problem of the good configurations for the free flow (that is, if T k +1 , − δ (cid:0) Z k +1 (cid:1) belongsto G k +1 ( ε ) or not, with Z k +1 = ( Z k , x k , v )). Again, if the two particles i and j are such that 1 ≤ i < j ≤ k ,by hypothesis the distance between those particles will always be larger than ε . If i < k and j = k + 1, wecan again use the Shooting Lemma 4, so that up to exclude 2( k −
1) cylinders (for the velocity v ) of respectivevolume C ( d ) R (cid:0) ε /δ (cid:1) d − , the pair of particles ( i, k + 1) will stay at a distance larger than ε after a time δ .Now if i = k and j = k + 1, the condition (cid:12)(cid:12)(cid:0) T k +1 , − τ ( Z k +1 ) (cid:1) X,k − (cid:0) T k +1 , − τ ( Z k +1 (cid:1) X,k +1 (cid:12)(cid:12) ≤ ε (29)may happen either if only one of the two particles has bounced against the wall at time τ , or if none orboth have. In the last case, (29) writes (cid:12)(cid:12) ( x k − τ v k ) − ( x k − τ v ) (cid:12)(cid:12) = τ | v − v k | ≤ ε . If v / ∈ B ( v k , η ) with ε ≤ δη , this can never happen. In the first case, (29) writes (cid:12)(cid:12) τ (cid:0) v − S ( v k ) (cid:1) − (cid:0) S ( x k ) − x k (cid:1)(cid:12)(cid:12) ≤ ε . Wededuce then that for all unitary vector u orthogonal to e we have τ · (cid:12)(cid:12) u · (cid:0) v − S ( v k ) (cid:1)(cid:12)(cid:12) ≤ ε . Then, for τ ≥ δ and v / ∈ K (cid:0) S ( v k ) , e , ε /δ (cid:1) , the condition (29) cannot hold if only one of the particles has bounced.In both cases, up to exclude a subset of velocities of size C ( d ) (cid:0) η d + R (cid:0) ε /δ (cid:1) d − (cid:1) , (29) cannot hold. • It remains the post-collisional case to be investigated. Let us start with the recollisions for the hard spheredynamics. First, since the scattering does not modify the velocities of the particles i < k , the first recollisionobtained from Z ε, (cid:48) k +1 (with Z ε, (cid:48) k +1 = ( . . . , x k , v k (cid:48) , x k + εω, v (cid:48) )) cannot be between the particles i and j with1 ≤ i < j ≤ k − Z εk +1 ). In other words, if we get rid ofthe recollisions with the other pairs, it would imply that we will get rid also of the recollisions for the pairs1 ≤ i < j ≤ k −
1. If now i ≤ k − j = k , thanks to Lemma 4, up to exclude 4( k −
1) cylinders ofradius 12
Ra/ε for the velocity v k (cid:48) , we would make sure of the absence of recollision for those pairs. Similarly,for 1 ≤ i ≤ k − j = k + 1, by excluding the same 4( k −
1) cylinders as in the pre-collisional case(for v (cid:48) this time, and not v ), we would eliminate the recollisions for that case. However the conditionsdescribed here concern the post-collisonal velocities v k (cid:48) and v (cid:48) , not the adjunction parameters v and ω . Toconvert those conditions and obtain a control on those parameters, we will simply use Lemma 5. In summary,this lemma applied to the excluded cylinders together provides a set of adjunction parameters ( ω, v ) of size C ( d )( k − R d − (cid:0) a/ε (cid:1) d − / to exclude. It remains only the most delicate case to consider, when i = k and j = k + 1. Here one cannot repeat the argument of the pre-collisional case, for both post-collisional velocities v k (cid:48) and v (cid:48) depend on v (which prevent to define cylinders depending only on Z k ), and also on ω , which isanother source of difficulty. Now for that pair, since the configuration ( x k , v k (cid:48) , x k + εω, v (cid:48) ) is pre-collisional bydefinition of the scattering, no recollision can happen between those particles if none or both have bouncedagainst the wall. In the case when only one particle has already bounced at time τ , the condition of therecollision between the particles k and k + 1 writes (cid:12)(cid:12) x k · e e + εω − εe − τ (cid:0) v (cid:48) − S ( v k (cid:48) ) (cid:1)(cid:12)(cid:12) = ε . The naiveconsideration of the cylinder, as for the pre-collisional case, would provide a condition on v (cid:48) − S ( v k (cid:48) ), whichprevents to conclude directly. But writing v (cid:48) − S ( v k (cid:48) ) = v (cid:48) − v k (cid:48) − (cid:0) S ( v k (cid:48) ) − v k (cid:48) (cid:1) , and using the definitionof the post-collisional velocities, we have v (cid:48) − v k (cid:48) = (cid:0) v − v k (cid:1) − (cid:0) v − v k (cid:1) · ωω , showing that v (cid:48) − S ( v k (cid:48) )belongs to K (0 , e , Ra/ρ ) if and only if (cid:0) v − v k (cid:1) − (cid:0) v − v k (cid:1) · ωω belongs to the same cylinder. This is acondition on v (instead of v (cid:48) ), but depending also on ω . This condition is equivalent to that v − v k belongsto K (0 , e − e · ωω, Ra/ρ ). As a conclusion, up to exclude among the adjunction parameters ( ω, v ) thesubset (cid:0) (cid:91) ω ∈ S d − (cid:8) ω (cid:9) × K (0 , e − e · ωω, Ra/ρ ) (cid:1) , which has a size controlled by C ( d ) R (cid:0) Ra/ρ (cid:1) d − , we can claim that there will be no recollision between theparticles k and k + 1, which concludes the study of the recollisions in the post-collisional case.Let us finally consider the problem of the good configurations for the free flow in the post-collisional case.First, since only the velocities of the two last particles are modified with the scattering, no pair ( i, j ) ofparticles, with 1 ≤ i < j < k , can be closer than ε by hypothesis on Z k . Concerning the pairs ( i, k ) and( i, k + 1) for i < k , thanks to the Shooting Lemma 4, up to exclude 2( k −
1) cylinders of radius ε /δ among27he velocities v k (cid:48) and v (cid:48) , we can be sure that the particles will stay at a distance larger than ε one fromanother. The Scattering Lemma 5 enables to translate this condition on ω and v : it means that a subsetof size C ( d )( k − R d +1 / (cid:0) ε /δ (cid:1) d − / has to be excluded to keep the distance larger than ε between thoseparticles. Finally, for the particles k and k + 1, we have (cid:12)(cid:12)(cid:0) T k +1 , − τ ( Z , (cid:48) k +1 ) (cid:1) X,k − (cid:0) T k +1 , − τ ( Z , (cid:48) k +1 ) (cid:1) X,k +1 (cid:12)(cid:12) ≤ ε (with Z , (cid:48) k +1 = ( . . . , x k , v k (cid:48) , x k , v (cid:48) )) if and only if τ | v (cid:48) − v k (cid:48) | ≤ ε or (cid:12)(cid:12) τ (cid:0) v (cid:48) − S ( v k (cid:48) ) (cid:1) − (cid:0) S ( x k ) − x k (cid:1)(cid:12)(cid:12) ≤ ε . If | v (cid:48) − v k (cid:48) | > η with ε ≤ δη , that is, thanks to the conservation of the kinetic energy along thecollisions implying | v (cid:48) − v k (cid:48) | = | v − v k | , if | v − v k | ≥ η , the first condition cannot hold. The last condition (cid:12)(cid:12) τ (cid:0) v (cid:48) − S ( v k (cid:48) ) (cid:1) − (cid:0) S ( x k ) − x k (cid:1)(cid:12)(cid:12) ≤ ε is finally studied in the same fashion as for the post-collisional case ofthe hard sphere flow: up to exclude among the adjunction parameters ( ω, v ) the subset (cid:0) (cid:91) ω ∈ S d − (cid:8) ω (cid:9) × K (0 , e − e · ωω, ε /δ ) (cid:1) , of size C ( d ) R (cid:0) ε /δ (cid:1) d − , the second condition cannot hold.The subset B k ( Z k ) of Theorem 5 is now simply the collection of all the excluded subsets of the adjunc-tion parameters ( ω, v ) ∈ S d − × R described above: on its complement the adjunction parameters providenew configurations of k + 1 particles that are in a good configuration, and the size of B k ( Z k ) is controlled bythe sum of the size of the previous excluded subsets, which concludes the proof of Theorem 5. In Theorem 5, it was important that the particle undergoing the adjunction is far from the wall. Sadly, thereis no direct way to fulfill this condition, since the positions are not integration variables in the integrated intime collision operators, that is this condition cannot be obtained with a naive cut-off.However, there is a way to use the adjunction parameters to reach that goal. If a particle has not a grazingvelocity, it will not stay close to the wall for a long time. We will then remove the pseudo-trajectorieswith grazing collisions, and remove also the time intervals during which the particle chosen to undergo theadjunction is too close to the wall. To do so, an important (and quite technical) step is to control the effectof the scattering mapping on the grazing collisions.
Definition 10 (Adjunction parameters inducing grazing collisions after scattering) . For any v ∈ B (0 , R ) ,we will call the subset of the adjunction parameters inducing grazing collisions after adding a particle toanother one with velocity v and after scattering , and we will denote N ∗ ( R, α )( v ) ⊂ S d − × B (0 , R ) for theset defined by: N ∗ ( R, α )( v ) = ß ( ω, v ) ∈ S d − × B (0 , R ) / ( v − v ) · ω > , v (cid:48) ∈ {| v · e | ≤ α } or v (cid:48) ∈ {| v · e | ≤ α } ™ . It is possible then to obtain the following result.
Lemma 6.
There exist two strictly positive constants C ( d ) and c ( d ) depending only on the dimension d suchthat for all strictly positive numbers α ≤ c ( d ) and R ≥ , and all v ∈ B (0 , R ) , one has: (cid:12)(cid:12)(cid:12) N ∗ ( R, α )( v ) (cid:12)(cid:12)(cid:12) ≤ C ( d ) R α / (30) in the case d = 2 , and (cid:12)(cid:12)(cid:12) N ∗ ( R, α )( v ) (cid:12)(cid:12)(cid:12) ≤ C ( d ) R d − α (31) in the case d ≥ .Proof of Lemma 6. First, one needs an intermediate result about the measure of some subspace of a sphere,which follows. 28et α , r and R be three strictly positive numbers. In the case d = 2, if R ≥ α ≤ min (cid:110) , R / , (cid:0) √ −√ √ / √ (cid:1) (cid:111) ,2 √ α ≤ r ≤ R , and if (cid:12)(cid:12) x · e (cid:12)(cid:12) ≤ r − √ α , then (cid:12)(cid:12)(cid:12)(cid:8) y ∈ R / (cid:12)(cid:12) x − y (cid:12)(cid:12) = r, (cid:12)(cid:12) y · e (cid:12)(cid:12) ≤ α (cid:9)(cid:12)(cid:12)(cid:12) ≤ √ Rα / , (32)and in the case d ≥
3, there exists a constant C ( d ) depending only on the dimension d such that if r ≤ R ,then (cid:12)(cid:12)(cid:12)(cid:8) y ∈ R / (cid:12)(cid:12) x − y (cid:12)(cid:12) = r, (cid:12)(cid:12) y · e (cid:12)(cid:12) ≤ α (cid:9)(cid:12)(cid:12)(cid:12) ≤ C ( d ) r d − α. (33)This intermediate result is obtained after studying the three possible subcases concerning the position of thepoint x with respect to the hyperplane { x · e = 0 } . For this purpose, let us introduce the quantity p = x · e .Without loss of generality, let us assume that p ≥ • If r < p − α (the trivial case when the sphere does not cross {− α ≤ y · e ≤ α } ), then here of course (cid:12)(cid:12)(cid:12)(cid:8) y ∈ R / (cid:12)(cid:12) x − y (cid:12)(cid:12) = r, (cid:12)(cid:12) y · e (cid:12)(cid:12) ≤ α (cid:9)(cid:12)(cid:12)(cid:12) = 0. • If r ≥ p − α , and ( r ≤ p + α or r ≤ α ) (the case when the sphere crosses only a single plane delim-iting {− α ≤ y · e ≤ α } ), then either r ≤ α , and then here { y ∈ R / | x − y | = r, | y · e | ≤ α } ⊂ ∂B ( x, r ), sothat: (cid:12)(cid:12)(cid:12)(cid:8) y ∈ R / (cid:12)(cid:12) x − y (cid:12)(cid:12) = r, (cid:12)(cid:12) y · e (cid:12)(cid:12) ≤ α (cid:9)(cid:12)(cid:12)(cid:12) ≤ C ( d ) r d − , or r ≥ p − α and r < p + α , and in that case only one of thetwo apices (along the diameter parallel to e ) of the sphere is strictly in between the two planes delimiting {− α ≤ y · e ≤ α } . It is clear that in this particular case, the surface of the sphere is maximized when theapex is tangent to one of the two planes, so that one has here: (cid:12)(cid:12)(cid:12)(cid:8) y ∈ R d / | y − x | ≤ r, | y · e | ≤ α (cid:9)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:8) y ∈ R d / | y − ( r − α ) e | ≤ r, | y · e | ≤ α (cid:9)(cid:12)(cid:12)(cid:12) . • Finally, the only remaining case is when r > p + α and r > α , that is when the sphere is large enoughto have both its apices (along the diameter parallel to e ) outside {− α ≤ y · e ≤ α } . The question now isthen when the measure of the surface contained between the two planes y · e = ± α is maximal. We willinvestigate this using an explicit computation, and separating the two cases d = 2 and d ≥ d ≥
3. One has (cid:12)(cid:12)(cid:12)(cid:8) y ∈ R d / (cid:12)(cid:12) x − y (cid:12)(cid:12) ≤ r, (cid:12)(cid:12) y · e (cid:12)(cid:12) ≤ α (cid:9)(cid:12)(cid:12)(cid:12) = (cid:90) ( x + r S d − ) ∩{| y · e |≤ α } d S = (cid:90) [ p − r,p + r ] ∩ [ − α,α ] (cid:12)(cid:12)(cid:12)(cid:16) » r − ( p − z ) (cid:17) S d − (cid:12)(cid:12)(cid:12) d z. The hypotheses about p , r and α imply that [ − α, α ] ⊂ [ p − α, p + r ], so that (cid:12)(cid:12)(cid:12)(cid:8) y ∈ R d / (cid:12)(cid:12) x − y (cid:12)(cid:12) ≤ r, (cid:12)(cid:12) y · e (cid:12)(cid:12) ≤ α (cid:9)(cid:12)(cid:12)(cid:12) = (cid:90) α − α (cid:0) r − ( p − y ) (cid:1) ( d − / (cid:12)(cid:12) S d − (cid:12)(cid:12) d y = C ( d ) r d − (cid:90) ( α − p ) /r − ( α + p ) /r (1 − u ) ( d − / d u. The function p (cid:55)→ (cid:82) ( α − p ) /r − ( α + p ) /r (1 − u ) ( d − / is decreasing for p ≥ {− α ≤ y · e ≤ α } ), one has (cid:90) ( α − p ) /r − ( α + p ) /r (1 − u ) ( d − / d u ≤ (cid:90) α/r (1 − u ) ( d − / d u ≤ αr . We obtained in the end for the case d ≥ (cid:12)(cid:12)(cid:12)(cid:8) y ∈ R d / (cid:12)(cid:12) x − y (cid:12)(cid:12) ≤ r, (cid:12)(cid:12) y · e (cid:12)(cid:12) ≤ α (cid:9)(cid:12)(cid:12)(cid:12) ≤ C ( d ) r d − α,
29o that (33) is proved.When d = 2, the set (cid:8) y ∈ R / (cid:12)(cid:12) x − y (cid:12)(cid:12) ≤ r, (cid:12)(cid:12) y · e (cid:12)(cid:12) ≤ α (cid:9) corresponds to (cid:8) ( y , y ) ∈ R / ( y − p ) + y = r , − α ≤ y ≤ α (cid:9) , so that the explicit computation leads to (cid:12)(cid:12)(cid:12)(cid:8) y ∈ R / (cid:12)(cid:12) x − y (cid:12)(cid:12) ≤ r, (cid:12)(cid:12) y · e (cid:12)(cid:12) ≤ α (cid:9)(cid:12)(cid:12)(cid:12) = 2 (cid:90) α − α y − p ) r − ( y − p ) d y = 2 r (cid:16) arccos (cid:16) − ( α + p ) r (cid:17) − arccos (cid:16) α − pr (cid:17)(cid:17) (34)The goal now is to simplify the quantity (34), firstly by removing the dependency with respect to p , then to r , and lastly by obtaining a more convenient expression.First, the quantity (34), seen as a function of p , is increasing (as a simple computation of the derivative showsit), so after introducing a cut-off on the values of p that are close to the upper bound p = r , namely: p ≤ r − α a , (35)with a ∈ ]0 ,
1[ (so that α a − α > α small enough), we find2 r (cid:16) arccos (cid:16) − ( α + p ) r (cid:17) − arccos (cid:16) α − pr (cid:17)(cid:17) ≤ r (cid:16) arccos (cid:16) α a − αr − (cid:17) − arccos (cid:16) α a + αr − (cid:17)(cid:17) . (36)Now, we consider the right hand side of the inequality (36) as a function of r , and again the computation ofthe derivative proves that it is increasing if and only ifarccos (cid:16) α a − αr − (cid:17) − arccos (cid:16) α a − αr − (cid:17) ≥ α a + α (cid:112) r ( α a + α ) − ( α a + α ) − α a − α (cid:112) r ( α a − α ) − ( α a − α ) · (37)If one assumes in addition that α is small compared to r , that is, explicitely, such that 2 α a ≤ r and α ≤ α a − α − r ≤ α a + α − r ≤ α a − r ≤
0, so that the arguments inside the two arccosine functions in (37)are negative. Since the arccosine function is convex on [ − , (cid:16) α a − αr − (cid:17) − arccos (cid:16) α a + αr − (cid:17) ≥ (cid:16)(cid:16) α a + αr − (cid:17) − (cid:16) α a − αr − (cid:17)(cid:17) … − (cid:16) α a + αr − (cid:17) ≥ α (cid:112) r ( α a + α ) − ( α a + α ) · As a consequence, the following inequality2 α (cid:112) r ( α a + α ) − ( α a + α ) ≥ α a + α (cid:112) r ( α a + α ) − ( α a + α ) − α a − α (cid:112) r ( α a − α ) − ( α a − α ) (38)would imply (37). But since α a − α >
0, (38) is equivalent to2 r ( α a − α ) − ( α a − α ) ≤ r ( α a + α ) − ( α a + α ) or again α a ≤ r , which was assumed, so (38), and then (37) hold. As a consequence, for 2 α a ≤ r ≤ R , onehas r (cid:16) arccos (cid:16) α a − αr − (cid:17) − arccos (cid:16) α a + αr − (cid:17)(cid:17) ≤ R (cid:16) arccos (cid:16) α a − αR − (cid:17) − arccos (cid:16) α a + αR − (cid:17)(cid:17) . (39)Finally, the upper bound of (39) can be simplified when α is small, using basically the idea that arccos( x ) − π ∼ − − (cid:112) x + 1). More precisely, using the identity: π − » x + 1) ≤ arccos( x ) ≤ π − » x + 1) , x ∈ [ − , − / (cid:16) α a − αR − (cid:17) − arccos (cid:16) α a + αR − (cid:17) ≤ α a + α ) R − α a − α ) R ≤ α a/ √ R (cid:16) √ (cid:112) α − a − √ (cid:112) − α − a (cid:17) ≤ α a/ √ R (cid:0) √ α − a / − √ − α − a ) (cid:1) , as soon as α ≤ ( R/ /a , implying in particular 0 ≤ ( α a + α ) /R ≤ /
3. If one has in addition that( √ / √ α − a ≤ − ( √ − √ α ≤ (cid:0) (1 + √ − √ / ( √ / √ (cid:1) / (1 − a ) , then in the endarccos (cid:16) α a − αR − (cid:17) − arccos (cid:16) α a + αR − (cid:17) ≤ α a/ √ R .
Multiplying the difference of the two arccosines by 2 r and keeping in mind that r is bounded by R , we recoverthe result (32) for the dimension d = 2, with the restrictions that were described concerning α , when wechoose a = 1 / N ∗ ( R, α )( v ), we recall that, by definition, this set is composed ofthe adjunction parameters ω and v leading to at least one grazing post-collisional velocity v (cid:48) (that is suchthat − α ≤ v (cid:48) · e ≤ α ) between v (cid:48) and v (cid:48) . Here it is important to recall the elementary geometrical propetriesfulfilled by those post-collisional velocities: by definition of the scattering operator, for v and v fixed, thetwo velocities v (cid:48) and v (cid:48) lie in the boundary of the ball centered on ( v + v ) / | v − v | / v (cid:48) and v (cid:48) delimiting a diameter of this ball. In addition, the three vectors v (cid:48) − v , v (cid:48) − v and ω have alwaysthe same direction.Now that the general geometrical setting has been set, let us now describe the main argument for the controlof the size of N ∗ ( R, α )( v ). Assuming that v and v are both fixed, if v (cid:48) is prescribed to lie in a given partof the ball centered on ( v + v ) / | v − v | /
2, what can be said about the part in which ω lies?In the case of v (cid:48) , it is also important to notice that v (cid:48) − v and ω have, in addition to the same direction, thesame orientation (since ( v − v ) · ω >
0: we are in the situation in which the scattering has to be applied).The situation can then be pictured as in Figure 8 below. The key observation is that, in the two dimensionalFigure 8: The link between the constraints over the position of v (cid:48) and the position of ω .case, if the vector v (cid:48) − ( v + v ) / v (cid:48) − v will cover half of this angle, since v (cid:48) − ( v + v ) / v + v ) / | v − v | /
2, while the origin of v (cid:48) − v lies on the boundary of this ball. Fromthis point, the link between the surface covered by v (cid:48) − v and the one covered by ω is a simple question ofscaling. Let us now make this argument rigorous. 31e saw that the idea is to solve the problem when v is fixed, and then deduce the general result by releasingthe constraint on v . We introduce then N ∗ ( α )( v , v ) = (cid:8) ω ∈ S d − / − α ≤ v (cid:48) · e ≤ α or − α ≤ v (cid:48) · e ≤ α (cid:9) , so that |N ∗ ( R, α )( v ) | = (cid:82) v ∈ B (0 ,R ) |N ∗ ( R, α )( v , v ) | d v .For the dimension d ≥
3, we have to parametrize properly the problem. Denoting ˜ r the norm of v − v and u ∈ S d − its normalized projection on the hyperplane v · e = 0, we have v = v ( u, ˜ r, θ ) = v + ˜ r (cos θe +sin θu ), with θ ∈ [0 , π ]. The polar coordinates provide then (cid:12)(cid:12) N ∗ ( R, α )( v ) (cid:12)(cid:12) ≤ C ( d ) (cid:90) S d − (cid:90) R (cid:90) π ˜ r d − (cid:12)(cid:12) N ∗ ( α ) (cid:0) v , v ( u, ˜ r, θ ) (cid:1)(cid:12)(cid:12) d θ d˜ r d u. Therefore, considering x = ( v + v ) / r = | v − v | / ω belongs to N ∗ ( α )( v , v ) if and only if v (cid:48) or v (cid:48) belong to ∂B ( x, r ) ∩ {| y · e | ≤ α } .We can now use the argument based on the inscribed angle theorem, since for any two-dimensional plane P through v , the intersection of this plane with ∂B ( x, r ) is a circle which contains v , while the intersectionof P with the ball centered on v and of radius | v − v | / r is a circle of the same radius (so that we areexactly in the situation described in Figure 8). We have then (cid:12)(cid:12)(cid:12) P ∩ ∂B ( v , r ) ∩ (cid:8) z / ∃ λ ∈ R / λ ( z − v ) ∈ {− α ≤ y · e ≤ α } ∩ ∂B ( x, r ) (cid:9)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12) P ∩ (cid:0) ∂B ( x, r ) ∩ {− α ≤ y · e ≤ α } (cid:1)(cid:12)(cid:12) , (it means that the green line on Figure 8 measures half the length of the blue line), so that (cid:12)(cid:12)(cid:12) ∂B ( v , r ) ∩ (cid:8) z / ∃ λ ∈ R / λ ( z − v ) ∈ {− α ≤ y · e ≤ α } ∩ ∂B ( x, r ) (cid:9)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12) ∂B ( x, r ) ∩ {− α ≤ y · e ≤ α } (cid:12)(cid:12) . Finally, and it is an important point, the elements z of ∂B ( v , r ) ∩ (cid:8) z / ∃ λ ∈ R / λ ( z − v ) ∈ {− α ≤ y · e ≤ α } ∩ ∂B ( x, r ) (cid:9) are such that z − v has a norm equal to r , and is colinear to ω (since z − v is colinear to v (cid:48) − v by construction). So we can state in the end that (cid:12)(cid:12) N ∗ ( α ) (cid:0) v , v ( u, ˜ r, θ ) (cid:1)(cid:12)(cid:12) = 1 r d − (cid:12)(cid:12) ∂B ( x, r ) ∩ {− α ≤ y · e ≤ α } (cid:12)(cid:12) . One will now conclude using the intermediate results (32) and (33). For the case d ≥
3, one has (cid:12)(cid:12) N ∗ ( R, α )( v ) (cid:12)(cid:12) ≤ C ( d ) (cid:90) S d − (cid:90) R (cid:90) π ˜ r d − (cid:16) r d − ˜ r d − α (cid:17) d θ d˜ r d u ≤ C ( d ) R d − α, which is the sought result for the case when the dimension is higher than 2.When d = 2, one has to be more careful, due to the restrictions specific to this case. We start by cutting offthe small difference between the velocities v and v (to take into account the constraint 2 √ α ≤ r ≤ R ): todo so, let us consider a parameter b ∈ ]0 , (cid:12)(cid:12) N ∗ ( R, α )( v ) (cid:12)(cid:12) ≤ (cid:90) B ( v ,α b ) (cid:12)(cid:12) N ∗ ( α )( v , v ) (cid:12)(cid:12) d v + (cid:90) B ( v , R ) \ B ( v ,α b ) (cid:12)(cid:12) N ∗ ( α )( v , v ) (cid:12)(cid:12) d v . Let us rewrite and bound from above the second term using again the polar coordinates, which take themuch simpler expression in the case d = 2: (cid:12)(cid:12) N ∗ ( R, α )( v ) (cid:12)(cid:12) ≤ (cid:90) B ( v ,α b ) (cid:12)(cid:12) N ∗ ( α )( v , v ) (cid:12)(cid:12) d v + (cid:90) Rα b (cid:90) π ˜ r (cid:12)(cid:12) N ∗ ( α ) (cid:0) v , v (˜ r, θ ) (cid:1)(cid:12)(cid:12) d θ d˜ r.
32n order to apply the control (32), we need now to take into account the constraint | x · e | ≤ r −√ α , that is here | v · e +(˜ r cos θ ) / | ≤ ˜ r/ −√ α . Remembering also that if r < | x · e |− α , then |{ y ∈ ∂B ( x, r ) / | y · e | ≤ α }| = 0,we decompose then the integral over θ as: (cid:90) Rα b (cid:90) π ˜ r (cid:12)(cid:12) N ∗ ( α ) (cid:0) v , v (˜ r, θ ) (cid:1)(cid:12)(cid:12) d θ d˜ r ≤ (cid:90) Rα b (cid:90) ˜ r/ −√ α< | v · e +(˜ r cos θ ) / |≤ ˜ r/ α ˜ r (cid:12)(cid:12) N ∗ ( α ) (cid:0) v , v (˜ r, θ ) (cid:1)(cid:12)(cid:12) d θ d˜ r + (cid:90) Rα b (cid:90) | v · e +(˜ r cos θ ) / |≤ ˜ r/ −√ α ) ˜ r (cid:12)(cid:12) N ∗ ( α ) (cid:0) v , v (˜ r, θ ) (cid:1)(cid:12)(cid:12) d θ d˜ r. The first term will be small since the domain of integration is small, and (32) will hold for the second term.More precisely, the condition ˜ r/ − √ α < | v · e + (˜ r cos θ ) / | ≤ ˜ r/ α contains actually the two conditions1 − r (cid:0) √ α + v · e (cid:1) < cos θ ≤ r (cid:0) α − v · e ) or − − r ( α + v · e ) < cos θ ≤ − r (cid:0) √ α − v · e (cid:1) . Using now the fact that the second derivative of the arccosine function is non negative on [ − , − ≤ x ≤ y ≤ x ) − arccos( y ) ≤ arccos( − − arccos( − y − x ). This providesthen thatarccos (cid:16) min (cid:0) , − r ( √ α + v · e (cid:1)(cid:17) − arccos (cid:16) min (cid:0) ,
1+ 2˜ r ( α − v · e (cid:1)(cid:17) ≤ arccos( − − arccos (cid:0) − r ( α + √ α ) (cid:1) ≤ arccos( − − arccos (cid:0) − r √ α (cid:1) and similarlyarccos (cid:16) max (cid:0) − , − − r ( α + v · e (cid:1)(cid:17) − arccos (cid:16) min (cid:0) − , − r ( √ α − v · e (cid:1)(cid:17) ≤ arccos( − − arccos (cid:0) − r √ α (cid:1) . To get rid of the arccosines in the expression, we use in the end that π − (cid:112) x + 1) ≤ arccos( x ) for all x ∈ [ − , − / − − arccos (cid:0) − r √ α (cid:1) ≤ … r √ α, up to have √ α/ ˜ r ≤ /
12, hence (cid:12)(cid:12)(cid:8) θ ∈ [0 , π ] / ˜ r/ − √ α < | v · e + (˜ r cos θ ) / | ≤ ˜ r/ α (cid:9)(cid:12)(cid:12) ≤ √ α / − b/ . Back to the decomposition of (cid:12)(cid:12) N ∗ ( R, α )( v ) (cid:12)(cid:12) , we have, bounding roughly the integrands of the two first terms(keeping simply in mind that N ∗ ( α )( v , v ) is by definition a part of the sphere S d − ) and applying at last(32) in the last one: (cid:12)(cid:12) N ∗ ( R, α )( v ) (cid:12)(cid:12) ≤ (cid:90) B ( v ,α b ) (cid:12)(cid:12) S d − (cid:12)(cid:12) d v + C (cid:90) Rα b (cid:90) ˜ r/ −√ α< | v · e +(˜ r cos θ ) / |≤ ˜ r/ α R (cid:12)(cid:12) S d − (cid:12)(cid:12) d θ d˜ r + C (cid:90) Rα b (cid:90) | v · e +(˜ r cos θ ) / |≤ ˜ r/ −√ α ) ˜ r (cid:16) r √ Rα / (cid:17) d θ d˜ r ≤ C (cid:0) α b + R α / − b/ + R / α / (cid:1) . It is important to notice that (32) holds for R ≥ α small enough, with a condition which does notdepend on R , and such that √ α ≤ ˜ r/
4. If one takes b < /
2, and if ˜ r ≥ α b , then ˜ r/ (4 √ α ) ≥ α b − / / → + ∞ when α →
0, so that there exists a constant c ( b ) such that for α ≤ c ( b ), one has ˜ r/ ≥ √ α and √ α/ ˜ r ≤ / r ∈ [ α b , R ], which enables indeed to use (32), and in addition to control the size of the domain of thesecond term.As a conclusion, since (cid:12)(cid:12) N ∗ ( R, α )( v ) (cid:12)(cid:12) ≤ CR (cid:0) α b + α / − b/ (cid:1) , choosing b = 1 / N ∗ ( R, α )( v ) is going to zero for R fixed and as α goes to zero. This concludesthe proof of Lemma 6. 33 .6 The convergence of the pseudo-trajectories All the tools are now in place to obtain pseudo-trajectories without recollisions. Now, we can see on whichset of initial configurations the geometrical results may be used to construct such pseudo-trajectories. First,let us introduce the domain of local uniform convergence:
Definition 11 (Domain of local uniform convergence) . let s be a positive integer. We introduce the domainof local uniform convergence , denoted as Ω s , as the subset Ω s of the phase space of s particles defined as Ω s = (cid:84) j =1 Ω js with Ω s = (cid:8) Z s / ∀ i (cid:54) = j, x i (cid:54) = x j (cid:9) , Ω s = (cid:8) Z s / ∀ i, v i · e (cid:54) = 0 (cid:9) , Ω s = (cid:8) Z s / ∀ i (cid:54) = j, v j / ∈ v i + Vect ( x i − x j ) (cid:9) , Ω s = (cid:8) Z s / ∀ i (cid:54) = j, v j / ∈ S ( v i ) + Vect (cid:0) S ( x i ) − x j (cid:1)(cid:9) , together with the following subsets in the phase space D εs ⊂ R ds : we set ∆ s = ∆ s ( ε, R, ε , α, γ ) = (cid:84) j =1 ∆ js , where ∆ s = ∆ s ( ε ) = (cid:8) ∀ ≤ i ≤ s, x i · e > ε/ (cid:9) , ∆ s = ∆ s ( R ) = (cid:8) | V s | ≤ R (cid:9) , ∆ s = ∆ s ( ε ) = (cid:8) min ≤ i
This proposition signifies that the goal of removing the recollisions, except for small subsets ofadjunction parameters with a controlled size, has been fulfilled. In addition, the closedness of the two differentpseudo-trajectories starting from the same initial configuration and undergoing the same adjunctions has beenquantitavily obtained.The definition of the elements U j l (cid:0) Z s,l − ( t l − ) (cid:1) and E j l (cid:0) Z s,l − ( t l ) (cid:1) may look a bit intricated, but they areperfectly well defined by recursion: starting from Z s and choosing the first particle j to undergo an adjunctionenables to defines U j , so we can choose t outside, enabling now to define Z s, ( t ) , and then E j . From thispoint, one can just iterate the process, choosing the second particle j for the adjunction, defining then U j ,and so on.Proof of Proposition 5. The proof is a simple induction using the geometrical lemmas.The first step consists in noticing that once Z s belongs to ∆ s ( ε, R, ε , α, γ ) with max( ε /δ, Ra/ε ) ≤ γ ,the velocities of this configuration are outside the cylinders described in Lemma 4, and its results apply:we have Z s ∈ G εs ( ε ), and T s, − δ ( Z s ) ∈ G s ( ε ). In addition, all the velocities v i of the configuration Z s satisfy | v i · e | ≥ α . As a consequence, for j , the index of the first particle undergoing an adjunction, the set U j ( Z s )of times τ such that | x j ( τ ) · e | < ρ has a measure smaller than 2 ρ/α . Then, for any time t (chosen forthe first adjunction) outside U j ( Z s ), the configuration Z s, ( t ) = T s, − t ( Z s ) fulfills the condition of Theorem5 (in particular, it is easy to show that the positions of the particles of the configuration Z εs, ( t ) are closeto the corresponding particles of Z s, ( t ) since the trajectories are easily explicitable, and the velocities arethe same up to apply the symmetry with respect to the wall), so this theorem provides a set B s (cid:0) Z s, ( t ) (cid:1) forwhich, if the adjunction parameters ( ω , v s +1 ) are taken outside of, the new configurations of s + 1 particles T s +1 , − δ (cid:0) Z s, ( t ) (cid:1) = Z s, ( t − δ ) and Z εs, ( t ) belong respectively to G s +1 ( ε ) and G εs +1 ( ε ). In order to havenew configurations of s + 1 particles at t such that all the velocities are not grazing (such that | v i · e | ≥ α ),one has to remove only the set {| v s +1 · e | ≤ α } , of measure C ( d ) R d − α , if the adjunction with ( ω , v s +1 )provides a pre-collisional configuration, and one has to remove the set N ∗ ( R, α )( v j ), described in Lemma 6 ifthe adjunction provides a post-collisional configuration (so that in that case the scattering has to be appliedimmediately, before applying the transport). Those three exclusions define the set E j (cid:0) Z s, ( t ) (cid:1) , with thesize verifying the control stated in the proposition. Then on the new configuration of s + 1 particles Z s, ( t )can be applied again the same process, and so on until the last k -th adjunction. This concludes the proof ofthe controls on the size of U j l and E j l for all 1 ≤ l ≤ k , and the absence of recollision.The control on the divergence between the two pseudo-trajectories Z and Z ε can be obtained again byrecursion, starting from the initial configuration Z s and the explicit expressions of the trajectories (whichare now easy, thanks to the absence of recollision). The difference between the pseudo-trajectories comesfrom the adjunctions, but unlike the case of the domain without obstacle, it also comes from the bouncingsagainst the obstacles, both of them generating errors of order ε . Proposition 5 is proven.Now that we know which adjunction parameters have to be removed, it is important to see what isthe impact on the terms of the Duhamel formula of such removals. Let us first introduce the truncated inadjunction parameters elementary terms of the hierarchies. Definition 12 (Truncated in adjunction parameters operators and elementary terms of the hierarchies) . Forany integer ≤ j ≤ s , any function f ( s +1) ∈ C (cid:0) [0 , T ] × (Ω c × R d ) s +1 (cid:1) with f ( s +1) ( t, · ) ∈ X ,s +1 ,β , for any t ∈ [0 , T ] , and for any functions U : Z s (cid:55)→ U ( Z s ) ∈ P ([0 , t − δ ]) and E : Z s (cid:55)→ E ( Z s ) ∈ P ( S d − × R d ) ofmeasurable subsets U ( Z s ) and E ( Z s ) , we define the truncated in adjunction parameters, integrated in timecollision-transport operator of the Boltzmann hierarchy of type ( ± , j ) the function t ≥ δ (cid:90) t − δ U ( Z s ) ( u ) (cid:90) S d − ω × R dvs +1 E ( Z s, ( u )) (cid:104) ω · (cid:0) v s +1 − v ,js, ( u ) (cid:1)(cid:105) ± f ( s +1) (cid:0) u, Z s, (cid:0) u, j, ( ω, v s +1 ) (cid:1) ( u ) (cid:1) d ω d v s +1 d u, enoted as I ,δs ± ,j ( U, E ) f ( s +1) ( t, Z s ) .In the case when E ( Z s ) = S d − × R d for every Z s (the surgery occurs only on the time variable), we willsimply denote the operator as I ,δs ± ,j ( U ) f ( s +1) ( t, Z s ) .Then, for any positive integer k , any J k = ( j , J k − ) ∈ J sk , M k = ( ± , M k − ) ∈ M k , for any measurablefunction f ( s + k ) : [0 , T ] × (cid:0) Ω c × R d (cid:1) s + k → R with f ( s + k ) ( t, · ) ∈ X ,s + k,β for any t ∈ [0 , T ] , and for anyfamilies of functions U J k = ( U j , . . . , U j k ) = ( U j , U J k − ) and E J k = ( E j , . . . , E j k ) = ( E j , E J k − ) (with U j l : Z s + l (cid:55)→ U ( Z s + l ) ∈ P ([0 , t − δ ]) and E j l : Z s + l (cid:55)→ E ( Z s + l ) ∈ P ( S d − × R d ) for all ≤ l ≤ k ), we definethe truncated in adjunction parameters elementary term of the Boltzmann hierarchy of type ( M k , J k ) thefunction defined by recursion t ≥ kδ (cid:90) t − δ ( k − δ U j ( Z s ) (cid:90) S d − ω × R dvs +1 E j ( Z s, ( t )) (cid:104) ω · (cid:0) v s +1 − v ,j s, ( t ) (cid:1)(cid:105) ± × (cid:16) I ,δs +1 ,s + k − M k − ,J k − ( U J k − , E J k − ) f ( s + k ) (cid:17)(cid:0) t , Z s, (cid:0) t , j , ( ω, v s +1 ) (cid:1) ( t ) (cid:1) d ω d v s +1 d t , denoted as I ,δs,s + k − M k ,J k ( U J k , E J k ) f ( s + k ) .In the same fashion, we introduce the truncated in adjunction parameters elementary terms of the BBGKYhierarchy of type ( M k , J k ) , defined here for h ( s + k ) ∈ L ∞ (cid:0) [0 , T ] × D εs + k (cid:1) , denoted I N,ε,δs,s + k − M k ,J k ) ( U J k , E J k ) h ( s + k ) . Remark 7.
Considering the two subsets of pathological adjunction parameters U j l and E j l defined andstudied in Proposition 5 the objective is to perform the decomposition I · ,δs,s + k − M k ,J k ) = I · ,δs,s + k − M k ,J k ) ( U cJ k , E cJ k ) + remainder terms. More explicitely, each operator in the iterated elementary terms will be decomposed as I ,δs ± ,j = I ,δs ± ,j ( U j ) + I ,δs ± ,j ( U cj , E ) + I ,δs ± ,j ( U cj , E cj ) , where here only the last term will not be a remainder.Concerning the truncated in pathological adjunction parameters of the BBGKY hierarchy, it is importanthere to notice that the elementary terms are not defined as usual iterated integrals . The same work as inSection 3.1 has to be done, and the operator obtained in the limit will behave like an integral, in the sensethat for two subsets E and E of adjunction parameters in S d − × R d such that | E ∩ E | = 0 , one has I N,ε,δs ± ,j ( E ∩ E ) = I N,ε,δs ± ,j ( E ) + I N,ε,δs ± ,j ( E ) , enabling in particular the surgery on pathological adjunctionparameters. For more details on the definition of those objects, the reader may refer to [9] . Let us now introduce the relevant notations describing this surgery. We define the elementary BBGKYterm , denoted J N,ε,δs,s + k − M k ,J k ( U c , E c ) f ( s + k ) N, , as the function( t, Z s ) (cid:55)→ T s,εt (cid:16) I N,ε,δs,s + k − M k ,J k ( U cJ k , E cJ k ) f ( s + k ) N, ( t, Z s ) (cid:17) and the elementary Boltzmann term , denoted J ,δs,s + k − M k ,J k ( U c , E c ) f ( s + k )0 , as the function( t, Z s ) (cid:55)→ I ,δs,s + k − M k ,J k ( U cJ k , E cJ k ) (cid:16) u (cid:55)→ T s + k, u f ( s + k )0 (cid:17) ( t, Z s ) , where U and E are the notations for the families of subsets ( U , . . . , U k ) ⊂ [0 , T ] k and ( E , . . . , E k ) ⊂ (cid:0) S d − × R d (cid:1) k . With those elementary terms, taking into account the surgeries in adjunction parameters, wecan constitute approximations of the solutions: F n,R,δN ( U c , E c ) = t (cid:55)→ (cid:16) T s,εt f ( s ) N, ( · ) | V s |≤ R + n (cid:88) k =1 s ≤ N − k (cid:88) M k ∈ M k (cid:88) J k ∈ M sk J N,ε,δs,s + k − M k ,J k ( U c , E c ) (cid:0) f ( s + k ) N, | V s + k |≤ R (cid:1) ( t, · ) (cid:17) ≤ s ≤ N In particular, see the Section “Rigorous definition of the truncated in adjunction parameters, integrated in time transport-collision-transport of the BBGKY hierarchy”, starting page 416. F n,R,δ ( U c , E c ) = t (cid:55)→ (cid:16) T s, t f ( s )0 ( · ) | V s |≤ R + n (cid:88) k =1 (cid:88) M k ∈ M k (cid:88) J k ∈ J sk J ,δs,s + k − M k ,J k ( U c , E c ) (cid:0) f ( s + k )0 | V s + k |≤ R (cid:1) ( t, · ) (cid:17) s ≥ for the Boltzmann hierarchy.The purpose of the following lemma is then to measure the error produced by the surgery in the pathologicaladjunction parameters. Lemma 7 (Surgery with the adjunction parameters for the elementary terms of the hierarchies) . Let β > and µ be two real numbers. Then, there exist two strictly positive constants c ( d ) and C ( d, β , µ ) such thatthe following holds.Let N and n be two positive integers, ε , R and δ be three strictly positive numbers such that the Boltzmann-Grad limit N ε d − = 1 holds. For any pair F N, = ( f ( s ) N, ) ≤ s ≤ N and F = ( f ( s )0 ) s ≥ of sequences of initialdata belonging respectively to X N,ε,β ,µ and X ,β ,µ , and any strictly positive numbers a , ε , ρ , η and α such that R ≥ , η ≤ , ε ≤ a, √ a ≤ ε , ε ≤ ηδ, a ≤ ρ, α ≤ c ( d ) , one has for the BBGKY hierarchy: ∆ s ( Z s ) (cid:12)(cid:12)(cid:12) T s,εt (cid:0) H n,R,δN (cid:1) ( s ) − (cid:0) F n,R,δN ( U c , E c ) (cid:1) ( s ) (cid:12)(cid:12)(cid:12) ( t, Z s ) exp (cid:16) (cid:101) β λ ( t )2 | V s | (cid:17) ≤ C n ( s + n ) R (cid:16) ρα + η d + R d (cid:16) aρ (cid:17) d − + nR d − (cid:16) aε (cid:17) d − / + nR d +1 / (cid:16) ε δ (cid:17) d − / + R d − α + R d α / (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( f ( s ) N, ) ≤ s ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε,β ,µ (42) and for the Boltzmann hierarchy: ∆ s ( Z s ) (cid:12)(cid:12)(cid:12)(cid:0) F n,R,δ (cid:1) ( s ) − (cid:0) F n,R,δ ( U c , E c ) (cid:1) ( s ) (cid:12)(cid:12)(cid:12) ( t, Z s ) exp (cid:16) (cid:101) β λ ( t )2 | V s | (cid:17) ≤ C n ( s + n ) R (cid:16) ρα + η d + R d (cid:16) aρ (cid:17) d − + nR d − (cid:16) aε (cid:17) d − / + nR d +1 / (cid:16) ε δ (cid:17) d − / + R d − α + R d α / (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( f ( s )0 ) s ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,β ,µ (43) with ∆ s = ∆ s (cid:0) ε, R, ε , max(16 Ra/ε , ε /δ ) (cid:1) , and U and E being the two families of pathological adjunctionparameters described in Proposition 5. The proof of Lemma 7 is presented in [9].Now it is time to obtain the crucial result of this work. Can we show that the convergence of the pseudo-trajectories implies the convergence of the solutions of the hierarchies? Relying on the regularity of theinitial data, the following result answers affirmatively the question, casting a decisive bridge between the twohierarchies.For this result, let us investigate the explicit expressions of the elementary terms. For the Boltzmann hier-archy, the final term J ,δs,s + k − M k ,J k ( U c , E c ) f ( s + k )0 writes: t ≥ kδ (cid:90) t − δ ( k − δ U cj ( Z s ) (cid:90) S d − ω × R dvs +1 ( ± ) E cj ( Z s, ( t )) (cid:104) ω · (cid:0) v s +1 − v ,j s, ( t ) (cid:1)(cid:105) ± × t ≥ ( k − δ (cid:90) t − δ ( k − δ U cj ( Z s, ( t )) (cid:90) S d − ω × R dvs +2 ( ± ) E cj ( Z s, ( t )) (cid:104) ω · (cid:0) v s +2 − v ,j s, ( t ) (cid:1)(cid:105) ± . . . × t k − ≥ δ (cid:90) t k − − δ U cjk ( Z s,k − ( t k − )) (cid:90) S d − ωk × R dvs + k ( ± k ) E cjk ( Z s,k − ( t k )) (cid:104) ω k · (cid:0) v s + k − v ,j k s,k − ( t k ) (cid:1)(cid:105) ± k × | V s + k |≤ R f ( s + k )0 (cid:0) Z s,k (0) (cid:1) d ω k d v s + k d t k . . . d ω d v s +2 d t d ω d v s +1 d t . not defined as usual integrals. However,thanks to the surgery in adjunction parameters, we will see that a new writing can be provided for thoseterms. But there are two remaining differences between the elementary terms of the two hierarchies: adifferent sequence of initial data, and the presence of prefactors (the product of the ( N − s ) ε d − terms).Both points will be addressed in the next section, so here let us introduce finally an hybrid version of theelementary term of the BBGKY hierarchy, getting rid of those differences, denoted J ε,δs,s + k − M k ,J k ( U c , E c ), anddefined as: t ≥ kδ (cid:90) t − δ ( k − δ U cj ( Z s ) (cid:90) S d − ω × R dvs +1 ( ± ) E cj ( Z s, ( t )) (cid:104) ω · (cid:0) v s +1 − v ε,j s, ( t ) (cid:1)(cid:105) ± × t ≥ ( k − δ (cid:90) t − δ ( k − δ U cj ( Z s, ( t )) (cid:90) S d − ω × R dvs +2 ( ± ) E cj ( Z s, ( t )) (cid:104) ω · (cid:0) v s +2 − v ε,j s, ( t ) (cid:1)(cid:105) ± . . . × t k − ≥ δ (cid:90) t k − − δ U cjk ( Z s,k − ( t k − )) (cid:90) S d − ωk × R dvs + k ( ± k ) E cjk ( Z s,k − ( t k )) (cid:104) ω k · (cid:0) v s + k − v ε,j k s,k − ( t k ) (cid:1)(cid:105) ± k × | V s + k |≤ R f ( s + k )0 (cid:0) Z εs,k (0) (cid:1) d ω k d v s + k d t k . . . d ω d v s +2 d t d ω d v s +1 d t (the only difference between J ,δs,s + k − M k ,J k ( U c , E c ) and J ε,δs,s + k − M k ,J k ( U c , E c ) lies in the choice of the pseudo-trajectories). Lemma 8 (Error coming from the divergence of the trajectories) . Let s and n be two positive integers, β > and µ be two real numbers. Then, there exists a time T (cid:48) satisfying ≤ T (cid:48) < T (where T is given byTheorem 3) such that, for any nonnegative, normalized (for the L norm) function f ∈ X , ,β with √ f beingLipschitz-continuous with respect to the position variable uniformly in the velocity variable, and any compactset K of the domain of local uniform convergence Ω s , there exist five strictly positive numbers ε , ε , α , γ and R (depending only on K ) such that for every strictly positive numbers R , δ , ε , a , ε , ρ , η and α which satisfy ε ≤ ε, ε ≤ ε , α ≤ α, max (cid:0) Rε/ε , ε /δ (cid:1) ≤ γ, R ≥ R and √ ≤ ε , a ≤ ρ, ε ≤ ηδ, R ≥ , η ≤ , α ≤ c ( d ) and finally nε ≤ a , then in the Boltzmann-Grad limit N ε d − = 1 , for all the parameters fixed but ε and N , one has the following uniform convergence on K and on [0 , T (cid:48) ] , with the explicit control on the rate ofconvergence: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K n (cid:88) k =1 (cid:88) M k ∈ M k (cid:88) J k ∈ J sk (cid:16) J ,δs,s + k − M k ,J k ( U c , E c ) (cid:0) f ⊗ ( s + k )0 | V s + k |≤ R (cid:1) − J ε,δs,s + k − M k ,J k ( U c , E c ) (cid:0) f ⊗ ( s + k )0 | V s + k |≤ R (cid:1)(cid:17) ( Z s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ≤ C ( d )( s + n ) nR εα (cid:12)(cid:12) ∇ x (cid:112) f (cid:12)(cid:12) ∞ || F || ,β / ,µ . Proof of Lemma 8.
The first step consists in noticing that the elementary term J ε,δs,s + k − M k ,J k is defined as a usualintegral, thanks to the first point of Proposition 5: after surgery, the pseudo-trajectories of the BBGKY hi-erarchy do not produce any recollision, and therefore between two adjunctions they are described only usingthe free transport and symmetries. It is now possible to apply the plan used in Section 3.1 (relying on thefact that the hard sphere transport coincides locally with the free transport) to give a sense to the collisionoperator, but this time without any time restriction (since there is no recollision to be avoided, so that thehard sphere transport coincides with the free transport, globally in time).The rest of the proof relies on the explicit expression of J ,δs,s + k − M k ,J k − J ε,δs,s + k − M k ,J k .In particular, the velocities of the particles j l chosen for the adjunctions are the same at the times of adjunc-tion t j for the two pseudo-trajectories, that is v ε,j l s,l − ( t l ) = v ,j l s,l − ( t l ) for all 1 ≤ l ≤ k , as a consequence ofthe second point of Proposition 5, together with the fact that the adjunctions happen only when the particlechosen is far from the wall, and the relation between ε and ρ : either the particles j l of both hierarchies havebounced against the wall at t l , or none of them has. 38hen, the only difference between the expressions of J ,δs,s + k − M k ,J k and J ε,δs,s + k − M k ,J k lies in Z s,k (0) and Z εs,k (0), therespective arguments of the function f ( s + k )0 , corresponding to the final configurations of the two pseudo-trajectories. Taking now into account that the velocities for the two pseudo-trajectories at time 0 may differby the symmetry S , we denote j , . . . , j p the labels of the particles having different velocities at time 0, and t j l (for 1 ≤ l ≤ p ) the times such that x ,j l s,k ( t j l ) · e = 0. Let us describe the process for the easy case s + k = 2to fix the ideas. We decompose: f ⊗ (cid:0) X s,k (0) , V s,k (0) (cid:1) − f ⊗ (cid:0) X εs,k (0) , V εs,k (0) (cid:1) = (cid:16) f (cid:0) x , s,k (0) , v , s,k (0) (cid:1) − f (cid:0) x , s,k ( t j ) , v , s,k (0) (cid:1)(cid:17) f (cid:0) x , s,k (0) , v , s,k (0) (cid:1) (44)+ f (cid:0) x , s,k ( t j ) , v , s,k (0) (cid:1)(cid:16) f (cid:0) x , s,k (0) , v , s,k (0) (cid:1) − f (cid:0) x , s,k ( t j ) , v , s,k (0) (cid:1)(cid:17) (45)+ f (cid:0) x , s,k ( t j ) , v , s,k (0) (cid:1) f (cid:0) x , s,k ( t j ) , v , s,k (0) (cid:1) − f ⊗ (cid:0) X εs,k (0) , V εs,k (0) (cid:1) , (46)the terms (44) and (45) being controlled respectively by | x , s,k (0) − x , s,k ( t j ) | and | x , s,k (0) − x , s,k ( t j ) | , while wecan use the boundary conditions for the term (46), replacing v ,ls,k (0) by v ε,ls,k (0), and then using the divergencebetween the pseudo-trajectories (second point of Proposition 5).In order to make those arguments quantitative, considering that t j l ≤ (2 kε + ε/ /α (because t = 0 lies in theinterval delimited by t j l and t j l ε , and | t j ε − t j l | ≤ (2 k +1 / ε/α ), we have | x ,j l s,k (0) − x ,j l s,k ( t j l ) | ≤ (2 k +1 / εR/α ,and writing: (cid:12)(cid:12)(cid:12)(cid:16) f (cid:0) x , s,k (0) , v , s,k (0) (cid:1) − f (cid:0) x , s,k ( t j ) , v , s,k (0) (cid:1)(cid:17) s + k (cid:89) l =2 f (cid:0) x ,ls,k (0) , v ,ls,k (0) (cid:1)(cid:12)(cid:12)(cid:12) ≤ (cid:16)(cid:113) f (cid:0) x , s,k (0) , v , s,k (0) (cid:1) + (cid:113) f (cid:0) x , s,k ( t j ) , v , s,k (0) (cid:1)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x , s,k (0) − x , s,k ( t j ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ x (cid:112) f (cid:12)(cid:12) ∞ (cid:12)(cid:12)(cid:12) s + k (cid:89) l =2 f (cid:0) x ,ls,k (0) , v ,ls,k (0) (cid:1)(cid:12)(cid:12)(cid:12) , implies (because √ f ∈ X , ,β / ) (cid:12)(cid:12) f ( s + k ) (cid:0) Z s,k (0) (cid:1) − f ( s + k )0 (cid:0) Z εs,k (0) (cid:1)(cid:12)(cid:12) ≤ (cid:12)(cid:12) ∇ x (cid:112) f (cid:12)(cid:12) ∞ ( s + k ) (cid:16) s + k ) 2 k + 1 / α Rε + kε (cid:17) exp (cid:16) − β | v , s,k (0) | (cid:17) exp (cid:16) − β s + k (cid:88) l =2 | v ,ls,k (0) | (cid:17) . The conclusion is now obtained thanks to the conservation of the kinetic energy along the pseudo-trajectoriesexp (cid:16) − β | v , s,k (0) | (cid:17) exp (cid:16) − β (cid:80) s + kl =2 | v ,ls,k (0) | (cid:17) = exp (cid:16) − β | V s + k | (cid:17) , and the contracting property of theintegrated in time transport-collision operators. Lemma 8 is proven. To complete the proof of Lanford’s result and provide a quantitative version of Theorem 4, it remains to eval-uate the error produced with the replacement of the elementary terms J N,ε,δs,s + k − M k ,J k by the hybrid ones J ε,δs,s + k − M k ,J k .We recall that the two differences between those terms lie in the sequence of initial data and on the absenceof the prefactor.Let us start this last section with a result concerning the admissible Boltzmann initial data , proving thatthe tensorized initial data (cid:0) ( f ⊗ s ) s ≥ (cid:1) N of the Boltzmann hierarchy enable to define an associated sequenceof initial data (cid:0) f ( s ) N, (cid:1) ≤ s ≤ N for the BBGKY hierarchy (for any numbers of particles N ) which converges, for39ll s , locally uniformly on Ω s . For more details, the reader may refer to [9] , or especially to [10] for adetailed discussion and the proof of the following result. Proposition 6 (The tensorized initial data are Boltzmann admissible) . Let β > and µ be two realnumbers. For any nonnegative, normalized function f belonging to X , ,β such that e µ | f | , ,β ≤ , thesequence of chaotic configurations (cid:0) f ⊗ s (cid:1) s ≥ (defined for all s ≥ as f ⊗ s ( Z s ) = (cid:81) si =1 f ( z i ) ) is a sequenceof admissible Boltzmann data, that is, if for any N ∈ N ∗ we define: f ( N ) N, : Z N (cid:55)→ Z − N D εN f ⊗ N ( Z N ) with Z N = (cid:90) R dN D εN f ⊗ N d Z N , and f ( s ) N, ( Z s ) = (cid:90) R d ( N − s ) D εN f ( N ) N, d z s +1 . . . d z N , then we have: • for any N ∈ N ∗ with N ε d − = 1 , the sequence of initial data F N, = (cid:0) f ( s ) N, (cid:1) ≤ s ≤ N belongs to X N,ε,β ,µ , • sup N ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) f ( s ) N, (cid:1) ≤ s ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε,β ,µ < + ∞ , • for any positive integer s , we have f ( s ) N, −→ N → + ∞ f ( s )0 locally uniformly on Ω s , with the explicit rate ofconvergence: (cid:12)(cid:12) D εs f ⊗ s − f ( s ) N, (cid:12)(cid:12) ∞ ≤ C ( d ) sε | f | L ∞ (Ω c ,L ( R d )) . (47)It is now possible to draw the final link between the solutions of the two hierarchies: the next lemmaquantifies the difference J N,ε,δs,s + k − M k ,J k − J ε,δs,s + k − M k ,J k . Lemma 9 (Error coming from the substitution of the initial data and the prefactors) . Let s and n be twopositive integers, β > and µ be two reals numbers. Let f be a nonnegative, normalized function belongingto X , ,β and let K be a compact set of the domain of local uniform convergence Ω s .Then, there exists five strictly positive numbers ε , ε , α , γ and R , depending only on K , such that for everystrictly positive numbers R , δ , ε , a , ε , ρ , η and α which satisfy ε ≤ ε, ε ≤ ε , α ≤ α, max (cid:0) Rε/ε , ε /δ (cid:1) ≤ γ and R ≥ R , and √ ≤ ε , a ≤ ρ, ε ≤ ηδ, R ≥ , η ≤ , α ≤ c ( d ) and finally nε ≤ a , we have inthe Boltzmann-Grad limit N ε d − = 1 the uniform convergence on the compact set K , uniform on the timeinterval [0 , T ] , of the sequence of the sum of the elementary BBGKY terms with the sequence of initial data (cid:0) F N, (cid:1) N ≥ , associated to the sequence of tensorized initial data (cid:0) f ⊗ s (cid:1) s ≥ , towards the sum of the hybridterms with the sequence of tensorized initial data (cid:0) f ⊗ s (cid:1) s ≥ , that is: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K n (cid:88) k =1 (cid:88) M k ∈ M k (cid:88) J k ∈ J sk (cid:16) J ε,δs,s + k − M k ,J k ( U c , E c ) (cid:0) f ⊗ ( s + k )0 | V s + k |≤ R (cid:1) − J N,ε,δs,s + k − M k ,J k ( U c , E c ) (cid:0) f ( s + k ) N, | V s + k |≤ R (cid:1)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ ≤ C ( d )( s + n ) ε | f | L ∞ (Ω c ,L ( R d )) || F || ,β ,µ . Proof of Lemma 9.
This result is a direct consequence of the control (47) of Proposition 6, together with thefollowing bounds on the prefactors:1 ≥ ( N − s )!( N − s − k )! ε k ( d − ≥ (cid:0) ( N − s − k ) ε d − (cid:1) k ≥ k ln( N − s − k/N ) ≥ − k ( s + k ) N ≥ − s + n ) N · for N large enough (with respect to s and n , that are fixed here). Theorem 6 (Lanford’s theorem in the half-space) . Let β > and µ be two real numbers. There exist twotimes ≤ T (cid:48) < T such that the following holds:let f be a nonnegative normalized function belonging to X , ,β which satisfies | f | , ,β ≤ exp( − µ ) andsuch that √ f is Lipschitz with respect to the position variable uniformly in the velocity variable. See Section 16.1 starting page 503. See in particular Section 6.1 ”Quasi-independence”, starting page 43. hen if one considers the solution F = (cid:0) f ( s ) (cid:1) s ≥ on [0 , T ] of the Boltzmann hierarchy with the tensorizedinitial datum F = (cid:0) f ⊗ s (cid:1) s ≥ , and if one considers for every positive integer N the solution F N = (cid:0) f ( s ) N (cid:1) ≤ s ≤ N on [0 , T ] of the BBGKY hierarchy with the initial datum F N, = (cid:0) f ( s ) N, (cid:1) ≤ s ≤ N , where (cid:0) F N, (cid:1) N ≥ is thesequence of initial data associated to the tensorized initial datum (cid:0) f ⊗ s (cid:1) s ≥ (in the sense of Proposition 6),one has that, in the Boltzmann-Grad limit N ε d − = 1 , for every positive integer s , the following locallyuniform convergence on the domain of local uniform convergence Ω s , uniform on the time interval [0 , T (cid:48) ] ,holds: f ( s ) N −→ N → + ∞ f ( s ) , moreover, whatever the dimension d is, the rate of convergence is of order O ( ε γ ) , for all γ ∈ ]0 , / .Proof of Theorem 6. Collecting all the errors obtained along the different cut-offs, if one can express theparameters a, ε , η, ρ, α, δ, R and n as explicit functions of ε , fulfilling the conditions: nε/a −→ ε → , a/ε −→ ε → , ε / ( ηδ ) −→ ε → , ε /δ −→ ε → , a/ρ −→ ε → , α −→ ε → R −→ ε → + ∞ , we would recover an explicit rate of convergence of the solutions of the BBGKY hierarchy towards the solutionof the Boltzmann hierarchy.To do so, we may choose δ = ε / | ln( ε ) | , a = ε | ln( ε ) | , ε = ε / | ln( ε ) | , η = ε / ,ρ = ε / | ln( ε ) | and α = ε / | ln( ε ) | , providing the claimed rate of convergence, and concluding the proof of Theorem 6. Remark 8.
The bound T (cid:48) of the time interval on which the convergence holds is smaller than the time T of co-existence of the solutions of the two hierarchies (provided by Theorem 3). This loss comes from thequantitative control of the difference between the terms with different pseudo-trajectories, especially from theLipschitz control on √ f , implying to consider only powers of the initial data (with a power a smaller than ) on which the collision operator will act. But this power forces to relax the weight β of the space X , ,β inwhich f a lies, making the size of the time interval in which the collision operator is contracting smaller.The limitation in the rate of convergence comes essentially from the geometrical estimates, that are clearlynot optimal. This limitation is then of technical order, and specific to the proof presented in this work.Nevertheless, it would be surprising to recover the rate of convergence O ( ε ) by optimal means. Indeed,concerning the derivation of the Boltzmann equation without obstacle, one can find in [2] an improvementof the explicit rate of convergence previously obtained in [10], namely ε ln( ε ) , obtained by the integrationof the singularity in time. Therefore, in the case of particles evolving outside of an obstacle, except if thisobstacle has a regularizing effect (which is probably not the case), one shouldn’t expect a better convergencethan ε ln( ε ) . Appendices
A The proof of the (almost everywhere) well-posedness of the hardsphere transport
Proof of Proposition 1.
Let δ ∈ ]0 , ρ ≥ R ≥ x = 0, or with high energies (that is, norms of the velocities): we set D εN,ρ,R = D εN ∩ Ä ( B R d (0 , ρ )) N × B R dN (0 , R ) ä . For every integer 1 ≤ i ≤ N , and any pair 1 ≤ j (cid:54) = k ≤ N with j (cid:54) = i and k (cid:54) = i , we also consider A ij,k ( δ ) = (cid:8) Z N ∈ D εN,ρ,R / | x i − x j | ≤ δR + ε, | x i − x k | ≤ δR + ε (cid:9) . Z N does not belong to this subset of initial configurations, the particle i will not be able to collide withthe particles j and k on the time interval [0 , δ ]. We have also: (cid:12)(cid:12) A ij,k (cid:12)(cid:12) ≤ C ( d, N ) ρ d ( N − R dN (cid:16) d (cid:88) k =1 Ç dk å (cid:0) δR (cid:1) k ε d − k (cid:17) . (48)Similarly, for all integers 1 ≤ i (cid:54) = j ≤ N , we consider B ij ( δ ) = (cid:8) Z N ∈ D εN,ρ,R / | x i − x j | ≤ δR + ε, d (Ω , x i ) ≤ δR + ε/ (cid:9) . If Z N does not belong to B ij ( δ ), the particle i will not collide with the particle j nor bounce against theobstacle during the time interval [0 , δ ]. We have in addition: (cid:12)(cid:12) B ij (cid:12)(cid:12) ≤ C ( d, N ) ρ d ( N − − R dN +1 δ d (cid:88) k =1 Ç dk å (2 δR ) k ε d − k . (49)Considering D εN,ρ,R \ C εN,ρ,R ( δ,
1) with C εN,ρ,R ( δ,
1) = Å (cid:16) (cid:91) ≤ i ≤ N ≤ j (cid:54) = k ≤ N, j (cid:54) = i, k (cid:54) = i A ij,k ( δ ) (cid:17) ∪ (cid:16) (cid:91) ≤ i ≤ N ≤ j ≤ N, j (cid:54) = i B ij ( δ ) (cid:17) ã , we obtained a subset of D εN,ρ,R composed of initial configurations which are all leading to a dynamics welldefined on the whole time interval [0 , δ ], with: (cid:12)(cid:12) C εN,ρ,R ( δ, (cid:12)(cid:12) ≤ C ( d, N ) R dN +2 Ä ρ d ( N − ε d − + ρ d ( N − − ε d − ä δ (50)for δ small enough with ε , R and ρ fixed.Now let t be a given strictly positive constant, we choose δ > t/δ is a positive integer m . Bythe definition of the dynamics, and since Z N ∈ B R d (0 , ρ ), at time δ , the transport sends Z N in D εN,ρ + δR,R .Remembering that ε ≤
1, the bound (50) can be rewritten using: Ä ρ d ( N − ε d − + ρ d ( N − − ε d − ä ≤ ρ d ( N − − ε d − . (51)Then, following the same steps as above and thanks to the bound (50) rewritten with (51), and up to excludinga small subset C εN,ρ + δR,R ( δ,
1) of D εN,ρ + δR,R of size bounded by C ( d, N ) R dN +2 ( ρ + δR ) d ( N − − ε d − δ , thedynamics is well-defined until 2 δ . Since the hard sphere flow preserves the measure (see for example [8] ),it is possible to exclude from the set of the initial configuration a subset denoted C εN,ρ,R ( δ, C εN,ρ,R ( δ,
1) and T N,ε − δ (cid:0) C εN,ρ + δR,R ( δ, (cid:1) ∩ (cid:0) B R d (0 , R ) N × B R dN (0 , R ) (cid:1) , such that outside, the dynamics iswell-defined until 2 δ .By induction, up to excluding a subset C εN,ρ,R ( δ, m ) which has a size smaller than C ( d, N ) R dN +2 ε d − δ m − (cid:88) k =0 Ä ( ρ + kδR ) d ( N − − ä , the dynamics is well defined on the whole time interval [0 , t ]. Taking ρ = R , the size of the excluded set isthen bounded by (remembering that t = mδ ): C ( d, N ) R d (2 N − ε d − t (1 + ( m − δ ) d ( N − − δ ≤ C ( d, N, t, ε ) R d (2 N − δ. (52)Considering the subset F εN,R ( t ) = (cid:84) m ∈ N ∗ C εN,R,R ( t/m, m ), of measure zero, any initial configuration takenin D εN,R,R = D εN ∩ Ä ( B R d (0 , R )) N × B R dN (0 , R ) ä and outside F εN,R ( t ) produces a well-defined dynamics on[0 , t ]. In particular, see Appendix 4.A ”More About Hard-Sphere Dynamics”. R n ) n ∈ N going to infinity with R ≥
1, and F εN ( t ) = (cid:83) n ∈ N F εN,R n ( t ), of measurezero, we see that it contains all the initial configurations leading to an ill-defined dynamics before the time t .Finally, considering F εN = (cid:83) p ∈ N ∗ F εN ( pt ), any initial configuration Z N ∈ D εN taken outside this subset F εN ,leads to a well-defined dynamics, and the first point of Proposition 1 is proven.For the second point, we will consider an initial configuration of D εN , which is not an element of F εN . Wesaw that this subset is of measure zero, and the dynamics from this initial configuration is well-defined onthe whole time interval R + .On the one hand, if one assumes that E ( Z N ) admits an accumulation point, say t > t k ) k ∈ N such that t k −→ k → + ∞ t ), since there is only a finite number of particles (here : N ),there exists 1 ≤ N ≤ N and a subsequence ( t ϕ ( k ) ) k ∈ N of ( t k ) k ∈ N , such that all the events of this subsequencecorrespond to the particle N : each event t ϕ ( k ) is either a collision or a bouncing involving N .On the other hand, going back to the definition of the subset F εN , the fact that Z N / ∈ F εN exactly means that: ∀ t > , ∀ p ∈ N ∗ , ∀ n ∈ N , Z N / ∈ F εN,R n ( pt ) , where F εN,ρ n ,R n ( pt ) is defined above, that is: ∀ t > , ∀ p ∈ N ∗ , ∀ n ∈ N , ∃ m ∈ N ∗ / Z N ∈ C εN,R n ( pt/m, m ) . In particular, one chooses t = 2 t , where t is the accumulation point of the set of events E ( Z N ) mentionnedabove, so that t ∈ ]0 , t [, and then there exists k ∈ N so that for all k ≥ k , we have t ϕ ( k ) ∈ ]0 , t [. Since thesequence (cid:0) R n (cid:1) n ∈ N is increasing and tends to infinity, there exists n ∈ N such that Z N ∈ (cid:0) B R d (0 , R n ) (cid:1) N × B R dN (0 , R n ). Finally, for t = 2 t , n = n and p = 1, there exists m ∈ N ∗ such that Z N ∈ C εN,R n ( t/m , m ).So all the events concerning the particle N are separated by a time interval of length larger than t/m , whichis obviously a contradiction. Proposition 1 is entirely proven. B The proof of the contracting property of the BBGKY and Boltz-mann operators
B.1 Stability of the spaces ‹ X N,ε, (cid:101) β, (cid:101) µ α and ‹ X , (cid:101) β, (cid:101) µ α under the action of the operators E N,ε and E The first step to obtain Theorem 3 page 15 is to show that the BBGKY and the Boltzmann opera-tors E N,ε (cid:0) ( f ( s ) N, ) ≤ s ≤ N , · (cid:1) and E (cid:0) ( f ( s )0 ) s ≥ , · (cid:1) (for appropriate sequences of initial data ( f ( s ) N, ) ≤ s ≤ N and( f ( s )0 ) s ≥ ), send respectively the spaces ‹ X N,ε, (cid:101) β, (cid:101) µ α and ‹ X , (cid:101) β, (cid:101) µ α into themselves.Concerning the first terms f ( s ) N, and T s, t f ( s )0 of the two operators, on the one hand it is immediate thatif ( f ( s ) N, ) ≤ s ≤ N ∈ X N,ε, (cid:101) β (0) , (cid:101) µ (0) α , then the constant function with respect to time t (cid:55)→ ( f ( s ) N, ) ≤ s ≤ N belongsto the space ‹ X N,ε, (cid:101) β, (cid:101) µ α , and this for any time interval [0 , T ], as soon as the two weight functions (cid:101) β and (cid:101) µ are non increasing (as an obvious consequence of the embedding result Proposition 2). On the other hand,concerning the Boltzmann hierarchy, the same result can be proven (see [10] ), that is if ( f ( s )0 ) s ≥ belongs to X , (cid:101) β (0) , (cid:101) µ (0) α , then t (cid:55)→ ( T s, t f ( s )0 ) s ≥ belongs to ‹ X , (cid:101) β, (cid:101) µ α , up to assume in addition that the weight functions (cid:101) β and (cid:101) µ are decreasing (and not only non increasing).This additional hypothesis is another element showing that the conjugated hierarchies (as (12)) are moreregular than the ”usual” hierarchies (9) and (11).Concerning the second terms defining the BBGKY and the Boltzmann operators, that is, the integratedin time collision operators, the functional spaces are designed exactly so that the elements of the spaces ‹ X N,ε, (cid:101) β, (cid:101) µ α and ‹ X , (cid:101) β, (cid:101) µ α , under the action of those integrated in time collision operators, provide functionswith finite |||·||| N,ε, ˜ β, ˜ µ α and |||·||| , ˜ β, ˜ µ α norms. See Lemma 20 page 239 .2 The contracting property of the operators E N,ε and E Before Theorem 3, the choice of the time interval on which we defined the spaces ‹ X N,ε, (cid:101) β, (cid:101) µ α and ‹ X , (cid:101) β, (cid:101) µ α was arbitrary (and then could be chosen as long as one wants). Here, in order to use the fixed point theo-rem in the Banach spaces, we need to show that the operators E N,ε and E are contracting mappings in the |||·||| N,ε, (cid:101) β, (cid:101) µ α and |||·||| , (cid:101) β, (cid:101) µ α respectively. A time restriction will be necessary to deduce this contracting property.The original idea and computation are due to Ukai ([19]) and Uchiyama ([18]), but a recent version isof course presented in [10], and also in [9]. We first point out the fact that the computation of the normof those collision operators is the same for the BBGKY hierarchy and for the Boltzmann hierarchy. Wepresent only the case α = 1, for the integrated in time transport-collision-transport operator of the BBGKYhierarchy.For any t ∈ [0 , T ] and for any 1 ≤ s ≤ N −
1, and with Q denoting Q = (cid:12)(cid:12)(cid:12) (cid:90) t T s,ε − u C N,εs,s +1 T s +1 ,εu h ( s +1) N ( u, Z s ) d u (cid:12)(cid:12)(cid:12) , we have (thanks to Theorem 2): Q ≤ N − s ) ε d − | S d − | s (cid:88) i =1 (cid:90) t (cid:12)(cid:12)(cid:12) h ( s +1) N ( u, Z s +1 ) exp (cid:16) (cid:101) β ( u )2 s +1 (cid:88) j =1 | v j | (cid:17)(cid:12)(cid:12)(cid:12) L ∞ ( D εs +1 ) × (cid:90) R d (cid:0) | v i | + | v s +1 | (cid:1) exp (cid:16) − (cid:101) β ( u )2 s +1 (cid:88) j =1 | v j | (cid:17) d v s +1 d u. Remembering that we want to obtain a bound on the |||·|||
N,ε, (cid:101) β, (cid:101) µ norm of the function of sequences: t (cid:55)→ (cid:16) s ≤ N − (cid:90) t T s,ε − u C N,εs,s +1 T s +1 ,εu h ( s +1) N ( u, · ) d u (cid:17) ≤ s ≤ N , we consider the product of Q with exp (cid:16) (cid:101) β ( t )2 (cid:80) si =1 | v i | (cid:17) × exp (cid:0) s (cid:101) µ ( t ) (cid:1) and bound this product uniformly in t ∈ [0 , T ] and 1 ≤ s ≤ N −
1, which gives in the end (see [9] or [10] for more details): Q ( t, s, Z s ) ≤ C ( d, N, ε ) (cid:101) β λ ( T ) − d/ exp (cid:0) − (cid:101) µ λ ( T ) (cid:1) (cid:0) (cid:101) β λ ( T ) − / (cid:1) λ × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) h ( s +1) N (cid:1) ≤ s ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε, (cid:101) β λ , (cid:101) µ λ , (53)We obtained an explicit control of the |||·||| N,ε, (cid:101) β λ , (cid:101) µ λ norm of the integral term of the BBGKY operator. Itremains to choose wisely λ and T to obtain a contracting mapping.To finish showing that the integrated in time transport-collision-transport operator sends the space ‹ X N,ε, (cid:101) β λ , (cid:101) µ λ into itself, it is also important to check the left continuity for all t ∈ ]0 , T ] with respect to the | · | ε,s, (cid:101) β λ ( t ) norm, for any s . This verification is presented in [9] in details, and leads to the discussion page 13 aboutbalancing the strength of the continuity in time and the value of the parameter α .To perform this verification we consider, for each 1 ≤ s ≤ N fixed : Q exp (cid:16) (cid:101) β λ ( t )2 | V s | (cid:17) exp( s (cid:101) µ λ ( t )) with Q = (cid:12)(cid:12)(cid:12)(cid:90) tu T s,ε − τ C N,εs,s +1 T s +1 ,ετ h ( s +1) N ( τ, Z s ) d τ (cid:12)(cid:12)(cid:12) . Using the definition of thenorms | · | ε,s, (cid:101) β λ ( t ) , ||·|| N,ε, (cid:101) β λ , (cid:101) µ λ ( t ) and |||·||| N,ε, (cid:101) β λ , (cid:101) µ λ , and the fact that the functions (cid:101) β λ and (cid:101) µ λ are decreasing,we find, as above: Q exp (cid:16) (cid:101) β λ ( t )2 | V s | (cid:17) exp( s (cid:101) µ λ ( t )) ≤ C ( d, N, ε ) (cid:101) β λ ( T ) − d/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) h ( s ) N (cid:1) ≤ s ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε, (cid:101) β λ , (cid:101) µ λ exp( − (cid:101) µ λ ( T )) × (cid:16) s (cid:88) i =1 | v i | + s (cid:101) β λ ( T ) − / (cid:17) (cid:90) tu exp (cid:16) λ τ − t ) s (cid:88) i =1 | v i | + λs ( τ − t ) (cid:17) d τ. (cid:90) tu exp (cid:16) λ τ − t ) s (cid:88) i =1 | v i | + λs ( τ − t ) (cid:17) d τ ≤ (cid:90) tu d τ Ã (cid:90) tu exp (cid:16) λ τ − t ) s (cid:88) i =1 | v i | + λs ( τ − t ) (cid:17) d τ ≤ √ t − u » λ (cid:0) (cid:80) si =1 | v i | + 2 s (cid:1) , we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:90) tu T s,ε − τ C N,εs,s +1 T s +1 ,ετ h ( s +1) N ( τ, Z s ) d τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:16) (cid:101) β λ ( t )2 | V s | (cid:17) ≤ C ( d, N, ε ) (cid:101) β λ ( T ) − / exp( − ( s + 1) (cid:101) µ λ ( T )) (cid:0) (cid:101) β λ ( T ) − / (cid:1) √ λ √ s √ t − u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:0) h ( s ) N (cid:1) ≤ s ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N,ε, (cid:101) β λ , (cid:101) µ λ that is, for every 1 ≤ s ≤ N : (cid:12)(cid:12) h ( s ) N ( t ) − h ( s ) N ( u ) (cid:12)(cid:12) ε,s, (cid:101) β λ ( t ) −→ u → t − , hence the continuity in time (in the sense (15) of Definition 6 page 14) for the second term of the BBGKYoperator.Thanks to those controls, and following for example [10] , we can find, for any β > µ ∈ R , first aconstant λ (which is the decrease rate of the weights (cid:101) β and (cid:101) µ , that is, the speed of the loss of regularity), andthen a constant T (which is the length of the time interval on which (cid:101) β remains positive, so that the limitationon the size of the time interval on which we have existence of the solutions comes from this criterion) suchthat the |||·||| · , (cid:101) β λ , (cid:101) µ λ norm of the integrated in time collision operator is smaller than 1, hence the result ofTheorem 3 page 15 thanks to the fixed point theorem in the Banach spaces. Acknowledgements.
The author expresses his warm gratitude to Isabelle Gallagher and Laurent Desvil-lettes, for their constant support and invaluable advice, dispensed throughout the years spent preparing histhesis, from which this article is taken. The author is also deeply grateful to Chiara Saffirio and to theUniversity of Basel, for the ideal conditions and the serene atmosphere which made this work possible. Fi-nally, the author acknowledges the support of the Swiss National Science Foundation through the Eccellenzaproject PCEFP2 181153, and of the NCCR SwissMAP.
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