From Newton s law to the linear Boltzmann equation without cut-off

Transcription

1 From Newton s law to the linear Boltzmann equation without cut-off Nathalie Ayi (1) (1) Université Pierre et Marie Curie 27 Octobre 217 Nathalie AYI Séminaire LJLL 27 Octobre / 35

2 Organisation of the talk 1- Physical and historical context of the result 2- Difficulty in our framework 3- Presentation of the usual tools for the proof 4- Adaptation of the tools + new strategy in our context Nathalie AYI Séminaire LJLL 27 Octobre / 35

3 Context Kinetic theory of gases = Describe a gas as a physical system constituted of a large number of small particles. Statistical point of view : we are interested in the evolution of the density of particles f(t, x, v) where t = time x = position v = velocity For all infinitesimal volume dxdv around the point (x, v): f(t, x, v)dxdv = number of particles which have position x and velocity v at time t. Nathalie AYI Séminaire LJLL 27 Octobre / 35

4 Historical results: A fundamental example, the Boltzmann equation (1872) = the evolution equation for the density of particles of a sufficiently rarefied gas. t f + v. x f = }{{} Q(f, f) }{{} free transport localized binary collisions In the sixth problem of Hilbert (19), idea = Boltzmann equation as an intermediate step in the transition between atomistic and contiuous model for gas dynamics. Nathalie AYI Séminaire LJLL 27 Octobre / 35

5 Historical results: A fundamental example, the Boltzmann equation (1872) = the evolution equation for the density of particles of a sufficiently rarefied gas. t f + v. x f = }{{} Q(f, f) }{{} free transport localized binary collisions In the sixth problem of Hilbert (19), idea = Boltzmann equation as an intermediate step in the transition between atomistic and contiuous model for gas dynamics. Nathalie AYI Séminaire LJLL 27 Octobre / 35

6 The Boltzmann-Grad scaling Change of scale = passage to the limit on one precise parameter of the system Boltzmann-Grad scaling = N, Nε d 1 = 1. Rarefied gas : Nε d 1, not too much collisions. Nathalie AYI Séminaire LJLL 27 Octobre / 35

7 Historical Panorama Lanford proved the derivation of the Boltzmann equation from systems of particles in the context of hard-spheres (1975). Particles bounces off according to the laws of elastic reflection. Nathalie AYI Séminaire LJLL 27 Octobre / 35

8 Proof recently improved by Gallagher, Saint-Raymond and Texier, Pulvirenti, Saffirio and Simonella in the context of hard-spheres and short range potentials. (Figure: The Boltzmann equation and its application, Cercignani.) Our context: infinite-range potentials. Nathalie AYI Séminaire LJLL 27 Octobre / 35

9 The Boltzmann equation without cut-off The Boltzmann equation t f + v. x f = b(v v 1, ν) = cross-section, θ = deviation angle R d S d 1 (f f 1 ff 1 )b(v v 1, ν)dνdv 1 b(v v 1, ν) = b( v v 1, cos(θ)) integrability with respect to θ Boltzmann with cut-off non-integrability with respect to θ Boltzmann without cut-off Example Inverse-power law potentials: Φ(r) = 1 r s 1, s > 2. The cross-section satisfies b( v v 1, cos θ) = q(cosθ) v v 1 γ, sin d 2 θq(cos θ) Cθ 1 α, α > Nathalie AYI Séminaire LJLL 27 Octobre / 35

10 Hard-spheres, Short range potentials Boltzmann with cut-off Difficulty = infinite range potential singularity due to grazing collisions Boltzmann without cut-off Intuitive idea= use of compensation between gain and loss terms f(v)f(v 1 ) f(v )f(v 1) The Boltzmann equation t f + v. x f = (f f 1 ff 1 )b(v v 1, ν)dνdv 1 R d S d 1 Nathalie AYI Séminaire LJLL 27 Octobre / 35

11 The Hard-spheres case Microscopic Model: for i {1,..., N}, dx i = v i, x i, v i R d dt dv i =, x i (t) x j (t) > ε. dt If x i x j = ε, Appropriate quantities: v i = v i + ((v j v i ) ν)ν v j = v j ((v j v i ) ν)ν f N (t, Z N ) = distribution function of N particles with z i = (x i, v i ), Z N = (z 1,..., z N ). Marginal of order one : f (1) N (t, x 1, v 1 ) = f N dx 2... dx N dv 2... dv N. Nathalie AYI Séminaire LJLL 27 Octobre / 35

12 Formal justification of the limit Liouville equation + Green formula [ t f (1) N + v 1 x1 f (1) N = (N 1)εd 1 f (2) N (t, x 1, v 1, x 1 + εν, v 2) S d 1 R d N, t f + v 1 x1 f = S d 1 R d f (2) N (t, x 1, v 1, x 1 εν, v 2 ) ] ((v 2 v 1 ) ν) + dνdv 2. [ f (2) (t, x 1, v 1, x 1, v 2) ] f (2) (t, x 1, v 1, x 1, v 2 ) ((v 2 v 1 ) ν) + dνdv 2 If f (2) (Z 2 ) = f(z 1 )f(z 2 ) the Boltzmann equation. Key notion = Propagation of chaos. Nathalie AYI Séminaire LJLL 27 Octobre / 35

13 Technical proof The BBGKY hierarchy: for s < N, t f (s) N + V s Xs f (s) N = C s,s+1f (s+1) N Iterated Duhamel formula expression of the marginals as series. The Boltzmann hierarchy : g s (t, Z s ) = g(t, z 1 )g(t, z 2 )... g(t, z s ) with g solution of the Boltzmann equation is a solution of t g s + V s Xs g s = C s,s+1g s+1 Two steps to prove the result : - bounds for the series, - the termwise convergence of each term of the series. Key idea = Geometrical interpretation of the terms Nathalie AYI Séminaire LJLL 27 Octobre / 35

14 Iterated Duhamel s formula. Duhamel s formula: f (s) N The BBGKY series: f (s) N N s (t) = n= (t) = T s(t)f (s) N t t1 () + t T s (t t 1 )C s,s+1 f (s+1) N (t 1 )dt 1. tn 1... T s (t t 1 )C s,s+1 T s+1 (t 1 t 2 )C s+1,s+2... T s+n (t n ) f (s+n) N ()dt n... dt 1. The Boltzmann series: g (s) (t) = n t t1... tn 1 T s (t t 1 )C s,s+1t s+1(t 1 t 2 )C s+1,s+2... T s+n(t n ) g (s+n) ()dt n... dt 1 Strategy: Notion of pseudo-trajectories. Nathalie AYI Séminaire LJLL 27 Octobre / 35

15 v 3 v 1 v 2 FIGURE : t t1 T 1 (t t 1 )C 1,2 T 2 (t 1 t 2 )C 2,3 T 3 (t 2 )f (3) N ()dt 2dt 1 for the BBGKY hierarchy. t 2 t 1 t Representation of a pseudo-trajectory associated to the term Nathalie AYI Séminaire LJLL 27 Octobre / 35

16 v 3 v 1 v 2 FIGURE : t t1 3 (t 2 )f (3) ()dt 2dt 1 for the Boltzmann hierarchy. t 2 t 1 t Representation of a pseudo-trajectory associated to the term T 1 (t t 1 )C 1,2T 2 (t 1 t 2 )C 2,3T Strategy: Coupling of the pseudo-trajectories. N Nathalie AYI Séminaire LJLL 27 Octobre / 35

17 Recollision v 1 v 2 v 3 t t 2 Figure: An example of a recollision between particles 1 and 2 at time t. Strategy: Geometrical control of the recollisions t 1 t Nathalie AYI Séminaire LJLL 27 Octobre / 35

18 Strategy of the proof and limit Main term with no pathological situations converges to Boltzmann. Remainders associated to recollisions vanishes when passing to the limit. Limit : Short time validity of the results for the nonlinear Boltzmann equation due to the fact that compensation between gain and loss terms are ignored for the bounds of the series. Bodineau, Gallagher and Saint-Raymond (214): overcome the difficulty of the short time validity in the case of a fluctuation around the equilibrium derivation of the linear Boltzmann equation. Nathalie AYI Séminaire LJLL 27 Octobre / 35

19 The infinite range potential case Difficulty = The singularity prevents the single-use of Lanford s approach. Ideas = - framework of perturbation around the equilibrium, - Φ = φ >R + Φ <R, - new terms with presence of derivatives weak approach. Nathalie AYI Séminaire LJLL 27 Octobre / 35

20 Result Microscopic Model: for i {1,..., N}, x i T d, v i R d, dx i = v i, dt dv i = 1 Φ( x i x j ). dt ε ε j i Tagged particle in a gas at equilibrium: f N(Z N ) := M N,β (Z N )ρ (x 1 ) ρ density of probability on T d, M N,β Gibbs measure: for β > with H N (Z N ) := M N,β (Z N ) := 1 ( ) dn/2 β exp( βh N (Z N )) Z N 2π 1 i N 1 2 v i i<j N Φ( (x i x j ). ε Nathalie AYI Séminaire LJLL 27 Octobre / 35

21 Theorem (A., 216) Consider the initial distribution fn describing the density of a tagged particle in a gas at equilibrium, then the distribution f (1) N (t, x, v) of the tagged particle converges in D (T d R d ) when N goes to under the Boltzmann-Grad scaling Nε d 1 = 1 to M β (v)h(t, x, v) where h(t, x, v) is the solution of the linear Boltzmann equation without cut-off t h + v. x h = [h(v) h(v 1 )]M β (v 1 )b(v v 1 )dv 1 dν with initial data ρ (x 1 ) and where M β (v) := First partial result: Desvillettes and Pulvirenti (1999). ( ) d/2 ) β 2π exp ( β2 v 2, β >. Nathalie AYI Séminaire LJLL 27 Octobre / 35

22 The infinite range potential case Truncated marginals: f (s) N,R (t, Z s) := f N (t, Z s, z s+1,..., z N ) T d(n s) R d(n s) The BBGKY hierachy: for s < N, t f (s) N,R + s i=1 = C s,s+1 f (s+1) N,R + v i. xi f (s) N,R 1 ε +C s,s+1f (s+1) N,R (N s) ε i=1 s i,j=1 i j + 1 ε 1 i s s+1 j N Φ < ( x i x j (s) ). vi f N,R ε s i,j=1 i j Φ > ( x i x j (s) ). vi f N,R ε 1 { xi x j >Rε}dz s+1 s Φ( x i x s+1 ). vi f N (t, Z N ) T d(n s) R ε d(n s) 1 l s s+1 k N 1 { xl x k >Rε}dZ (s+1,n) Nathalie AYI Séminaire LJLL 27 Octobre / 35

23 Duhamel s formula: f (s) N,R (s) (t, Zs) = Ss(t) f (, Zs) + + t t + 1 ε N,R (s+1) S s(t t 1)C s,s+1 f (t1, Zs)dt1 N,R S s(t t 1)C s,s+1f (s+1) N,R (t 1, Z s)dt 1 s t i,j=1 i j (N s) + ε S s(t t 1) s i=1 t [ Φ > ( S s(t t 1) [ xi xj (s) ). vi f N,R ε 1 l s s+1 k N ] (t 1, Z s)dt 1 xi xs+1 Φ( ε T d(n s) R d(n s) 1 { xl x k >Rε}dZ (s+1,n) ). vi f N (t 1, Z s)dt 1 Nathalie AYI Séminaire LJLL 27 Octobre / 35

24 Obstacles to the convergence Four possible obstacles to the convergence: - the very long-range interactions, - clusters (or multiple simultaneous interactions), - the presence of recollisions, - super-exponential collision process. Nathalie AYI Séminaire LJLL 27 Octobre / 35

25 The pruning process Definition Let τ > and denote t := Kτ for some large integer K. We split [, t] into 1 k K [(k 1)τ, kτ]. We call a collision tree of controlled size a collision tree such that it has less than n k = 2 k branch points on the interval [t kτ, t (k 1)τ]. We call a collision tree with super-exponential growth a collision tree which does not satisfy the above property. Nathalie AYI Séminaire LJLL 27 Octobre / 35

26 Figure: A super-exponential tree (Source: The Brownian Motion as the limit of a deterministic system of hard-spheres, Bodineau et al). Nathalie AYI Séminaire LJLL 27 Octobre / 35

27 Obstacles to the convergence Four possible obstacles to the convergence: - the very long-range interactions, - clusters (or multiple simultaneous interactions), - the presence of recollisions, - super-exponential collision process. Iteration on the term: t S s (t t 1 )C s,s+1 f (s+1) N,R (t 1, Z s )dt 1. New strategy = truncations at each iteration step. Nathalie AYI Séminaire LJLL 27 Octobre / 35

28 Truncations to avoid pathological situations Elimination of recollisions: notion of good configurations. Definition The set of good configuration G k (ε ) is defined as follows: G k (ε ) := {Z k T dk R dk u [, t] i j d(x i uv i, x j uv j ) ε }. Bad sets to remove geom(s+k) : s + k 1 particles in a good configuration after a delay δ, the s + k particles are in a good configuration. Nathalie AYI Séminaire LJLL 27 Octobre / 35

29 Truncations associated to the elimination of recollisions: - cutting off the energy of the system via a smooth function such that { 1 if x E 2 χ E2 (x) = if x E χ E2 (H k (Z k )) := χ {Hk (Z k ) E 2 } for all integer k, - separation of the collision times by δ. Additional truncation: - small relative velocities via a smooth function such that { χ η if x η/2 (x) = 1 if x η χ { i {1,...,k} vi v k+1 η} := k χ η (v i v k+1 ). i=1 Nathalie AYI Séminaire LJLL 27 Octobre / 35

30 One iteration: t S s(t t 1 )C s,s+1 f (s+1) N,R (t 1, Z s)dt 1 t = t δ t δ + t δ + t δ + + t δ S s(t t 1 )C s,s+1 f (s+1) N,R (t 1, Z s)dt 1 ( ) (s+1) S s(t t 1 )C s,s+1 1 χ{hs+1 (Z s+1 ) E 2 } f N,R (t 1, Z s)dt 1 S s(t t 1 )C s,s+1 χ {Hs+1 (Z s+1 ) E 2 } χ (s+1) geom(s+1) f N,R (t 1, Z s)dt 1 ( ) ( ) S s(t t 1 )C s,s+1 χ {Hs+1 (Z s+1 ) E 2 } 1 χgeom(s+1) 1 χ{ i {1,...,s} vi v s+1 η} f (s+1) N,R (t 1, Z s)dt 1 S s(t t 1 )C s,s+1 χ {Hs+1 (Z s+1 ) E 2 } ( 1 χgeom(s+1) ) χ{ i {1,...,s} vi v s+1 η} f (s+1) N,R (t 1, Z s)dt 1. Nathalie AYI Séminaire LJLL 27 Octobre / 35

31 Definition of the operators: Q s,s (t) := S s (t) t δ t1 δ tn 1 δ ( ) Q s,s+n (t) :=... S s(t t 1 )C s,s+1 χ Hs+1 1 χgeom(s+1) χηs+1... ( )... S s+n 1 (t n 1 t n)c s+n 1,s+n χ Hs+n 1 χgeom(s+n) χηs+n S s+n (t n)dt n... dt 1. Remainders associated to the long-range part: m r P ot,a s,m+1 (, t, Zs) := n= t nδ Q s,s+n (t t n+1 ) 1 ε s+n i,j=1 i j [ Φ > ( x i x j ε ] (s+n) ). vi f N,R (t n+1, Z s)dt n+1 Nathalie AYI Séminaire LJLL 27 Octobre / 35

32 r P ot,b s,m+1 (, t, Z s) := (N (s + n)) ε m n= t nδ Q s,s+n (t t n+1 ) s+n [ Φ( x i x s+n+1 ). vi f N i=1 T d(n (s+n)) R ε d(n (s+n)) 1 l s+n s+n+1 k N 1 { xl x k >Rε}dZ (s+n,n) (t n+1, Z s )dt n+1. Nathalie AYI Séminaire LJLL 27 Octobre / 35

33 Control of the new terms Maximum Principle for the Liouville equation A priori estimates on the truncated marginals. Problem: Control on the marginals but not on the derivatives of the marginals Weak approach. Nathalie AYI Séminaire LJLL 27 Octobre / 35

34 Advantage of the iteration method no pathological situations easy to pass from Z m (t m ) to z (state of particle 1 at time t) via changes of variables such as T d R d [, t δ] S d 1 R d [, t m 1 δ] S d 1 R d T (m+1)d R (m+1)d (z, t 1, ν 2, v 2,..., t m, ν m+1, v m+1) Z m+1 = Z m+1(t m). Key point: z Lipschitz function of ( x 1, ṽ 1,..., x m+1, ṽ m+1 ) Tools to prove it: study of the reduced dynamics - bound on the microscopic time of interaction thanks to the lower bound on relative velocites, - use of the Cauchy-Lipschitz theorem, Φ being Lipschitz, - Lipschitz character of the collision times. Nathalie AYI Séminaire LJLL 27 Octobre / 35

35 Proposition The function ( x 1, ṽ 1,..., x k+1, ṽ k+1 ) z = z( x 1, ṽ 1,..., x k+1, ṽ k+1 ), where ( x 1, ṽ 1,..., x k+1, ṽ k+1 ) is the state of the k + 1 particles after k collisions at time t + k+1 (i.e. before the k + 1-th collision) on a pseudo-trajectory is a (C R,η,ε ) k CRe CR3 /η -Lipschitz function with C R,η,ε = η cos ( π 2 ε) and C is a constant ε which can only depend on Φ. Lipschitz control associated to the pseudo-trajectories = bound of the remainder associated to the long-range part controlled by ) 2 K+1 (e C R3 η Φ >. Limit of the approach: Very decreasing potentials. Nathalie AYI Séminaire LJLL 27 Octobre / 35

36 Thank you for your attention. Nathalie AYI Séminaire LJLL 27 Octobre / 35