STOCHASTIC APPROXIMATIONS OF THE SOLUTION OF
A FULL BOLTZMANN EQUATION WITH SMALL INITIAL
DATA.
SYLVIE MELEARD
Abstract. This paper gives an approximation of the solution of the Boltzmann equation by stochastic interacting particle systems in a case of cut-o collision operator and small initial data. In this case, follow- ing the ideas of Mischler and Perthame, we prove the existence and uniqueness of the solution of this equation and also the existence and uniqueness of the solution of the associated nonlinear martingale prob- lem.Then, we rst delocalize the interaction by considering a mollied Boltzmann equation in which the interaction is averaged on cells of xed size which cover the space. In this situation, Graham and Meleard have obtained an approximation of the mollied solution by some stochastic interacting particle systems. Then we consider systems in which the size of the cells depends on the size of the system. We show that the associ- ated empirical measures converge in law to a deterministic probability measure whose density ow is the solution of the full Boltzmann equa- tion. That suggests an algorithm based on the Poisson interpretation of the integral term for the simulation of this solution.
1. Introduction
In the upper atmosphere, the gas is rareed and is described by the non- negative density f(txv) of particles which at timetand pointxmove with velocityv. Thenf(txv) is positive and normalized so thatR f(txv)dxdv is equal to one and satises the Boltzmann equation
@
t
f(txv)+v rxf(txv) = Q(ff)(txv) on 0+1R3R3
= Q+(ff)(txv);Q;(ff)(txv)
f(0xv) = f0(xv) is a density of probability(1.1) where
Q
+(ff)(txv) =
Z
S 2
dn Z
R 3
dv
B(v;vn)f(txv0)f(txv0) (1.2) and
Q
;(ff)(txv) =f(txv)Lf(txv)
URL address of the journal: http://www.emath.fr/ps/
Received by the journal April 11, 1997. Revised November 12, 1997. Accepted for publication December 22, 1997.
c Societe de Mathematiques Appliquees et Industrielles. Typeset by LATEX.
with
Lf(txv) = Af(tx:)(v) =
Z
S 2
Z
R 3
B(v;vn)f(txv)dvdn
A(z) =
Z
S 2
B(zn)dn: (1.3)
The nonnegative cross-section B(zn) depends only on jzj and on jhz:nij. The velocitiesv0andv0 represent the post-collisional velocities of two parti- cles of velocities v andv having collided in a position in which their centers are on a line of direction given by the unit vector n belonging to the unit sphere S2. Conservation of kinetic energy and momentum for binary colli- sions implies that
v
0=v+ ((v;v) n)n v0 =v+ ((v;v) n)n:
We refer to Cercignani et al. (1994) for physical comments on this model.
The Boltzmann equation presents many important diculties, due to the unboundedness of B and to the localization in space in the quadratic colli- sion term (the interaction is not mean-eld). In the general case, uniqueness in not proved and existence of renormalized solutions is showed in the fa- mous paper of DiPerna and Lions (1989). On the other hand, existence and uniqueness have been studied by many authors under restrictive assump- tions on the cross-section B (principally a cut-o assumption) and results have been obtained in particular in small time or under small initial data, as it can be found in Kaniel and Shinbrot (1978), Bellomo and Toscani (1985), Hamdache (1985), Toscani (1986), Bellomo et al. (1988) and more recently in Mischler and Perthame (1997). One follows the ideas of Mischler and Perthame (1997), obtained in a more general situation of innite energy, in order to prove by a xed point argument the existence and uniqueness of the solution in a well chosen functional space B, in a case of cut-o collision operator and small initial data. This existence and uniqueness result (The- orem 2.1) is very close to "a priori" assumptions in the paper of Babovsky and Illner (1989).
The Boltzmann equation is an integro-dierential equation, in which the integral term comes from the randomness in the geometry of collisions. It is natural to study its probabilistic interpretation. One associates with the equation a nonlinear martingale problem and one obtains the existence and uniqueness of the solution of this martingale problem in the space of prob- ability measures having a measurable version of densities in B.
Our aim is then to give a stochastic approximation of the solution of the Boltzmann equation, obtaining thus a theoretical justication of the Nanbu and Bird algorithms in this case (cf. Babovsky and Illner (1989)).
The interaction appearing in the collision term is localized in space and is not mean-eld. So we do not know how to construct directly approximat- ing particle systems. One rst delocalizes the interaction by considering a mollied Boltzmann equation in which the mollifying kernel is issued from a grid method. The space is shared in disjoint cells of size in which the interaction is averaged. In this mean-eld case, Graham and Meleard (1997) prove some stochastic approximations of the solution of the mollied Boltz- mann equation by interacting particle systems and obtain a precise rate of
convergence in O(exp(3)=n), where n is the size of the particle system.
Moreover, a unied approach for systems with simple or binary mean-eld interactions is given.
In this paper, one considers such systems in which the size of the cells of the grid depends on the size of the system. More precisely, we assume that depends on n in the asymptotic exp( (n)K3)=n ! 0 (when n tends to innity). Then one proves that the empirical measures of the associated interacting particle systems converge in law to a deterministic probability measure whose density ow is the solution of the full Boltzmann equation.
The convergence is obtained for probability measures on the path space and convergence results for functionals of the paths can be deduced.
At our knowledge, this result (Theorem 5.4) seems to be the rst pathwise approximation result in a non mollied case and in dimension 3. Let us quote Caprino and Pulvirenti (1995) and Rezakhzanlou (1996), who obtain the convergence of stochastic particle systems to a one-dimensional Boltzmann equation at xed times. Our approach is unied for simple or binary mean- eld systems, and allows to understand the similarity between Bird's and Nanbu's algorithms. Moreover one gives a precise asymptotic betweenand
n that was an open question in Babovsky and Illner (1989).
One nally suggests an algorithm based on the Poisson interpretation of the integral term to simulate the solution of the Boltzmann equation, which avoids to discretize in time and exactly follows the pathwise history of the particles.
2. The existence and uniqueness result
Let us now prove the existence and uniqueness result obtained for the Boltzmann equation in a case of bounded collision operator and small initial data with nite energy.
Theorem 2.1. Let>0 andT be a positive time. Let us assume that (H1): A2L1(R3)
(H2): f0 is a density function satisfying
0f0(xv) C6 exp(0 ;jvj2) (2.1) where C0 is a real number such that C0 < kAk(1p)3
( p
) 3
T = C1T.
Then there exists a unique function f 2L1(0T]R3R3) solution of the Boltzmann equation (1.1) satisfying
0f(txv) C(t)
6 exp(;jvj2) (2.2)
where C(t) is a positive and bounded function on 0T] dened by C(t)1 =
1
C
0
;C
t.
Proof. The proof is completely inspired of the proof of Theorem 2 in Mischler and Perthame (1997) given in a case of innite energy. It consists rst in introducing an upper solution related to the Boltzmann equation, and second in obtaining a xed point theorem in a functional space related to this upper solution.
Let us considerg(tv) =C(t)h(v), whereh(v) = exp(;jvj ). One would like
@
t
g(tv) = _C(t)h(v) Q+(gg)(tv) = C2(t)Q+(hh)(v)
= C2(t)h(v)L(h)(v) since Q(hh) = 0 =Q+(hh);Q;(hh) =Q+(hh);hL(h).
Then one is looking for C such that _C C2supvL(h)(v). Therefore let us consider C 2 C1(0T]R) such that C(0) = C0 and solving _C(t) =
kAk
1(=)3=2C2(t). By denoting C =kAk1(=)3=2, one nally obtains that (C(t));1 = (C0);1;Ctand the function
^
h(tv) = C(t)
6 h(v) = C(t) 6 e;jv j
2 (2.3)
satises
@
t^h(tv) CC(t)^h(tv) 6Q+(^h^h)(tv): (2.4) Now one considers the set
B
=f'2L1(0T]R3R3)0'(txv)^h(tv)g
with the norm k'k = ess suptxvfj'(txv)j=jC(t)6 exp(;jvj2)jg for which it is complete. The global existence and uniqueness of the solution of the Boltzmann equation is deduced from a xed point theorem in this space. As in Mischler and Perthame (1997), let us dene the mapping : '2B !
= ('), where is the solution of
@
t
+v rx+CC(t) = Q+('') + (CC(t);L('))'
(0xv) = f0(xv): (2.5)
By a maximum principle one observes that sendsB intoBand by (2.4) that moreover
8'
1 '
2 2B
k'1;'2k 5
6k'1;'2k:
Hence is a contraction and admits a unique xed point in B, which is solution of the Boltzmann equation.
3. The nonlinear martingale problem associated with the Boltzmann equation
The weak form of the equation (1.1) is given for a function 'in Cb1(R6) by
@
t hf
t
'i;hf
t v r
x 'i
= hft(xv)dxdvZ ('(xv+ ((v;v):n)n);'(xv))
B(v;vn)f(txv)dndvi: (3.1) (Remark that the mapping (vv)!(v0v0) has a determinant equal to 1).
We associate with this evolution equation a nonlinear martingale problem for which every solution is a probability measure on the path space whose marginals are solutions of the equation (1.1).
Let us denote by ~P(D(0T]R6)) the space of probability measures on
D(0T]R6) having for everyt20T]a density with respect to the Lebesgue
measure. Let us remark that following Meyer (1966) p. 193-194, there exists forP in ~P(D(0T]R6)) a measurable functionp(txv) on 0T]R6such that for any t 2 0T], p(t:) is a density of Pt. We call such a function a measurable version of the densities of P.
Definition 3.1. The probability measureP 2P~(D(0T]R6)) is solution of the nonlinear martingale problem (M) if for every function'2Cb1(R6), for (X V) the canonical process on D(0T]R3R3),
'(XtVt);'(X0V0);
Z
t
0 V
s :r
x
'(XsVs)ds
; Z
t
0 Z
S 2
Z
R 3
('(XsVs+ ((v;Vs):n)n);'(XsVs))
B(Vs;vn)p(sXsv)dvdnds
is a P-martingale, where p(t:) is a measurable version of the densities of the ow of marginals (Pt)t0,P0(dxdv) =f0(xv)dxdv.
Clearly this denition does not depend on the choice of the measurable version of the densities of P.
Let us denote by ~P(D(0T]R6)) the subspace of ~P(D(0T]R6)) such that a measurable version of the densities p satises 0p(txv)^h(tv) for almost every (txv)20T]R6, ^h being dened in (2.3). Then it is true for every measurable version of the densities.
Theorem 3.2. Under assumptions (H1) and (H2), the nonlinear martin- gale problem (M) has a unique solution P in ~P(D(0T]R6)). Every mea- surable version of the densities of P is almost surely equal to the solution f of the Boltzmann equation (1.1) dened in Theorem 2.1.
Let us rst observe that if P is solution of (M), then by taking the expectations in the martingale problem, each measurable version p of its densities is solution of the Boltzmann equation (1.1). Moreover if we assume that 0p(txv)^h(tv), then p is almost surely equal tof by Theorem 2.1. Therefore, we rst study the following classical martingale problem associated with the function f.
Definition 3.3. A probability measure P 2 P(D(0T]R6)) is a solution of the martingale problem (Mf) if for every function '2Cb1(R6),
'(XtVt);'(X0V0);
Z
t
0 V
s :r
x
'(XsVs)ds
; Z
t
0 Z
S 2
Z
R 3
('(XsVs+ ((v;Vs):n)n);'(XsVs))
B(Vs;vn)f(sXsv)dvdnds (3.2) is a P-martingale andP0(dxdv) =f0(xv)dxdv.
Proposition 3.4. Under assumptions(H1) and (H2), the martingale prob- lem(Mf) has a unique solutionPf absolutely continuous with respect to the Lebesgue measure. Its density q is solution of the evolution equation
q(txv) =f0(x;tvv) +
Z
t
0
(St;s)Q(qf)(sxv)ds (3.3)
where St is the semigroup associated with the ow solution of@tq+v:rxq = 0, and (St) is the dual operator.
Proof. 1) (H1) and (2.2) imply the jump kernelB(Vs;vn)f(sXsv)dvdn has a nite total mass uniformly in sXsVs. Moreover, the drift part in (3.2) has a Lipschitz continuous coecient. In this case, the existence and uniqueness of a solution Pf of (Mf) is well known.
2) Let us now prove that the solutionPf has a densityq. Let (Tn)n2Nbe the sequence of random jumps of the process Z under Pf there is a nite number of random jumps on the time interval 0T]. Following Jacod and Shiryeav (1987) p. 136 and, since the jump measureB(Vs;vn)f(sXsv)
dv
dnds is absolutely continuous with respect to time, the law of the rst jump T0 conditionally to X0 =xV0 =v has a density with respect to the Lebesgue measure. Since the law of (X0V0) has the density f0, then the triplet (X0V0T0) has a density with respect to the Lebesgue measure. Of course, it is the same for (XT0;VT0;T0) = (X0+T0V0V0T0). Moreover, conditionally to (XT0;VT0;T0), the law of the jump "VT0 has clearly a density and we deduce that the law of (XT0VT0T0) has a density. By the Markov property, we then obtain that for everyTn, the law of (XTnVTnTn) has a density, and so that Pf has a density q with respect to the Lebesgue measure. By taking expectation in (3.2), we obtain moreover that the ow (qt) satises for'in Cb1(R6)
@
t hq
t
'i;hq
t v r
x 'i
= hqt(xv)dxdv
Z
('(xv+ ((v;v):n)n);'(xv))
B(v;vn)f(txv)dndvi: (3.4) We can extend this formula to functions (tx) which are inCb1(0T]R6) by It^o's formula. Let St be the semigroup associated with the ow @tq +
v:r
x
q = 0 and St the dual operator, St = S;t. Of course, St'(xv) =
'(x+tvv). For a xed tin 0T] and' in Cb1(R6), we choose (sxv) =
S
t;s
'(xv) ='(x+(t;s)vv). Then@s+v:rx= 0 and(t:) ='. The equation (3.4) extended to implies that for every function'in Cb1(R6),
Z
'(xv)q(txv)dxdv
=
Z
S
t
'(xv)f0(xv)dxdv +
Z
t
0 Z
S
t;s
'(xv)f(sxv0)q(sxv0);f(sxv)q(sxv)]
B(v;vn)dndvdxdvds
=
Z
'(xv)Stf0(xv)dxdv +
Z
'(xv)
Z
t
0 S
t;s(f(sxv0)q(sxv0);f(sxv)q(sxv)]
B(v;vn)dndv)dxdvds
and then we deduce that for every t20T],dxdvalmost surely,
q(txv) =f0(x;vtv) +
Z
t
0
(St;s)Q(qf)(sxv)ds: (3.5)
Proposition 3.5. Under assumptions (H1) and (H2), the evolution equa- tion (3.5) has a unique solution in L1(0T]L1(dxdv)).
Proof. Letq0 be another solution of the evolution equation. Then
kq(t);q0(t)kL1(dxdv )
= k
Z
t
0
(St;s)Q(qf);Q(q0f)](sxv)dskL1(dxdv )
Z
t
0
k(St;s)Q(qf);Q(q0f)](sxv)kL1(dxdv )ds
=
Z
t
0
kQ(qf);Q(q0f)](sxv)kL1(dxdv )ds
Z
t
0 Z
fjf(sxv0)jjq(sxv0);q0(sxv0)j
+jf(sxv)jjq(sxv);q0(sxv)jgB(v;vn)dndvdxdvds
C
C(T)
6
Z
t
0
kq(s);q0(s)kL1(dxdv )ds
by (H1) and (2.2). We deduce by usual Gronwall's Lemma that the solution of the evolution equation is unique in L1(0T]L1(dxdv)).
We now prove Theorem 3.2.
Proof. We rst consider2Cb1(0T]R6) with compact support. We can then prove by using Fubini's theorem and the integration by part formula that
@
t hf
t
i;hf
t v r
x
+@ti
= hft(xv)dxdv
Z
((txv+ ((v;v):n)n);(txv))
B(v;vn)f(txv)dndvi:
We obtain by approximation the same formula for every function 2
C 1
b(0T]R6), and considering (sxv) = St;s'(xv), ' 2 Cb1(R6), we obtain as before that the solution solution f of the full Boltzmann equation (1.1) is solution of the evolution equation (3.5). The uniqueness proved in Proposition 3.5 implies thatq =f. Then the solution of (Mf) is in fact a solution of (M).
Let us now consider two solutions P and Q of (M) with measurable versions of the densities bounded by ^h. Thus these densities are solutions of (1.1) bounded by ^h and so are almost surely equal, and equal to f, by Theorem 2.1. Since the nonlinearity in the martingale problem just depends on this common owf, we now get a (classical) martingale problem in which the jump measure is given and bounded. The uniqueness in this martingale problem implies that P =Q, hence we get Theorem 3.2.
The ow of densities is equal tof and satises moreover for eacht20T]
f(txv) =f0(x;vtv) +
Z
t
0
(St;s)Q(ff)(sxv)ds a.s. in xv: (3.6)
Let us now give a regularity result for the function f, useful later in the proof of Proposition 5.2. The property stated below, as well as Theorem 2.1, are very close to properties presented as conjectures in Babovsky and Illner (1989).
Proposition 3.6. Let us assume that
(H3): There exists K >0 such that for every h2R3, ess supx2R3v 2R3jf0(x+hv);f0(xv)j
e
;jv j 2
Kjhj (3.7)
then the same property holds for f: there exists KT >0 such that ess supt20T]x2R3v 2R3jf(tx+hv);f(txv)j
e
;jv j 2
K
T
jhj 8h2R 3
:(3.8) Proof. f is solution of the evolution equation (3.6) and
f(tx+hv);f(txv)
= f0(x+h;tvv);f0(x;tvv) +
Z
t
0
Z
f(sx+h;(t;s)vv0)f(sx+h;(t;s)vv0)
;f(sx;(t;s)vv0)f(sx;(t;s)vv0)
;
f(sx+h;(t;s)vv)f(sx+h;(t;s)vv)
;f(sx;(t;s)vv)f(sx;(t;s)vv)
B(v;vn)dndvds:
Then
jf(tx+hv);f(txv)j
e
;jv j 2
jf
0(x+h;tvv);f0(tx;tvv)j
e
;jv j 2
+
Z
t
0 Z
f(sx+h;(t;s)vv0)
e
;jv j 2
jf(sx+h;(t;s)vv0);f(sx;(t;s)vv0)j +f(sx;(t;s)vv0)
e
;jv j 2
jf(sx+h;(t;s)vv0);f(sx;(t;s)vv0)j
B(v;vn)dndvds
+
0
f(sx+h;(t;s)vv)
e
;jv j 2
jf(sx+h;(t;s)vv);f(sx;(t;s)vv)j +f(sx;(t;s)vv)
e
;jv j 2
jf(sx+h;(t;s)vv);f(sx;(t;s)vv)j
B(v;vn)dndvds:
Let "ht(f) = ess supxvjf(tx+hv );f(txv )j e
; jv j
2 . We have
jf(tx+hv);f(txv)j
e
;jv j 2
Kjhj+C(T) 6
Z
t
0 Z
e
;jv 0
j
2"hs(f)e;jv0j2
e
;jv j
2 + e;jv
0
j
2"hs(f)e;jv0j2
e
;jv j 2
+e;jvj2"hs(f) +e;jvj2"hs(f)
B(v;vn)dndvds:
We use the conservation of energy: jvj2+jvj2=jv0j2+jv0j2. So we get
jf(tx+hv);f(txv)j
e
;jv j 2
Kjhj+ 4C(T) 6
Z
t
0 Z
"hs(f)e;jvj2B(v;vn)dndv
Kjhj+ 2C(T) 3 C
Z
t
0
"hs(f)ds and nally
"ht(f)Kjhj+ 23C(T)C
Z
t
0
"hs(f)ds: (3.9) Gronwall's Lemma allows to conclude.
4. The mollified problem
4.1. The mollified nonlinear martingale problem
Mollifying consists in delocalizing in space the interaction appearing in the Boltzmann equation in order to obtain a mean-eld model. We cover
R
3 by a grid of cubic, uniform disjoint cells " of volume j"j=3, and we introduce the regularizing kernel
I
(xy) = 1
3
X
1Ix21Iy 2: (4.1)
The kernel Q is replaced byQ, dened by
Q
(ff)(txv) =
Z
f(txv0)f(tyv0);f(txv)f(tyv)]
B(v;vn)I(xy)dydndv
