45067 Orleans Cedex 2 Stephane Mischler UNIVERSITE PARIS 6 Laboratoire d'Analyse Numerique

BOLTZMANN AND B.G.K. EQUATIONS

Laurent Desvillettes UNIVERSITE D'ORLEANS Departement de Mathematiques

BP 6759

45067 Orleans Cedex 2 Stephane Mischler UNIVERSITE PARIS 6 Laboratoire d'Analyse Numerique

4, Place Jussieu 75005 Paris January 6, 2004

Abstract

We prove the convergence of splitting algorithmsfor Boltzmann and B.G.K. equations. The proof in the case of the Boltzmann equation is made in the framework of renormalized solutions.

1 Introduction

A rareed gas is usually described by the Boltzmann equation (Cf. [Ce], [Ch, Co], [Tr, Mu]). In this model, the dynamics of the gas is given by the nonnegative density ^f(^t;x;v) of particles which at time^t2[0^;T] and point

x2IR

3, move with velocity ^v2IR3, where ^T is a stricly positive number.

Such a density satises the Boltzmann equation,

@f

@t

+^vrx^f =^Q(^f)^; (1^:1)

f(0^;x;v) =^f0(^x;v)^; (1^:2) where^Qis a quadratic collision kernel acting only on velocities and dened (with the notations of [DP, L]) by

Q(^f) =^Q+(^f) ^Q (^f)^; (1^:3)

Q

+(^f)(^v) =^Z_v

2IR³

Z

!²S^2f(^v0)^f(^v0)^B(^{v v;!})^d!dv; (1^:4)

A(^z) =^Z_!

2S^2B(^z;!)^d!; (1^:5)

L(^f) =^Av^f; (1^:6)

Q (^f)(^v) =^f(^v)^L(^f)(^v)^: (1^:7) In formula (1.4), the post{collisional velocities ^v0 and ^v0 are parametrized by

v

0=^v+ ((^{v v})^!)^!; (1^:8)

v 0

=^v ((^{v v})^!)^!; (1^:9) where^! is a unit vector varying in the sphere ^S2.

Finally, the nonnegative cross section^B is assumed to satisfy the follow- ing properties, rst introduced in [DP, L]:

Assumption 1

: The function ^B(^z;!) belongs to ^L1loc(^{IR3 S2}) and depends only on ^jzjand ^jz!j.

Moreover, the function ^A satises for all ^R>0, (1 +^jzj2) ^1Z_v

2B^RA(^z+^v)^{dv !}

jz^j!10^; (1^:10) where ^BR (or^B_Rv) is the set ^fv2IR3;^jvj<Rg.

Finally, we assume that the nonnegative initial datum ^f0 satises the following physically relevant assumption:

Assumption 2

: The function ^f0 is such that

Z

x²IR³

Z

v²IR^3f0(^x;v)^f1 +^jxj2+^jvj2+^jlog^f0(^x;v)^jgdvdx<+^1: (1^:11) R.J. DiPerna and P-L. Lions proved in [DP, L] that under assumptions 1 and 2, the Boltzmann equation (1.1) { (1.9) admits a nonnegative renor- malized solution in ^C([0^;T]^;L1(^IR3IR3)).

The proof uses the averaging lemmas introduced by F. Golse, B. Perthame and R. Sentis in [G, P, S], and developed by F. Golse, P-L. Lions, B.

Perthame and R. Sentis in [G, L, P, S] and by R.J. DiPerna, P-L. Lions and Y. Meyer in [DP, L, M].

Note that a new and simpler proof was given by P-L. Lions in [L 1].

We shall also consider in the sequel a simpler model of rareed gases, namely the B.G.K. model, rst introduced in [Bh, Gr, Kr].

The gas is still described by a nonnegative density ^f(^t;x;v), but the equation satised by^f now becomes

@f

@t

+^vrx^f =^M[^f] ^f; (1^:12)

f(0^;x;v) =^f0(^x;v)^; (1^:13) where^M[^f](^t;x;v) is a Maxwellian function of ^v:

M[^f](^t;x;v) = (^t;x)

(2^T(^t;x))³⁼²exp ^{jv u}(^t;x)^j2 2^T(^t;x)

; (1^:14) and^;u;T are the respective density, global velocity and temperature of the gas. More precisely,

(^t;x) =^Z_v

2IR^3f(^t;x;v)^{dv ;} (1^:15)

(^t;x)^u(^t;x) =^Z_v

2IR^3vf(^t;x;v)^dv; (1^:16)

(^t;x)^fju(^t;x)^j2+ 3^T(^t;x)^g=^Z_v

2IR^3jvj

2

f(^t;x;v)^{dv :} (1^:17)

Note that the previous quantities are not well{dened when= 0, therefore we dene^M[0] = 0.

The existence of a global nonnegative solution for the B.G.K. system (1.12) { (1.17) under assumption 2 on the initial datum was proved by B. Perthame in [Pe]. The proof was based on a dispersion lemma. Another proof was given by E. Ringeissen in [Ri], allowing to take into account a gas in a bounded domain with boundary conditions.

Equation (1.1) { (1.9) as well as (1.12) { (1.17) can be written in the form

@f

@t

=^Af +^{B f;} (1^:18)

f(^t= 0) =^f0; (1^:19)

where

A= ^vrx^; (1^:20)

and ^Bis a nonlinear operator acting only on the variable ^v.

Therefore, in order to compute numerically their solution, it is usual to solve equations

@f

@t

=^Af (1^:21)

and

@f

@t

=^{B f} (1^:22)

one after another and to apply Trotter's formula

et^{(A+B )}= lim_n

!+1

(^entAentB)^n: (1^:23) This procedure is known as a splitting method for system (1.18), (1.19) and it is said to converge if Trotter's formula (1.23) holds when^Aand^Bare the operators introduced in (1.18). A large amount of splitting algorithms involving discretization in time can be found in [L, M].

We intend to prove that the splitting method converges for the Boltz- mann and B.G.K. equations in the cases described earlier.

Note that this method is actually used in the numerical computation of both equations (Cf. [De, Pr]).

Note also that we proved in an earlier work the convergence of the split- ting algorithm in the simpler cases of the \grey" radiative transfer equation and of Vlasov{Maxwell system (Cf. [De 1] and [De 2]). The proofs of exis- tence of global solutions for the Boltzmann equation (Cf. [DP, L]) and for

the B.G.K. model (Cf. [Pe]) were already known at that time, but it seemed dicult to prove the convergence of the splitting algorithm in the context of those works. Namely, the analysis of sub- and supersolutions in [DP, L] did not seem well{adapted to the splitting algorithm, and the dispersion lemma of [Pe] seemed also inoperant in this context.

However, the new proof of existence for the Boltzmann equation of [L 1], and the proof of existence for the B.G.K. model of [Ri] are better{adapted to the method of splitting and can therefore be followed, as will be seen in the sequel.

Therefore, in section 2, we prove the convergence of Trotter's formula for the Boltzmann equation, and the corresponding result for B.G.K. model in section 3.

2 Splitting for Boltzmann equation

2.1 Introduction and main result

In this section, we introduce the splitting algorithm for equation (1.1){ (1.9).

We dene

Af = ^vrx^f; (2^:1^:1)

B f =^Q(^f)^; (2^:1^:2) and we intend to prove Trotter's formula (1.23) in this context.

Therefore, we dene for every ⁿin ^IN and^k in [0^;n 1] two sequences

fkn and^g_kn by the following procedure:

we note

^T = ^T

n

; tk =^kT; (2^:1^:3) and the functions^f_kn and ^g_kn are dened on [^tk^;tk⁺¹] by induction on^k:

f

n0(0) =^f0; (2^:1^:4)

@fkn

@t

=^Af_kn^; (2^:1^:5)

fkn(^tk) =^g_n^{k 1}(^tk) when^k>0^; (2^:1^:6)

@gkn

@t

=^{B g}_kn^; (2^:1^:7)

gkn(^tk) = ^f_kn(^tk⁺¹)^: (2^:1^:8)

This denition is meaningful because the solutions of equations (2.1.5), (2.1.6) and (2.1.7), (2.1.8) belong to^C([^tk^;tk⁺¹]^;L1(^IR3IR3)).

Then, we dene

fn(^t) =^f_kn(^t)^; (2^:1^:9)

gn(^t) =^g_kn(^t)^; (2^:1^:10) for every^tlying in [^tk^;tk⁺¹[.

The functions ^fn and^gn are therefore piecewise continuous with respect to the time variable on [0^;T] with values in^L1(^IR3IR3), and their discon- tinuities appear at points^tk for each^k in [1^;n].

The main result of this section is the following:

Theorem 1

: Under assumptions 1 and 2 on the cross section and ini- tial datum, the sequences ^fn and ^gn dened in (2.1.4) { (2.1.10) converge up to extraction to the same nonnegative limit^f in^L1([0^;T];^L1(^IR3IR3)) weak *, and this limit satises equation (1.1) { (1.9) in the sense of renor- malized solutions. More precisely,

Q (^f)

1 +^{f 2L1loc}([0^;T]^IR3IR3)^; (2^:1^:11) and

f

@

@t

+^vrx^glog(1 +^f) = ^Q(^f)

1 +^f (2^:1^:12) in the sense of distributions.

Remark

: This property exactly means that Trotter's formula (1.23) holds for^Aand ^B dened in (2.1.1), (2.1.2).

The proof of this theorem is given in subsections 2.2 to 2.7.

2.2 Equation satised by

^fn

and

^gn

For all nonnegative and smooth function such that

(0) = 0^{; j0}(^s)^j 1

1 +^s; (2^:2^:1) we compute:

@(^fn)

@t

=^Xn

i⁼¹

@(^fn)

@t

1^](_i ¹⁾_T;iT^[+^Xn

i⁼¹

(^f_in)(^iT) (^f_n^{i 1})(^iT)iT

=^Xn

i⁼¹

vrx(^fn) 1^](i ¹⁾T;iT^[

+^Xn

i⁼¹

(^g_n^{i 1})(^iT) (^gi_n ¹)((ⁱ 1)^T)iT

= ^vrx(^fn) +^Xn

i⁼¹

Z iT

(i ¹⁾T

@(^gn)

@t

(^s)^dsiT

= ^vrx(^fn) +^Xn

i⁼¹

Z iT

(i ¹⁾T

0(^gn)(^s)^Q(^gn)(^s)^ds

iT^:

Therefore, we obtain the following equation for^fnand ^gn:

@(^fn)

@t

+^vrx(^fn) =^Xn

i⁼¹

Z iT

(i ¹⁾T

0(^gn)(^s)^Q(^gn)(^s)^ds

!

iT^: (2^:2^:2) In order to pass to the limit in equation (2.2.2), we need estimates for the sequences ^fn and ^gn.

2.3 Estimates on

^fn

and

^gn

Lemma 1

: The sequences^fn and^gndened in Theorem 1 are nonnegative and satisfy for some nonnegative constant^CT:

t^2[0sup;T^]

Z

x²IR³

Z

v²IR^3fn(^t;x;v)^f1 +^jxj2+^jvj2+^jlog^fn(^t;x;v)^jgdxdvCT^;

(2^:3^:1)

t^2[0sup;T^]

Z

x²IR³

Z

v²IR^3gn(^t;x;v)^f1 +^jxj2+^jvj2+^jlog^gn(^t;x;v)^jgdxdvCT^:

(2^:3^:2) Moreover, the quantity^e(^gn) dened by

e(^gn)(^t;x;v) =^Z_v

2IR³

Z

!²S^2fgn(^t;x;v0)^gn(^t;x;v0) ^gn(^t;x;v)^gn(^t;x;v)^g

log^gn(^t;x;v0)^gn(^t;x;v0)

gn(^t;x;v)^gn(^t;x;v)

B(^{v v;!})^d!dv (2^:3^:3) satises the following estimate:

Z T

0 Z

x²IR³

Z

v²IR^3e(^gn)(^t;x;v)^dvdxdtCT^: (2^:3^:4)

Proof

: The total density and total energy are conserved in the two steps of the splitting algorithm, therefore

Z Z

IR³IR^3fn(^t)(1 +^jvj2)^dxdv=^{Z Z}_IR

3

IR^3gn(^t)(1 +^jvj2)^dxdv

=^{Z Z}_IR

3

IR^3f0(1 +^jvj2)^dxdv: (2^:3^:5) During the rst step we also get

Z Z

IR³IR^{3jx vtj}

2

fkn(^t)^dxdv=^{Z Z}_IR

3

IR^{3jx vt}k^j2fkn(^tk)^dxdv (2^:3^:6) for all ^t2[^tk^;tk⁺¹[, whereas during the second step we have

Z Z

IR³IR^{3jx vt}k^j2gkn(^t)^dxdv=^{Z Z}_IR

3

IR^{3jx vt}k^j2gkn(^tk)^dxdv (2^:3^:7) for all ^t2[^tk^;tk⁺¹[. Therefore,

Z Z

IR³IR^{3jx vtj}

2

fn(^t)^dxdv=^{Z Z}_IR

3

IR^3jxj

2

f

0

dxdv (2^:3^:8) for all ^t2[0^;T[, and

t^2[0sup;T^]

Z Z

IR³IR^3gn(^t)^jxj2dxdv sup

t^2[0;T^]

Z Z

IR³IR^3fn(^t)^jxj2dxdv

Z Z

IR³IR³2^f0(^jxj2+^T2jvj2)^dxdv: (2^:3^:9) Finally, we prove the estimate on the entropy production:

d

dt Z Z

IR³IR^3fkn log^f_kn^dxdv= 0^; and

d

dt Z Z

IR³IR^3gkn log^g_kn^dxdv= 14^{Z Z}IR³IR^3e(^g_kn)^dxdv0 for all ^t2[^tk^;tk⁺¹[. Therefore,

Z Z

IR³IR^3fn log^fn^dxdv(^t)^{Z Z}_IR

3

IR^3f0 log^f0dxdv; (2^:3^:10)

and ^{Z Z}

IR³IR^3gn log^gn^dxdv(^t) ^{Z Z}_IR

3

IR^3f0 log^f0dxdv

14

Z T

0 Z

IR³IR^3e(^gn)^dxdvdt (2^:3^:11) for all ^{t 2} [0^;T[. Finally, it is now classical (Cf. [DP, L]) that estimates (2.3.5) and (2.3.9) { (2.3.11) ensure the existence of a constant ^CT such

that sup

t^2[0;T^]

Z Z

IR³IR^3fn^jlog^fn^jdxdvCT^; (2^:3^:12)

t^2[0sup;T^]

Z Z

IR³IR^3gn^jlog^gn^jdxdvCT^; (2^:3^:13)

and ^Z _T

0 Z

IR³IR^3e(^gn)^dxdvdtCT^; (2^:3^:14) which ends the proof of lemma 1.

According to lemma 1, we can extract from the sequences ^fn and ^gn

subsequences still denoted by ^fn and ^gn, which converge respectively to ^f and ^gin ^L1([0^;T];^L1(^IR3IR3)) weak *.

2.4 Weak compactness of the renormalized collision terms

We present here the main estimate on the collision term:

Lemma 2

: The sequences ^Q1+(_g^gnⁿ⁾ and ^1+Q+_L^((g_gⁿ⁾ⁿ⁾ belong to a weakly compact set of^L1([0^;T]^IR3_x^B_Rv), for all ^R>0.

Proof

: We only prove here that the sequences are bounded in ^L1. The reader will nd in [DP, L] the proof of weak compactness.

For all ^R>0, we compute

Z

x²IR³

Z

v²B^R

Q (^gn)

1 +^gn ^dvdx

Z

x²IR³

Z

v²B^RL(^gn)^dvdx

Z

x²IR³

Z

v²IR^3gn(^v)^Z_v

2B^RA(^{v v})^dvdvdx

sup

z²IR³

(1 +^jzj2) ^1Z_v

2B^RA(^{v z})^dv

Z

x²IR³

Z

v²IR^3gn(^v)(1+^jvj2)^dvdx; (2^:4^:1) which is bounded because of assumption 1 and estimate (2.3.2).

Then, the boundedness of ^Q1++(_g^gnn) and ^1+Q+_L^((g_gⁿⁿ⁾⁾ comes out of the bound- edness of ^Q1+(_g^gnn) and of estimate (2.3.4).

More precisely, we recall that for all ^K>0,

Q

+(^gn)^KQ (^gn) + ^e(^gn)

log^K: (2^:4^:2)

2.5 The sequences

^fn

and

^gn

converge to the same limit

In order to pass to the limit in equation (2.2.2) we need to know that(^gn) and (^fn) converge to the same limit, and that the same holds for ^fn and

gn.

lemma 3

: Up to extraction, the sequences^fnand^gnsatisfy the following properties:

i) For all nonnegative and smooth function such that (2.2.1) holds,

(^fn) and (^gn) converge to the same limit in ^L1([0^;T]^{IR3 IR3}) weak.

ii) the sequences^fnand^gn have the same limit^f in^L1([0^;T]^IR3IR3) weak.

Proof

:

Step 1 : We prove i). Because of lemma 1, we just have to show that

(^fn) and(^gn) converge to the same limit in the sense of distributions. Let

'belong to^D(]0^;T[^IR3IR3) and ^K be its compact support.

We compute

j Z T

0 Z

x²IR³

Z

v²IR³((^fn) (^gn))(^t;x;v)^'(^t;x;v)^dvdxdtj

n^X1 j⁼⁰

j Z

(j⁺¹⁾T jT

Z

x²IR³

Z

v²IR³((^fn)(^t;x;v)

(^f_jn)((^j+ 1)^{T; x;v}))^'(^t;x;v)^dvdxdtj

+ ^nX1

j⁼⁰

j Z

(j⁺¹⁾T jT

Z

x²IR³

Z

v²IR³((^gn)(^t;x;v)

(^g_jn)(^{jT; x;v}))^'(^t;x;v)^dxdvjdt

n^X1 j⁼⁰

Z

(j⁺¹⁾T jT

Z

x²IR³

Z

v²IR³

Z

(j⁺¹⁾T

t ^vrx(^fn)(^s;x;v)^ds

'(^t;x;v)^dxdvjdt + ^nX1

j⁼⁰

Z

(j⁺¹⁾T jT

Z

x²IR³

Z

v²IR^3j

Z t jT

0(^gn)(^s)^Q(^gn)(^s)^dsdxdvjdt

^Tkvrx^'kL¹⁽K^)k(^fn)^k_L^1([0_;T^]_IR³_IR³⁾

+ ^Tk'kL¹⁽K^)k0(^gn)^Q(^gn)^kL¹⁽K^); (2^:5^:1) which clearly tends to 0 when ^T = ^T_n tends to 0.

Step 2 : We prove ii). Taking (^s) = ^1+s_s, we note that

0^s (^s)^Rs+^s1s>R^: (2^:5^:2) Therefore,

0^gn (^gn)^Rgn+^gn^jlog^gn^j

log^{R ;} (2^:5^:3) and the same estimate holds for ^fn. Using then estimates (2.3.12) and (2.3.13), we get

nsup²IN_tsup

2[0;T^]kfn (^fn)^k_L¹⁽_IR³_IR^3)!

!0

0^; (2^:5^:4)

and _nsup

2IN_tsup

2[0;T^]kgn (^gn)^kL¹⁽IR³IR^3)!0!0^: (2^:5^:5) According to step 1 and estimates (2.5.4), (2.5.5), we get ii).

2.6 Strong compactness for velocity averages

In this section we get some informations on the limits of the sequence^Q(^gn) which follow from the strong compactness of the velocity averages of^gn.

lemma 4

: For all ^' in ^L1([0^;T]^IR3IR3), the sequence

jn(^t;x) =^Z_v

2IR^3gn(^t;x;v) (^t;x;v)^dv (2^:6^:1) lies in a strongly compact set of ^L1([0^;T]^IR3).

Proof

: The proof is divided in six steps. During the ve rst steps, we x a function in^L1_c (^IR3), a nonnegative and smooth function satisfying (2.2.1). Denoting for every function^h in ^L1_loc([0^;T]^IR3IR3),

~

h(^t;x) =^Z_IR

3

h(^t;x;v) (^v)^dv (2^:6^:2) we prove that^g(^gn) lies in a strongly compact set of ^L1([0^;T]^IR3).

Step 1 : We compute

@

@t

(^gn) = ⁰(^gn)^Q(^gn) +^nX1

j⁼¹

n

(^g_jn)(^jT) (^g_n^{j 1})(^jT)^ojT(^t)

=⁰(^gn)^Q(^gn) +^nX1

j⁼¹

n

(^f_jn)((^j+ 1)^T) (^f_jn)(^jT)^ojT(^t)

=⁰(^gn)^Q(^gn) +^nX1

j⁼¹

(

Z

(j⁺¹⁾T

jT ^vrx(^fn)(^s;x;v)^ds

)

jT(^t)^: (2^:6^:3) Therefore, for all function in ^C_c¹(^IR3_x), we have

d

dt Z

x²IR³

g

(^gn)^'(^x)^dx=^{Z Z}_IR

3

IR³

0(^gn)^Q(^gn) (^v)(^x)^dvdx +^nX1

j⁼¹

Z

(j⁺¹⁾T jT

Z

x²IR³

Z

v²IR³(^fn) (^v)^vrx(^x)^dxdvdsjT(^t)^; (2^:6^:4)

which is a bounded sequence of measures in [0^;T], thanks to (2.3.1) and lemma 2.

Thus, the quantity

an(^t) = ^Z_x

2IR³

g

(^gn)(^t;x)^'(^x)^dx (2^:6^:5) is bounded in ^BV([0^;T]).

Considering now a sequence of (compactly supported) molliers "(^x), the previous statement implies that for every xed ^{" >} 0, the sequence

g

(^gn)x" is strongly compact in ^L1([0^;T]^IR3).

Thanks to the identity

g

(^gn) =^g(^gn)x"+^ng(^gn) ^g(^gn)x"

o

; (2^:6^:6) we only need to prove that the second term is uniformly (in ⁿ) small in

L

1([0^;T]^IR3) when^"tends to 0 to get the strong compactness in^L1([0^;T]

IR

3) of ^g(^gn).

This will in turn be true if we prove that

Ih = sup_n

2IN

Z T

0 Z

IR^3j

g

(^gn)(^t;x+^h) ^g(^gn)(^t;x)^jdxdt (2^:6^:7) tends to 0 when ^h tends to 0. Steps 2 to 5 are devoted to the proof of this estimate.

Step 2 : We compute

Ih _nsup

2IN n^X1

j⁼⁰

Z

IR³

Z

(j⁺¹⁾T jT ^j

g

(^gn)(^t;x+^h) ^g(^gjn)(^jT;x+^h)^jdxdt + sup_n

2IN n^X1 j⁼⁰

Z

IR³

Z

(j⁺¹⁾T jT ^j

g

(^gjn)(^jT;x+^h) ^g(^gjn)(^jT;x)^jdxdt + sup_n

2IN n^X1 j⁼⁰

Z

IR³

Z

(j⁺¹⁾T jT ^j

g

(^gjn)(^jT;x) ^g(^gn)(^t;x)^jdxdt

2^{Tjj jj}_L¹⁽_IR^3v) _nsup

2IN ^jj

0(^gn)^Q(^gn)^jj_L^1([0_;T^]_IR³_Supp ⁾+^Jh^; (2^:6^:8)