Solutions with increasing energy for the spatially homogeneous Boltzmann equation

Nonlinear Analysis: Real World Applications 3 (2002) 243 – 258 www.elsevier.com/locate/na

Solutions with increasing energy for the spatially homogeneous Boltzmann equation Xuguang Lua , Bernt Wennbergb; ∗ a Department

b Department

of Mathematical Sciences, Tsinghua University, Beijing 100084, China of Mathematics, Chalmers University of Technology, S-41296 G%oteborg, Sweden Received 24 December 1999; accepted 23 March 2001

Keywords: Boltzmann equation; Non-uniqueness; Weak solutions; Moment estimates; Povzner inequality; Energy growth

1. Introduction The spatially homogeneous Boltzmann equation: @ f(v; t) = Q(f; f)(v; t); @t

(v; t) ∈ R3 × (0; ∞);

(1)

describes the time evolution of the velocity distribution of a spatially homogeneous dilute gas of particles. We look for positive solutions f ∈ C([0; ∞); L1 (R3 )), of Eq. (1) such that for all t ∈ [0; ∞); f(·; t) is a density. In (1), Q is the collision operator acting on functions of velocity v:

B(v − v∗ ; !)[f(v )f(v∗ ) − f(v)f(v∗ )] d! dv∗ ; (2) Q(f; f)(v) = R3 ×S2

which describes the rate of change of f due to a binary collision (Q(f; f)(v; t) means Q(f(·; t); f(·; t))(v); the time enters only as a parameter in (2)). Here v; v∗ are the velocities of two particles before they collide, and v ; v∗ are their velocities after the collision. Leaving the details for later, we only note here that strictly hard potential interaction is the only case considered in this paper. This is not only for technical reasons: the results depend strongly on the fact that the B(v−v∗ ; !) grows unboundedly ∗ Corresponding author. Tel.: +46-31-772-5326; fax: +46-31-16-1973. E-mail addresses: [email protected] (X. Lu), [email protected] (B. Wennberg).

c 2002 Elsevier Science Ltd. All rights reserved. 1468-1218/02/$ - see front matter
PII: S 1 4 6 8 - 1 2 1 8 ( 0 1 ) 0 0 0 2 6 - 8

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with |v − v∗ |. In the general case considered here, B(v − v∗ ; !) also may be unbounded as a function of !, but that does not have any impact on the result. In the following section, all details about the collision operator, as well as the concept of weak solutions for Eq. (1) will be discussed. The remaining part of this section contains a short discussion and a background. In a gas, each binary collision conserves the total mass, momentum and energy of the participating particles, and it is natural to expect from the solutions of the Boltzmann equation that the total energy should be conserved throughout the evolution (this is of course a very non-mathematical statement; one essential property of the Boltzmann equation is that, even though the microscopical dynamics is reversible, the Boltzmann equation by itself, is not). Under suitable conditions, the energy actually is conserved. It is known that in the case of hard cutoD potentials, there is a unique solution to Eq. (1) that conserves energy. It is also known that there are no solutions for which the energy is decreasing [10,11]. However, still under the cutoD condition, it is shown in [16], that under very general conditions, there are always solutions for which the energy jumps, at time t = 0 or at any other time. Clearly it is then possible to construct a solution with an arbitrary number of jumps, and one can quite arbitrarily also choose the size of the jump (always with the restriction that it be positive). The purpose of this paper is two-fold: Frst to show that the energy behaves in the same way for weak solutions to the non-cutoD equation, i.e. that it is non-decreasing but may increase; secondly to construct solutions for which the energy increases continuously with time. The main results are given as Theorems 2, 3 and 5. It is in its place to say that the solutions constructed here, and in [16] are not the Frst examples of solutions to the Boltzmann equation where energy is increasing. In [12], so-called homo-energetic solutions are constructed. These are special solutions to the full (spatially dependent) Boltzmann equation, for which the temperature is space independent. There are examples of homo-energetic shear Gows, for which the temperature is increasing. The paper is organised as follows. First, in Section 2, the Boltzmann equation is presented in the generality needed for our purpose. This includes a precise deFnition of the collision operator, the deFnition of weak solutions to (1) and we include some known results concerning such solutions, including a weak stability result. In Section 3, we prove that the energy of the weak solutions may not decrease, just as in the case of mild solutions to the Boltzmann equation with cutoD, and this implies the existence of solutions for which the energy is conserved. Contrary to the case of cutoD potentials, however, this does not entail uniqueness of solutions, as we only deal with solutions which are weak limits of solutions to truncated problems. For this particular class of solutions, we also prove that moments are generated, as in the cutoD case. Section 4, Fnally, contains the construction of solutions for which the energy is strictly (and continuously in time) increasing at quite arbitrary rates. This shows in particular, that without additional assumptions, the solutions are not unique.

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245

2. The spatially homogeneous Boltzmann equation The spatially homogeneous Boltzmann equation (1) has been studied thoroughly for a long time, and we give here only the results that are relevant for the subsequent sections. In the collision operator in (2), the velocities before and after a collision are related by v = v − v − v∗ ; !!; v∗ = v∗ + v − v∗ ; !!;

! ∈ S2 ;

where ·; · is the inner product in R3 ; |v|2 = v; v and S2 = {! ∈ R3 | |!| = 1}. This implies that v +v∗ = v+v∗ ; |v |2 +|v∗ |2 = |v|2 +|v∗ |2 , i.e. mass and energy are conserved in each collision. The rate at which each possible collision occurs is given by the collision kernel B(v−v∗ ; !), which is a nonnegative function of |v−v∗ | and |v−v∗ ; !| only. For the interaction potentials of inverse power laws, B(v − v∗ ; !) takes the form (see, e.g. [5] or [12]): B(v − v∗ ; !) = b( )|v − v∗ | ;

= arccos(|v − v ∗ |−1 |v − v∗ ; !|);

(3)

where the function b( ) is strictly positive in (0; =2) and the exponent is related to the potentials of interacting particles: for the so-called soft potentials: −3 ¡ ¡ 0; for pseudo-Maxwellian molecules: = 0; for the hard potentials: 0 ¡ ¡ 1; and for the hard sphere model: = 1, b( ) = const: cos . Except for the case of hard spheres, the angular function b( ) has a non-integrable singularity at = =2: b( ) = O((=2 − )−(3− )=2 );

(4)

as → =2. In general, however, the function b( ) cos2 sin is integrable on (0; =2). The so-called cutoD-assumption means that b( ) is truncated so as to make
=2 b( ) sin d ¡ ∞: 0

For hard potentials with the cutoD assumption, the collision operator can be split into two parts, Q+ (f; f)(v), and Q− (f; f)(v), corresponding to the positive and negative part of (2), and solutions to (1) satisfy the mild form of the equation:
t Q(f; f)(v; ) d; t ¿ 0: f(v; t) = f0 (v) + 0

For these solutions it is known that:   • R3 f(v; t) dv = R3 f0 (v) dv (i.e. mass is conserved, so that f(·; t) for all t is a density). • E(f(t)) ¿ E(f0 ); t ¿ 0 (i.e. the energy is non-decreasing), where
f(v; t)|v|2 dv: E(f(t)) = R3

• There is one and only one solution for which the energy is conserved.

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• H (f(t)) 6 H (f0 ); t ¿ 0 (i.e. the kinetic entropy is non-increasing) where
H (f(t)) = f(v; t) log(f(v; t)) dv: R3

A rather complete discussion of this, including numerous references, can be found in [1,10,11]. In the non-cutoD case, the two parts of the collision operator are not deFned separately, and one is restricted to the study of weak solutions to Eq. (1). These can be deFned in diDerent ways depending on the singularity of the collision kernel B. There is always a singularity in !, as described in (4), but for soft potentials there is also a singularity where |v − v∗ | → 0. For very soft potentials ( 6 − 1) the latter singularity is the more severe. A concept of weak solutions for this situation has been considered by Goudon [9] and Villani [13]. As our main object here is the study of hard potentials, we give the deFnition of weak solutions as given by Arkeryd [2], which is suitable when −1 ¡ 6 1, i.e.
=2 0¡ b( ) cos sin d ¡ ∞: (5) 0

Denition 1. A non-negative function f : [0; ∞) → L1 (R3 ) is a weak solution to (1) if it satisFes


t
@ f(v; t)’(v; t) dv = f(v; 0)’(v; 0) dv + f(v; ) ’(v; ) dv d @ 3 3 3 0 R R R
t


B(v − v∗ ; !)f(v; t)f(v∗ ; t) + R3 ×R3 ×S2

0



× (’(v ; t) − ’(v; t)) d! dv∗ dv d; for all ’ ∈ C 1; ∞ . Here  C 1; ∞ = ’ ∈ C 1 ([0; ∞)R3 )| |’|1; ∞ ≡

sup

(|’(v; t)| + |@t ’(v; t)| + |∇v ’(v; t)|) ¡ ∞

(v; t)∈R3 ×[0;∞)

(6)



:

Note that any mild solution to the Boltzmann equation with cutoD satisFes Eq. (6), which guarantees the existence of weak solutions in that case. Arkeryd used a weak stability result to prove that weak solution also exist in the general case. Here, and always in the sequel of the paper, we assume that the initial data satisfy the L12 Log bound:
0¡ f0 (v)(1 + |v|2 + |log f0 (v)|) dv ¡ ∞: (7) R3

We also deFne the weighted L1 -spaces L1s with norm
fL1s = |f(v)|(1 + |v|2 )s=2 dv: R3

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247

Theorem 1 (weak stability and existence of solutions) (Arkeryd [2]). Assume that the collision kernel B(v − v∗ ; !) satisfy (3) and (5); and let Bn (v − v∗ ; !) be a sequence of cuto; kernels converging point-wise to B(v − v∗ ; !) and satisfying Bn (v − v∗ ; !) 6 B(v − v∗ ; !). Assume that f0n (v) is a sequence of initial data for the Boltzmann equation with kernel Bn (v − v∗ ; !); and f0n * f0

weakly in L1 (R3 ):

Let fn (v; t) be the corresponding (weak) solutions that conserve mass and momentum; and satisfy
sup (8) fn (v; t)(1 + |v|2 + |log(fn (v; t))|) dv ¡ ∞: n¿1; t¿0

R3

Then there is a subsequence fnj such that for all t; fnj (·; t) * f(·; t) weakly in L1 (R3 ); where f(v; t) is a weak solution to the Boltzmann equation with the kernel B and f(·; 0) = f0 . This weak solution conserves mass and momentum; and H (f(t)) 6 lim inf H (fnj (t)); j→∞

E(f(t)) 6 lim inf E(fnj (t)); n→∞

t ¿ 0;

(9)

t ¿ 0:

(10)

Remark. The statement about the weak lower semi-continuity of entropy (9) is proven in [7]. This together with the bound (10) on the energy implies that also the limit satisfy (8). For the remaining part, we restrict ourselves to the case ¿ 0, i.e. to hard potentials and hard sphere model.

3. Energy and moment estimates for weak solutions Moment estimates for the Boltzmann equation are usually obtained by the use of a type of estimates that were Frst obtained by Povzner; later more elaborate versions have been obtained in several contexts. See e.g. [4,11,14] or [10] for references. The following lemma gives a sharpened version of the Povzner inequality, which is essentially used for hard potentials without cutoD. Lemma 1. Let c ¿ 0; s ¿ 2 be constants. Then (c + |v |2 )s=2 + (c + |v∗ |2 )s=2 − (c + |v|2 )s=2 − (c + |v∗ |2 )s=2 6 2s+1 [(c + |v|2 )(s−1)=2 (c + |v∗ |2 )1=2 + (c + |v∗ |2 )(s−1)=2 (c + |v|2 )1=2 ] cos sin − Ks [(c + |v|2 )s=2 + (c + |v∗ |2 )s=2 ] cos2 sin2 ; where = arccos(|v − v∗ |−1 |v − v∗ ; !|);

 Ks = min

 1 s(s − 2); 2 : 4

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Proof. The proof of this lemma relies on a suitable parameterisation of the sphere S2 and on writing r 2 = r 2 sin2 + r∗2 cos2 + 2rr∗ sin cos cos %; r∗2 = r 2 cos2 + r∗2 sin2 − 2rr∗ sin cos cos %; where r = |v|; r  = |v | etc., cos = |v − v∗ |−1 v − v∗ ; !; ∈ [0; ], and || 6 1. By symmetry, we can assume that ∈ [0; =2]. Then   c + r 2 6 [ c + r 2 sin + c + r∗2 cos ]2 ;   c + r∗2 6 [ c + r 2 cos + c + r∗2 sin ]2 : Sine s ¿ 2, applying the elementary inequality (a + b)s 6 as + bs + 2s (as−1 b + a bs−1 );

a; b ¿ 0;

to each of the right-hands of the above two inequalities we have (c + r 2 )s=2 + (c + r∗2 )s=2 6 [(c + r 2 )s=2 + (c + r∗2 )s=2 ](sins + coss ) + 2s+1 [(c + r 2 )1=2 (c + r∗2 )(s−1)=2 + (c + r 2 )(s−1)=2 (c + r∗2 )1=2 ] cos sin : Also, since s ¿ 2, it is easily shown that sins + coss 6 1 − Ks cos2 sin2 : This proves the lemma. In [11] and [10] it was proven that the energy of the mild solutions to the cutoD hard potential Boltzmann equation is non-decreasing. Here, we use the approach of Lu to extend this result to weak solutions, but still only for hard potentials. Theorem 2. Let the initial data to the Boltzmann equation satisfy 0 6 f0 ∈ L12 (R3 ); and let the collision kernel B(v − v∗ ; !) satisfy Eqs. (3) and (5) with 0 6 6 1. Then for any weak solution f ∈ L∞ ([0; ∞); L12 (R3 )) of the Boltzmann equation (1); the energy is always non-decreasing on [0; ∞): E(f(t)) ¿ E(f(s));

t ¿ s ¿ 0:

(11)

In particular; it follows from Theorem 1 that there is a solution to Eq. (1) that conserves the energy. Proof. For & ¿ 0; ' ¿ 0, let  &|v|2 1 ; ’&; ' (v) = log 1 + & 1 + '|v|2

’& (v) =

1 log(1 + &|v|2 ): &

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249

√ It is easily seen that ’&; ' ∈ C 1; ∞ : ’&; ' (v) 6 1=' and |grad v ’&; ' (v)| 6 1= & so that for t ¿ s ¿ 0,

f(v; t)’&; ' (v) dv = f(v; s)’&; ' (v) dv R3

R3

+




1 2

s

t




d

R3 ×R3 ×S2

F&; ' (v; v∗ ; !; ) d! dv∗ dv;

where F&; ' (v; v∗ ; !; ) = B(v − v∗ ; !)f(v; )f(v∗ ; )[’&; ' (v ) + ’&; ' (v∗ ) − ’&; ' (v) − ’&; ' (v∗ )]: The absolute value of the expression within square brackets is smaller than 2 2 √ |v − v∗ ; !| = √ |v − v∗ | cos ; & &  =2 and, with A0 = 4 0 b( ) cos sin d ¡ ∞;
t


2 d B(v − v∗ ; !)f(v; )f(v∗ ; ) √ |v − v∗ ; !| d! dv∗ dv & s R3 ×R3 ×S2 
2
t 2 6 √ A0 d f(v; )(1 + |v|2 )(1+ )=2 dv ¡ ∞: & R3 s Moreover, ’&; ' (v) 6 ’& (v); ’&; ' (v) → ’& (v) (' → 0 +), and so the dominated convergence theorem implies that
t


d |F& (v; v∗ ; !; )| d! dv∗ dv ¡ ∞; s

R3 ×R3 ×S2

and that
R3


f(v; t)’& (v) dv =

R3

+

1 2

f(v; s)’& (v) dv
s

t




d

R3 ×R3 ×S2

F& (v; v∗ ; !; ) d! dv∗ dv;

where F& (v; v∗ ; !; ) = F&; 0 (v; v∗ ; !; ). Next using the conservation of energy, |v |2 + |v∗ |2 = |v|2 + |v∗ |2 and the fact that %& is a logarithm, we can write  1 &2 (|v |2 |v∗ |2 − |v|2 |v∗ |2 )+ ’& (v ) + ’& (v∗ ) − ’& (v) − ’& (v∗ ) = log 1 + & (1 + &|v|2 )(1 + &|v∗ |2 )  1 &2 (|v|2 |v∗ |2 − |v |2 |v∗ |2 )+ − log 1 + ; (1 + &|v |2 )(1 + &|v∗ |2 ) &

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X. Lu, B. Wennberg / Nonlinear Analysis: Real World Applications 3 (2002) 243 – 258

where (y)+ = max{y; 0}. Neglecting the positive part we obtain

f(v; t)’& (v) dv ¿ f(v; s)’& (v) dv R3

R3




− where J& (v; v∗ ) ≡

1 2&

s

t



d

R3 ×R3

f(v; )f(v∗ ; )J& (v; v∗ ) dv∗ dv;

(11*)

 &2 (|v|2 |v∗ |2 − |v |2 |v∗ |2 )+ d!: B(v − v∗ ; !) log 1 + (1 + &|v |2 )(1 + &|v∗ |2 ) S2




Using the identity |v|2 |v∗ |2 − |v |2 |v∗ |2 = 12 (|v |4 + |v∗ |4 − |v|4 − |v∗ |4 ) and Lemma 1 (with c = 0; s = 4) we have (|v|2 |v∗ |2 − |v |2 |v∗ |2 )+ 6 24 (|v|2 + |v∗ |2 )|v| |v∗ | cos sin : Thus by the inequality (1=&) log(1 + &2 y) 6 &y we Fnd that J& (v; v∗ ) 6 24 A0

&(|v|2 + |v∗ |2 )|vv∗ v − v∗ | 6 C (1 + |v|2 )(1 + |v∗ |2 ): 1 + &(|v|2 + |v∗ |2 )

Since lim&→0+ J& (v; v∗ ) = 0; for all (v; v∗ ) ∈ R3 × R3 , it follows from the dominated convergence theorem that
t

d f(v; )f(v∗ ; )J& (v; v∗ ) dv∗ dv = 0; lim &→0+

and

s

R3 ×R3


lim

&→0+

R3

f(v; t)’& (v) dv = E(f(t));

for all t ¿ 0. Therefore (10) follows from (11) by letting & → 0 +. The proof of Theorem 2 is completed. An early result on moments of solutions to the Boltzmann equation is that of Elmroth [8], who showed that all moments that are initially bounded remain bounded. It was then shown, Frst by Desvillettes [6], that solutions of the Boltzmann equation for hard potentials gain moments. His result was then improved upon in e.g. [15]. Here we use the estimates from the latter paper together with the weak convergence result from Section 2 to prove that also in the non-cutoD case, there are solutions that behave similarly. Theorem 3 (moment estimates for weak solutions). Let the kernel B(v−v∗ ; !) satisfy (3) and (5) for 0 ¡ 6 1; and let f0 ¿ 0 satisfy (7). Then the Boltzmann equation (1) has a weak solution that conserves mass; momentum and energy; and satis>es

X. Lu, B. Wennberg / Nonlinear Analysis: Real World Applications 3 (2002) 243 – 258

251

H (f(t)) 6 H (f0 ); t ¿ 0: Moreover; for all s ¿ 2 we have the estimate f(·; t)L1s 6 Ws (t; E(f0 ); f0 L10 ); where

t ¿ 0;



aLs Ws (t; E(f0 ); f0 L10 ) = 1 − exp(− =(s − 2)as t)

(s−2)= ;

t ¿ 0;

and; for any >xed s ¿ 2; the coe@cients as = as (E(f0 ); f0 L10 ) and aLs = aLs (E(f0 ); f0 L10 ) are strictly positive and continuous functions of E(f0 ) and f0 L10 . Proof. We begin by choosing a sequence of truncated collision kernels, and a sequence of truncated initial data as follows: Bn (v − v∗ ; !) = |v − v∗ | min(b( ); n); 2

f0n (v) = e−|v| =n f0 (v): Since the full initial data f0 satisFes the estimate (7), the same holds for all f0n , uniformly in n. In addition, all the functions f0n satisfy bounds on higher moments, f0n L1s 6 Cf0 ;n;s . Also we have lim f0n − f0 L12 = 0;

n→∞

lim H (f0n ) = H (f0 ):

n→∞

(12)

Let fn be the mild solutions of the Boltzmann equation (1) with these initial data and collision kernels. From the results in [11] or [10], those are unique in the class of solutions that conserve energy. And we also know that H (fn (t)) 6 H (f0n ); ∀t ¿ 0: These imply that {fn } has the L12 Log bound (8). Thus the results from Theorems 1 and 2 together with the limits in (9), (10) and (12) imply that there is a subsequence (which we still denote by fn ) that converges weakly to a weak solution f of (1) with the initial f|t=0 = f0 and satisfying the entropy inequality H (f(t)) 6 H (f0 ) and the conservation of energy E(f(t)) = E(f0 );

t ¿ 0:

Now, we wish to prove that the limiting function f at any positive time possesses moments of all orders. The method is the same as in [4,15] (see also [10]); we only need to verify that all estimates for fn can be obtained in a uniform way with respect to n. For any given s ¿ 2, we begin by multiplying Eq. (1) (with kernel Bn (v − v∗ ; !)) by (1 + |v|2 )s=2 , and then integrate. This gives d fn (·; t)L1s dt


1 = fn (v; t)fn (v∗ ; t)bn ( )|v − v∗ | ((1 + |v |2 )s=2 2 R3 ×R3 ×S2 + (1 + |v∗ |2 )s=2 − (1 + |v|2 )s=2 − (1 + |v∗ |2 )s=2 ) d! dv dv∗ ;

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X. Lu, B. Wennberg / Nonlinear Analysis: Real World Applications 3 (2002) 243 – 258

which by Lemma 1 together with an obvious symmetry argument implies d fn (·; t)L1s dt

6 Cs

R3 ×R3

fn (v; t)fn (v∗ ; t)|v − v∗ | (1 + |v|2 )(s−1)=2 (1 + |v∗ |2 )1=2 dv dv∗



−Cs; n

R3 ×R3

fn (v; t)fn (v∗ ; t)|v − v∗ | (1 + |v|2 )s=2 dv dv∗ ;

where s+1

Cs = 2


Cs; n = Ks A1; n ;

A0 ;

A1; n = 4

0

=2

(13)

bn ( ) cos2 sin3 d :

Using the inequality |v − v∗ | ¿ (1 + |v|2 ) =2 − (1 + |v∗ |2 ) =2 we have

fn (v; t)fn (v∗ ; t)|v − v∗ | (1 + |v|2 )s=2 dv dv∗ Cs; n R3 ×R3

¿ Cs; n f0n L10 fn (·; t)L1s+ − Cs; n f0 L12 fn (·; t)L1s :

(14)

Furthermore, by the HMolder inequality we have [fn (·; t)L1s ]1+ =(s−2) 6 f0n L =(s−2) fn (·; t)L1s+ : 1 2

Therefore by (11) and (14) we obtain d fn (·; t)L1s 6 as fn (·; t)L1s − as; n [fn (·; t)L1s ]1+ =(s−2) ; dt

t¿0

(15)

where as = 2Cs f0 L12 ;

as; n = Cs; n f0n L10 [f0n L12 ]− =(s−2) :

The conclusion that can be drawn from (15) is that (s−2)=

as =as; n fn (·; t)L1s 6 ; 1 − exp(− =(s − 2)as t)

t ¿ 0:

(16)

Since limn→∞ f0n L1k = f0 L1k ¿ 0 for k = 0; 2, it follows that lim as; n = Ks A1 f0 L10 [f0 L12 ]− =(s−2) ;

n→∞

where A1 = 4 for (12).

 =2 0

b( ) cos2 sin3 d : The conclusion follows by taking weak limit

4. Solutions with continuously increasing energy We begin this section by proving that Eq. (1) allows for solutions for which the energy makes an arbitrary number of jumps (but always increasing). This is an extension

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253

of the result from [16] to weak solutions of the non-cutoD Boltzmann equation. Then we go on to prove that the energy may continuously increase at an arbitrary rate in a way to be formulated precisely in Theorem 5. One of the main steps of our proof is to use the moment estimates (Theorem 3). Since for all solutions f ∈ L∞ ([0; ∞); L12 (R3 )), the mass f0 L10 is always conserved, we often simply write Ws (t; E(f0 ); f0 L10 ) = Ws (t; E(f0 ));

t ¿ 0:

Theorem 4. Let the collision kernel B(v − v∗ ; !) satisfy (3) and (5) with 0 ¡ 6 1; and let the initial datum f0 satisfy (7). Then for any positive integer N; and any set of times 0 = t0 ¡ t1 ¡ t2 ¡ · · · ¡ tN −1 ¡ tN ¡ ∞ and any set of numbers (“energy jumps”) 0 6 e1 6 e2 6 · · · 6 eN 6 eN +1 ¡ ∞; there is a weak solution f to the Boltzmann equation (1) with the initial f0 such that mass and momentum are conserved; the entropy is non-increasing: H (f(t)) 6 H (f0 ); t ¿ 0; and such that the energy jumps at the given times tk E(f(t)) = E(f0 ) +

N

ek 5(tk−1 ;tk ] (t) + eN +1 5(tN ;∞) (t);

t ¿ 0:

k=1

Moreover; on each time-interval; f satis>es the momentum estimate: for any s ¿ 2;

Ws (t − tk−1 ; E(f0 ) + ek ); t ∈ (tk−1 ; tk ]; k = 1; 2; : : : ; N; f(·; t)L1s 6 Ws (t − tN ; E(f0 ) + eN +1 ); t ∈ (tN ; ∞): Proof. For any n ¿ 1, let n (v) =

1 |v|−(5+1=n) /{|v|¿1} ; 4n

f0n (v) = f0 (v) + e1 n (v):

Then for all 0 ¡ 2, limn→∞ f0n − f0 L10 = 0, and E(f0n ) = E(f0 ) + e1 ; n = 1; 2; : : : ;
sup f0n (v)(1 + |v|2 + |log f0n (v)|) dv ¡ ∞:

n¿1

(17)

R3

By Theorem 3, there exist weak solutions fn of (1) with fn |t = 0 = f0n satisfying: 1. fn conserve the mass, momentum and energy; 2. H (fn (t)) 6 H (f0n ); t ¿ 0; 3. for any s ¿ 2, fn (·; t)L1s 6 Ws (t; E(f0n ); f0n L10 );

∀t ¿ 0; ∀n ¿ 1:

(18)

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By weak stability, there exist a subsequence {fnj }∞ j=1 and a weak solution f1 of (1) 1 with the initial datum f1 |t = 0 = f0 and having the L2 Log bound (8) and conserving the mass and momentum, such that fnj (·; t) * f1 (·; t) (j → ∞) weakly in L1 (R3 ); ∀t ¿ 0: Because limn→∞ H (f0n ) = H (f0 ) (which is easily proven by limn→∞ f0n −f0 L11 = 0), it follows from the weak lower semi-continuity of the entropy that the solution f1 also satisFes H (f1 (t)) 6 H (f0 ), t ¿ 0. Moreover, by weak convergence and because of the identity (17), we obtain from Eq. (18) that f1 (·; t)L1s 6 Ws (t; E(f0 ) + e1 );

∀t ¿ 0; s ¿ 2:

(19)

Also by (17), (18) and by the weak convergence, it follows that E(f1 (·; t)) = E(f0 ) + e1 ;

∀t ¿ 0:

(20)

Replacing f0 = f(·; 0) and e1 by f(·; t1 ) and e2 − e1 , respectively, and using the same argument, we see that there is a weak solution f2 on [t1 ; ∞) with the initial datum f2 (·; t1 ) = f1 (·; t1 ) and conserving the mass and momentum and satisfying H (f2 (t)) 6 H (f1 (t1 )) 6 H (f0 ) for t ¿ t1 , such that E(f2 (t)) = E(f1 (t1 )) + e2 − e1 = E(f0 ) + e2 ;

t ¿ t1 ;

and satisfying for all s ¿ 2, f2 (·; t)L1s 6 Ws (t − t1 ; E(f0 ) + e2 );

t ¿ t1 :

By induction using the same arguments, we obtain solutions fk of (1) on [tk−1 ; ∞) with initial data fk (·; tk−1 ) = fk−1 (·; tk−1 ) (f0 (·; 0) = f0 ) and conserving the mass and momentum and satisfying H (fk (t)) 6 H (f0 ) for all t ¿ tk−1 , such that E(fk (t)) = E(f0 ) + ek ;

∀t ¿ tk−1 ;

and for any s ¿ 2 fk (·; t)L1s 6 Ws (t − tk−1 ; E(f0 ) + ek );

∀t ¿ tk−1 ;

k = 1; 2; : : : ; N + 1. Now, we deFne

fk (·; t); t ∈ [tk−1 ; tk ]; k = 1; 2; : : : ; N; f(·; t) = fN +1 (·; t); t ∈ [tN ; ∞): Then by the integral form (6) of weak solutions of Eq. (1), it is easily veriFed that f is a desired weak solution having all properties stated in the theorem. Remark. Note that this construction does not set any eDective limit as to what energy could be reached in a Fnite time. Take a sequence tn → T and a sequence en → ∞. Then the above construction can be carried out for any set t1 ¡ t2 ¡ · · · ¡ tn and corresponding energies. Then one can go on for larger and larger n. The estimate in Theorem 3 is not so useful when the interval lengths are allowed to 2n−1 shrink. For instance, if the time-interval is subdivided as (0; ∞) = k=1 (2−n (2k − 1); 2−n 2k] ∪ (1; ∞); then the momentum estimates Ws (t − 2−n (2k − 1); E(f0 ) + · · ·) on

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255

the intervals (2−n (2k − 1); 2−n 2k] will all blow-up because the interval lengths 2−n tend to 0 as n → ∞. This poses a problem when trying to construct a solution with continuously increasing energy. Hence we construct here a solution for which the energy is a continuous Cantor-like function, where the energy is constant in every (Fxed) complementary interval of a Cantor set. Before formulating the theorem, we Frst recall the construction of a general symmetric Cantor set (see e.g. [3]). For convenience we only consider the Cantor set that has positive measure. The only diDerence between such a set and the original Cantor set is that rather than removing “middle thirds”, we remove smaller fractions of the remaining intervals. Apart from that, this Cantor set has all the usual properties: it is perfect, totally disconnected and so on. In the following, we always denote by I , J etc., the left-open and right-closed intervals, and, for instance for J = (a; b], we denote by JL = [a; b] the corresponding closed interval. Let 3 ∈ (0; 1) and {3n }∞ n = 1 ⊂ (0; 1=2) satisfy (see [3, p. 18]) lim 2n 31 32 · · · 3n = 3:

n→∞

First, the half-open interval (0; 1] is subdivided as (from left to right) (0; 1] = J1(0) = J1(1) ∪ I1(1) ∪ J2(1) ;

with 4(J1(1) ) = 4(J2(1) ) = 31 :

Here 4(A) denotes the Lebesgue-measure of a set A. Again, let J1(1) = J1(2) ∪ I1(2) ∪ J2(2) ;

J2(1) = J3(2) ∪ I2(2) ∪ J4(2) ;

with 4(Ji(2) ) = 31 32

where i = 1; 2; : : : ; 4. Then (from left to right) (0; 1] = J1(2) ∪ I1(2) ∪ J2(2) ∪ I1(1) ∪ J3(2) ∪ I2(2) ∪ J4(2) : Inductively, each set Jk(n−1) (1 6 k 6 2n−1 ) is decomposed as (n) (n) ∪ Ik(n) ∪ J2k ; Jk(n−1) = J2k−1

with 4(Ji(n) ) = 31 32 · · · 3n ;

where i = 1; 2; : : : ; 2n : At each n-level, we have decomposition m−1

(0; 1] =

n 2 

m=1 k=1

Next, let K (n) = deFned by C3 =

∞ 

n

Ik(m)

 2n

∪

2 

Ji(n) :

(21)

i=1

(n) i=1 Ji .

Then K (1) ⊃ K (2) ⊃ K (3) ⊃ · · · . The Cantor set is then

K (n)

n=1

which has the measure n

4(C3 ) = lim 4(K n→∞

(n)

) = lim

n→∞

2 i=1

4(Ji(n) ) = lim 2n 31 32 · · · 3n = 3: n→∞

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X. Lu, B. Wennberg / Nonlinear Analysis: Real World Applications 3 (2002) 243 – 258

The corresponding Cantor function (which we shall denote by 6) is deFned by
t 5C3 (x) d x; t ¿ 0; (22) 6(t) ≡ 3−1 0

(m) which is Lipschitz continuous on [0; ∞): Denote Ik(m) = (l(m) k ; rk ]. Then m−1

[0; ∞) \ C3 =

∞ 2 

m=1 k=1

(m) (l(m) k ; rk ) ∪ (1; ∞);

(23)

from which we see that the Cantor function 6 takes constant on each interval Ik(m) and [1; ∞): 6(t) = 6(rk(m) ); t ∈ Ik(m) ; 6(t) = 1; t ∈ [1; ∞): We also note that if F and G are both non-decreasing functions on [0; ∞) with properties that F(0) = G(0), F(t) = G(t) for all t ∈ [0; ∞) \ C3 , and G is continuous on [0; ∞), then F(t) ≡ G(t). This is a direct consequence of the fact that the Cantor set C3 is totally disconnected, and that if the set {F(t) | t ∈ [0; 1]} is dense in [F(0); F(1)], then F is continuous on [0; 1]. Theorem 5. Let the collision kernel B(v − v∗ ; !) satisfy (3) and (5) with 0 ¡ 6 1; and assume that f0 satis>es (7). Take any T ¿ 0; and let 9 be a continuously increasing function on [0; ∞) with 9(0) = 0. Then there exists a weak solution f of the Boltzmann equation (1) with f|t=0 = f0 for which the mass and momentum are conserved; H (f(·; t)) 6 H (f0 ); and such that the energy grows as E(f(t)) = E(f0 ) + 9(6T (t));

t ∈ [0; ∞);

(24)

where 6T (t) = T6(T −1 t); and 6 is the function de>ned in (22). (m) (n) Proof. Without loss of generality, we may suppose T = 1. Let Ik(m) = (l(m) = k ; rk ]; Ji (n) (n) ˜ (li ; r˜i ] be the left-open and right-closed intervals constructed in the above. Using Theorem 3, for each n ¿ 1, we may construct a weak solution to Eq. (1) with fn |t=0 = f0 , such that mass and momentum are conserved, and H (fn (·; t)) 6 H (f0 ); t ¿ 0, such that (using the decomposition (21)). m−1

n

E(f (t)) = E(f0 ) +

n 2 m=1 k=1

9(6(rk(m) ))5I (m) (t); k

n

+

2 i=1

9(6(r˜i(n) ))5J (n) (t) + 9(1)5(1; ∞) (t); i

t ¿ 0:

(25)

These imply Frst that {fn } has the L12 Log bound (8). By weak stability (Theorem 1) and the non-decrease of energy (Theorem 2), there exists a subsequence {fnj }∞ j=1 , and a weak solution f ∈ L∞ ([0; ∞); L1 (R3 )) of the Boltzmann equation (1) with the

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257

initial f|t=0 = f0 , such that fnj (·; t) * f(·; t) weakly in L1 (R3 ) for all t ¿ 0, and the solution f conserves the mass and momentum, and satisFes H (f(t)) 6 H (f0 ) for all t ¿ 0, and the energy E(f(t)) is non-decreasing on [0; ∞): Next, on each interval Ik(m) ; Ji(n) and (1; ∞), fn satisFes moment estimates as in Theorem 3. For instance, for any s ¿ 2, 1 6 k 6 2m−1 ; m ¿ 1 and any n ¿ m, (m) fn (·; t)L1s 6 Ws (t − l(m) k ; E(f0 ) + 9(6(rk )));

(m) t ∈ Ik(m) = (l(m) k ; rk ]

(26)

and fn (·; t)L1s 6 Ws (t − 1; E(f0 ) + 9(1));

t ∈ (1; ∞):

(27)

Now, we prove that the solution f satisFes (24). We Frst prove that E(f(t)) = E(f0 ) + 9(6(t));

t ∈ [0; ∞) \ C3 :

(28)

Let t ∈ [0; ∞) \ C3 . By (23) we Frst assume that t ∈ Ik(m) . Since 6 is constant on Ik(m) : 6(t) = 6(rk(m) ), this implies by (25) that for any nj ¿ m, E(fnj (t)) = E(f0 )+9(6(t)): Since fnj (·; t) converges weakly to f(·; t), it follows that E(f(t)) 6 E(f0 ) + 9(6(t)): On the other hand, for s ¿ 2, the estimate (26) implies that for any M ¿ 0,
fnj (v; t)|v|2 5{|v|6M } dv R3


= E(fnj (t)) −

R3

fnj (v; t)|v|2 5{|v|¿M } dv



1

fnj (·; t)L1s M s−2 1 (m) ¿ E(f0 ) + 9(6(t)) − s−2 Ws (t − l(m) k ; E(f0 ) + 9(6(rk ))): M ¿ E(f0 ) + 9(6(t)) −

Thus Frst letting j → ∞ and then letting M → ∞, we obtain E(f(t)) ¿ E(f0 ) + 9(6(t)). Therefore E(f(t)) = E(f0 ) + 9(6(t)); t ∈ Ik(m) : With the same argument (using (27)), we also have E(f(t)) = E(f0 ) + 9(6(t)); t ∈ (1; ∞): This proves (28), and hence Theorem 5, at least for this set of t. But the energy E(f(t)) is non-decreasing, and hence, by the comment just before the statement of Theorem 5, the energy must take the correct values for all t. Acknowledgements Part of this work was carried out while the second author was visiting Tsinghua University, Beijing, and he would like to express his gratitude to the university, and in particular to professor T.Q. Chen for the great hospitality. We would like to thank C. Cercignani for discussions concerning homo-energetic Gows. B.W. also acknowledges support from the TMR network “Asymptotic methods in Kinetic Theory” (contract no. ERB FMBX-CT97-0157), from the Swedish Natural Sciences Research Council, and the Swedish Royal Academy of Sciences.

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