Solutions of the Boltzmann Equation

Patterns and Waves-Qualitative Analysis of Nonlinear Differential EquationsPP. 37-96 (1986)

Solutions of the Boltzmann Equation By Seiji UKAI Abstract. This note is devoted to the review of the major results obtained in these ten years on the existence problem for the Boltzmann equation. The topics cover the local and global existence theorems for a variety of the Cauchy and initial boundary value problems, the existence and stability of the stationary solution for an exterior problem and the relation between the Boltzmann equation and fluid dynamics. They will be presented with emphasis on mathematical ideas of proofs. Also, important open problems will be pointed out. Key words: Boltzmann equation, Cauchy problem, initial boundary value problem, local solution, global solution, stationary flow, fluid dynamical limit Contents

1. Introduction 2. Local solutions, I. The case of cutoff potentials 3. Local solutions, 11. The case of non-cutoff potentials 4. Global solutions for the Cauchy problem 5. The initial boundary value problem 6 . The stationary flow 7. The Euler limit and the initial layer

37

44 52 57 74 82 89

1. Introduction The Boltzmann equation which describes the motion of a gas has been a fascinating subject for many mathematicians since its discovery by L. Boltzmann in 1872, because its physical significance is well-established and the mathematical problems are various. It is one of the nice nonlinear partial differential equations in mathematical physics, providing many challenging mathematical problems with fruitful results. However, the early development of the equation’s theory was rather slow. No existence theorems had been known until 1932 when Carleman [13] solved the spatially homogeneous case globally in time. The physically more realistic spatially non-homogeneous case was solved much later. The first solution, although local in time, was Received April 9, 1985.

S . UKAI

38

given by Grad [19] in 1965 and the global one by Ukai [37] in 1974. The subsequent progress, however, has been remarkable and is still lively. This note is intended as a review of the major existence theorems and properties of solutions known so far, with emphasis on the presentation of mathematical ideas of proofs. Also, important open problems are pointed out. There are two important topics which are not touched upon here, the Chapman-Enskog expansion and the one-dimensional shock profile solution for which the reader is referred to the review article [ l l ] . In this section, we give a brief explanation of the Boltzmann equation and the formulation of problems associated with it.

The Boltzmann equation The unknown of this equation is a scalar function f = f ( t , x , E ) which describes the number (or probability) density of gas particles having the position x = ( x l , x 2 , -,x , ) € R" and velocity E=(E,, E,, -,E,) € R" at time t € R. The specific phenomenon in a gas is the collision of gas particles. Boltzmann has derived his equation assuming (i) that collisions of more than two particles rarely occur and (ii) that between successive collisions, each particle moves according to Newton's law of motion. It is, 1.1.

-

--

(1.1.1) Here, the sum of the first two terms on the right-hand side describes the rate of change o f f due to (ii) in which . stands for the inner product of R", V , for the gradient (a/&,, ajax,, a/ax,) and similarly for V t , and a ( x , E) denotes the external force acting on particles such as gravity, electric and magnetic fields. The last term Q [ f ] is the rate of change of f d u e to binary collisions (i) and given by

.-

(1.1.2)

Q[fl=

RnxSn-1

a ,

d v , 0) l f ( ~ ) f -f(E)f(S')WE'du (~')

where v=15-E'1, cosO=(E-E').w/v, v) etc., with

o E S n - ' (the unit sphere in

+

R")andf(v)=

f(t, x ,

(1.1.3)

p = ~ - ( cos ~ e)w ,

v ' = ~ ' +cos ( ~ 010,

which denote velocities after a collision of particles having velocities E, E' before the collision and vice versa. The function q(u, 0) is called the collision cross section and determined by the interaction law which is specific to the species of gas particles. Two classical examples are the hard ball gas for which (1.1.4)

q(u, ~ ) = ~ , ~ Ielc ,o s

where uo is the surface area of the ball, and the inverse power law potential

39

Solutions of the Boltzmann Equation

of order s for which ( n = 3 , s > l ) , (1.1.5)

q(v, 8)=u’lcos Si-’‘qo(S)

,

r=l-- 4S ,

2 r’=l+-, S

where qo(S)>O is bounded and does not vanish near 8=lr/2. Extensive literature is available on the derivation of (1.1.1)-( 1.1.9, see e.g. [13, 14, 361, in which the following important properties of Q are also proved. Denote the bilinear symmetric operator induced from the quadratic operator Q by Q[., -1;

Further, define the inner product,

f(E)s(E)dS €= .

(1.1.7)

Theorem 1.1.1.

R”

(i)

Define thefunctions,

Then,for a n y f , g and O < j < n + l , (1.1.9)


Q[f, gl>~=O

*

( i i ) For a n y f 2 0 , (1.1.10) (iii) Q[fo]=O (1.1.11)

< k f , Q[fl>c
if and only i f , f0(E)=p(21rT)-n’z exp ( - 1C--vI2/(2T))

with any p > 0 , v E R”,T>O independent of

6.

The functions h,’s are called collision (summational) invariants, because from (1.1.3) it holds that, for O l j l n f l , (1.1.12)

hj(E)+hj(E’)=hj(p)+hj(q’)

9

which are the conservation laws of the number (j=O), momentum ( l s j l n ) and energy ( j = n + l ) of particles during a collision. The function f o of (1.1.11) is called a Maxwellian and describes the equilibrium state of a gas having the mass density p, flow velocity v and tem-

40

S. UKAI

perature T . If p, v, T are constant in t , x, then f O is called a n absolute Maxwellian, and otherwise, a local Maxwellian. Besides its physical importance, f ois in the special situation that it happens to be a stationary solution to (1.1.1) (e.g. a n absolute Maxwellian for the case a ( x , E ) = O ) , thanks to Theorem 1.1.1 (iii). The statements in Theorem 1 . 1 . 1 are all true, of course, only when relevant integrals converge. Cutoff and non-cutoff potentials The simplest problem associated with (1.1.1) is the local (in t ) existence of solutions to the Cauchy problem. However, this is not a trivial problem because the nonlinear operator Q is unbounded, which comes from two different singular behaviors of q(v, 0). One is the strong singularity at 8=7r/2 in (1.1.5). Since f > l , the integral over 9-' in (1.1.2) does not converge i f f is only bounded, but it does i f f is smooth because p = E , pf=E' at 0=7r/2, so the quantity { - . .} in (1.1.2) then vanishes there, thereby cancelling the singularity of q(v, 8). Thus, Q is well-defined only for smooth f and behaves like a (pseudo-) differential operator, an unbounded operator. In $ 3 we present a method to control this unboundedness, which is analogous to, but much simpler than, that used for the abstract nonlinear Cauchy-Kowalewski theorem of Nishida-Nirenberg-Obsjannikov [30]. However, this method has a limited application, and at present we can only go farther in removing this singularity by cutoff approximations. The most successful cutoff is Grad's angular cutoff [18], with which the local and global existence has been established for a variety of the Cauchy and initial boundary value problems for (1.1.1). This will be seen in $$ 4-7. It is Grad's estimates for Q and related operators with this cutoff that largely promoted the study of (1.1.1). Grad's cutoff is equivalent to replacing q by its angular cutoff, 1.2.

(1.2.1)

q"v, 8 ) = x " M v , 0)

1

where xc(0)=l (l8-~/21 > E ) , = O ( ~ O - R / ~ / < E ) . Obviously, the ambiguity in choosing the value of e arises unless the convergence is established in some way. This will be discussed in S 3.2. Observe that the hard ball gas (1.1.4) has no singularities and is considered as a cutoff potential case. Even with cutoff, Q may be unbounded since q(v, 0)-w ( v - > m ) , as seen in (1.1.4) and (1.1.5) with s > 4 . This unboundedness is handled in $ 2 by a method similar to that of S 3.

1.3. Conservation laws Once local solutions are found, global solutions can be constructed if nice a priori estimates are available. The Boltzmann equation has L' estimates be the inner coming from physically natural conservation laws. Let

<-, ->

Solutions of the Boltzmann Equation

41

product (1.1.7) but over R Z n x R E n . Suppose a ( x , E ) = O and let f be a smooth solution to (1.1.1). Then, by integration by parts and by (1.1.9), we have, (1.3.1)

=, ~ E R ,O < i < n + l ,

which are the conservation laws of the total number (j=O), momentum ( l < j < n ) and energy ( j = n + l ) of particles. In $ 2 . 4 , we see that f ( t ) 2 O provided the initial datum f , = f ( 0 ) 2 0 (and q(v, 0)20), a physically natural conclusion because f denotes the density. Hence, (1.3.1) is L‘ a priori estimates. But it is not useful because it seems difficult to find L1-local solutions by the reason stated in s 2 . 2 (see also [33]), which presents a contrast to the spatially homogeneous case where (1.3.1) is essential in costructing global solutions [2, 31. In $ 4 , we propose a method to find global solutions, which takes advantage of nice asymptotic behaviors of solutions of the linearized problem at the cost of the restrictive condition that the initials are small in some sense.

1.4. Boundary conditions When the gas is contained in a vessel 52, the boundary condition which describes the reflection of gas particles by the wall 852 is imposed on f. Denote the outward normal to 852 at x€i?52 by n ( ~and ) set, (1.4.1)

s*= {(x, E) € a52 x W” I n(x).E2O} ,

(1.4.2)

r’f=fIs.

,

(same signs). are trace operators to S” whose existence will be established in S 5.1. Since particles at (x, E) E S+ (resp. S - ) are particles incident to (resp. reflected by) the wall as, r i f represent the corresponding densities. Therefore, the boundary condition must take the form, 7-f=M r +f

(1.4.3)

t

where M is the boundary operator determined by the reflection law. Classical examples are; (1.4.4) ( i ) (ii) (iii) (iv)

(perfect absorption) M=O, (specular reflection) Mr+f=f(t, x, E-2(n(x)-E)n(x)), (reverse reflection) My+f=f(t, x, -[), (diffuse reflection)

Mr+f=g,(E)

n(z).e’>o

n(x)*E’f(t, x, E’)dE’

sw(E)=p,(2aTw)-“/2exp ( - lElz/(2Tw)), where T, is the temperature of the wall.

9

~ , = ( T ~ / ( 2 7 d ) ”, ~

M may be any (convex) linear com-

S. UKAI

42

bination of (i)-(iv). Also, (ii), (iii) may be generalized as ([24]), ( 1.4.5)

My+f=f(r, x , m ( x , E ) ) ,

m(x, E) E S-

,

and (iv) as

dux being the measure on 352, and so on. With these boundary conditions, we are given initial boundary value problems for ( l . l . l ) ,the subject of SS 5-6.

1.5. Large initial data In solving the Cauchy and initial boundary value problems globally in t , a n unpleasant restriction is required for initial data f , . Whereas the local existence can be established for arbitrarily large f , (SS 2,3,5), the global one requires the smallness condition on f,, as stated at the end of S 1.3. More precisely, f , should be near a n equilibrium described by a n absolute Maxwellian go (§§ 4-6). Of course, this is a technical restriction specific to our method of $ 4 . 1 , but at present, n o other methods are available. Thus, the global existence for f , far from go is a big open problem. For such f,, it is expected that the shock appears after a finite time, however smooth f , may be. Notice that all known solutions, local or global, are classical (smooth) ones, and so d o not present the shock. Therefore, the shock solution, if any, should be a weak one, but up to the present no definitions of weak solutions have been found which promise the global existence. Recently, Arkeryd [4] introduced a new concept to this problem, constructing global ‘non-standard’ solutions for arbitrary L1 initials f,. More precisely, his solutions solve (1.1.1) in the sense of the non-standard analysis that the variable t is considered in *R (the set of non-standard real numbers) and x , E in *Rn while the Lebesgue integral $ dE’do over R ” x Sn-’ is replaced by the Loeb integral $ L(*dE’ *do) over *Rnx*Sn-l where L(*d[’*do) is the Loeb measure induced from the Lebesgue measure dt’do. Although this result is remarkable, its conversion to the standard result seems difficult. For the spatially homogeneous case which will not be discussed in this note, global solutions exist for any f , , with and without cutoff, see [2, 31.

- -.

.-

Fluid dynamical limits The motion of a gas can be described not only by the Boltzmann equation but also fluid dynamical equations such as the Euler and Navier-Stokes equations. The former gives a microscopic description in the sense that the gas

1.6.

Solutions of the Boltzmann Equation

43

is looked at as a many-body system, while the latter gives a macroscopic one because the gas is regarded as a continuum like a fluid. Evidently, the former provides more precise information, but if the number N of gas particles is very large as is usually the case (N=1OZs,Avogadro's number), then the gas will behave much like a fluid, and at the limit N=oo both descriptions are expected to coincide. The Hilbert and Chapman-Enskog expansions reveal this. In 1912 the celebrated Hilbert, the first mathematician to attack the Boltzmann equation as well as many other big problems, gave a solution of (1.1.1) in a formal power series,

(1.6.1)

...

f=f'O'+Ef(1)+E2f'2)+

of € E N - ' (the mean free path). To determine the coefficients, assume a ( x , f ) = O and write ( 1 . 1 . 1 ) as 1

Df=, Q[fl ,

(1.6.2)

D=ac+E*V,,

which is obtained dividing (1.1.1) by N and writing f/N as f (the normalized probability density). Substituting (1.6.1) and equating the coefficients, he then obtains,

(1.6.3)

Q[f(o'l=o,

(1.6.4)

Df")=2 Q[f (O), f'"] ,

( 1.6.5)

D f C r= )

C Q [ f ( I ) f, ' k ' l - j ) ] , k+l

J=O

k> 1

.

By Theorem 1.1.1 (iii), f t o ) = f o , the Maxwellian ( l . l . l l ) ,with fluid dynamical quantities p, v, T still undetermined. Once they are known, (1.6.4) is a linear integral equation for f"). Applying his theory of integral equations, and by Theorem 1.1.1 (i), Hilbert has the solvability conditions,

(1.6.6)

E=O,

O
.

The integrals can be calculated explicitly, giving a system of nonlinear hyperbolic conservation laws for p, v, T ;

pt+V.(pv)=O, (1.6.7)

+v - ( p 'vv)+v p =0 ,

(pv)t

(pE),fV.(pEv+pv)=O ,

tvv = ( v,vr) , 1 E=e+- 1vI2, 2

supplemented by the equation of state of the ideal gas,

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44

where V=V,, p is the pressure and e the internal energy per unit mass. (1.6.7) is the compressible Euler equation. Similarly, (1.6.5) give inhomogeneous Euler equations which determine higher terms f(". The Chapman-Enskog expansion is not a n asymptotic expansion like (1.6.1); the procedure is complicated and is referred to [ l l ] . The first term of this expansion is again a local Maxwellian, but its p, u, T should solve the compressible Navier-Stokes equation, not the Euler equation (1.6.7). Although the relations thus revealed between the kinetic theory of gases and the fluid dynamics confirm a physical validity of the Boltzmann equation, these expansions are not yet justified fully at rigorous mathematical levels. In particular, the convergence is not known, so they d o not serve as the existence theorem. Only partial justifications have been given [7, 10, 11, 16, 19, 26, 31, 431. In S 7, we will discuss (1.6.1) to the first term. 2.

Local Solutions, I.

The case of cutoff potentials

We begin by solving the Cauchy problem for (1.1.1) locally in t , with a cutoff assumption general enough to include (1.1.4) and (1.1.5) cut by (1.2.1). This is the simplest case in which we meet the unboundedness of Q ($2.1). The main idea to control it is to set the Cauchy problem in a scale of Banach spaces, in order that the integration with respect to t can play a role of a smoothing operator (S 2.3). The same idea will be used in later sections where the problem of unboundedness arises in different situations. Two types of solutions are presented, one in a different function class from that of initials f, ($2.3) and the other in the same class but with further restrictions on q andf, (S 2.4). 2.1.

An estimate for Q We assume that q(v, 0) is measurable in (0, m) x [0,IC]and that,

with a constant q,>_O. For example, this is satisfied if (2.1.2)

M v , ~ ) I I ~ , ( ~ + ~ - ~ + z I ~ ) I c o s ~,~ - ~ O' < p < n ,

O17<2,

f
.

Since 7'>1 in (1.1.5), this may be taken as the weakest cutoff assumption. Here, q is not assumed nonnegative. Let a , /3 E R and set,

, =(I+ I W ) ~,' ~ - c , p = I f = f ( E ) I P a , a f € L"(R")I , Ilf lIa,p=s;P P,,p(E) I f ( E ) I . ,o,,a(t)=e-"(E)2J

Solutions of the Boltzmann Equation

45

The main results of this section is. Theorem 2.1.1. For any a>O, there is a constant C=C,>O, decreasing in a , and 3 0 3 (a-0), -0 (a+m), such that f o r any p > O and f , g € L;,p,

II Qrf, gl l l , , ~ - ~ ~ ~ I I f I I ~ ~ p I I ~ I I ~ , ~

(2.1.3) Proof.

Define the operators Q j , j = 1 , 2 by

(2.1.4) and similarly for Q2 with f ( v ) g ( f )replaced by f(E)g(E’). Then, (2.1.5)

Qrf,

1 gl=l

{Ql[f,

gl+Qi[g, f l - Q z [ . f , ~ l - Q z [ g , f l I

9

so that it suffices to prove (2.1.3) for each Q j . By (1.1.3) or (1.1.12) f o r j = n f 1, < ~ > 2 + < l j ’ > 2 = < E > 2 + < [ ’ > 2 holds, which gives, (2.1.6) for p 2 0 .

Pa,&$Pa,B(?’)

>-Pa,p(E)Pa,o(E’)

9

Hence we have,

I Q i [ f ,gl(E)I ~ ~ ~ t ~ ( E ) ~ l l f l l a . ~ l l ~ l l a , ~ 7

Then, Ir, by (2.1.1) and by (2.1.7)

1

R”

1E-E’[le“(E’’2dE’IC,1

,

which is valid for R > -n. This proves the case j = 1 , and similarly forj-2, though more straight-forward. Remark 2.1.2. Estimates similar to (2.1.3) are possible in the spaces LP,,,, because the transformation ( E , E’)+(?, 7’) by (1.1.3) has the unit Jacobian, [13].

p<

00,

The estimate (2.1.3) says that if r>O in (2.1.1), Q i s unbounded in L:,,. This can be improved partially, under a more restrictive condition. Theorem 2.1.3. a>Oandp>l,

(2.1.8)

Assume (2.1.2) with an additional condition r’=O.

IlQi[f,

~ l I l ~ , p + ~ - r I ~ ~ l l f I l ~ , -p l l ~ l l ~ , p

Then, i f

S. UKAI

46

For the proof, it suffices to proceed as before, using

1

(2.1.9)

Sn-l

-F-fldo< C-fl-l ,

instead of (2.1.6), which is valid for ,8>1 (see the appendix of [19] where f = - 1 is assumed but the result holds for ,821). Note that (2.1.8) is not true for Q z . The linear Cauchy problem The linear part of (1.1.1) can be treated fairly simply in the case of the Cauchy problem. Let us solve the problem,

2.2.

(2.2.1) where k = k ( t , x, E ) and f o = f o ( x ,E ) are given functions. According to the theory of first order partial differential equations, we shall first solve the characteristic equation dX/dt= B ,

(2.2.2)

dE/dt=a(X , 3) ,

under the initial condition (X,E ) = ( x , 5) at t = O . If a ( x , E ) € C1,a unique solution exists at least locally in t near t=O, denoted by (2.2.3)

( X , B)= S ,(x, E ) = ( W t , x, El, &t, x, 5 ) )

1

and (2.2.1) is solved by Jo This is a classical solution if f o and k are in C', and exists as long as (2.2.3) exists. A simple sufficient condition, convenient for a later purpose, for (2.2.3) to exist globally in t for all initials (x, E ) is (2.2.5)

a(x, E)

has the form

a(x, E)=-V,b(x)+a,(x,

E) ,

where ( i ) b ( x )E C2(Rn)is a scalar function, and a , ( x , E ) € C'(R"x R") is a n nvector function, (ii) b(x)>l, (iii) E . a , ( x , E ) = V e . a l ( x , E ) = O . Here, b ( x ) is the potential and (ii) implies merely that it is lower bounded, because the potential is unique only up to additive constants. An example statisfying (iii) is the magnetic field for which a , = ~ x B ( x ()n = 3 ) .

47

Solutions of the Boltzmann Equation

Theorem 2.2.1. Suppose (2.2.5) be fulfilled. Then the solution (2.2.3) exists for all t E R , for any initial (x, 6). Further, for each fixed t , the mapping S, is (i) C1-diffeomorphicand (ii) measure preserving, on R" x R". Proof. We introduce the one particle energy,

and note that (2.2.2) is Newton's equation of motion. conserved, that is,

In view of (2.2.5), E is

E(S,(X,6))= H x , E )

(2.2.7)

holds, as long as (2.2.3) exists. But, since b ( x ) > l , (2.2.7) provides a n a priori estimate for X and 9,and thereby assures the global existence. Now (i) is obvious and (ii) comes from the fact that div,,€ ( E , a , ( x , E ) ) = O . If a ( x , E ) = O , then S,(x, E)=(x+tE, E ) .

Remark 2.2.2.

Define the operator U(t), t E R , by (2.2.8)

(U(tlfo)(x,E ) =fo(X( - t, x , E l , b(- t, x, E ) ) =fo(S-t(x, E ) ) .

Then (2.2.4) can be rewritten as f ( t ) = U ( t y o + ] ' U(t-s)k(s)ds .

(2.2.9)

0

We shall discuss U ( t ) in function spaces suitable for the nonlinear problem. Let E ( x , f ) be as in (2.2.6) and set (2.2.10)

pu,&x, f)=eaE(">E)E(x, E)b'2

.

Define the space (2.2.11)

X,.,,={f=f(x,

E ) I p a , p f E W R n xR")}

Ilfll..u.a=IIPu,pfll~~

9

.

In view of Theorem 2.2.1 and (2.2.7), we readily have the

Theorem 2.2.3.

For any p E [l, 001 and

IIU(t)fllp,u,p=

llfllP,u,p

7

(Y,

3/ E R , it holds that

t f sR

t

f E

x:.p .

Obviously U ( t ) has the group property U ( t + s ) = U ( t ) U ( s ) ,s, t E R . As for the continuity in t , we can prove the

S . UKAI

48

Theorem 2.2.4. Let p~ [l, 00) and a , P E R . X,”,a. Its generator, denoted by A , is given by

Then U(t) is a C,-group on

Af= -(E.Vz+4x, E ) . V e ) f , whose domain is maximal in X{,,+ sense of distribution.

Here the derivatives are to be taken in the

This theorem is not valid for p = m , but it is if U ( t ) is restricted in a suitable subset of X : , p . For example, we have, Theorem 2.2.5. For p = m , the previous theorem remains true i f X:,, is replaced by

z:,,={fE x,,n where x R = x R ( x , E ) is such that

x Rn)I l l ~ ~ f l l - , ~ , ~ - -( -R>-Om ) } ,

xR=

1 if 1x1

+ IEi > R , and =O

otherwise.

Another example will be obtained by @-operation of [21J, noting that the dual of X2a,-B is X ; , p and the dual of U ( - t ) is U(t).

2.3.

Construction of local solutions We are now in a position to solve the Cauchy problem for (1.1.1);

(2.3.1) Iff is a solution, it must solve the integral equation (2.3.2) which comes from (2.2.9). We shall solve this in the space X:,B of (2.2.10), but need to vary the index LY with time t to control the unboundedness of Q which is seen in the Lemma 2.3.1.

Theorem 2.1.1 remains true i f L:,p is replaced by X : , p

For the proof, it suffices to repeat the proof of Theorem 2.1.1 with the assumption (2.2.5) (ii) taken into account. In spite of Remark 2.1.2, this lemma fails in X,”,, if p < 00. Define the bilinear operator (2.3.3) and the nonlinear operator

Solutions of the Boltzmann Equation

49

Then (2.3.2) can be written as f = N [ f ] , so we shall find a fixed point of N . The simplest but powerful tool for this is the contraction mapping principle. However, if r > O , it does not work if a is kept fixed, because due to the above lemma, N loses the weight ErIz a t each step of iteration. O n the other hand, if a is varied with I , then it does work well since the integration in (2.3.3) with respect to t turns to a smoothing operator. T o be precise, introduce the space (2.3.5)

x=x,,.,p([-T, TI)=If(t, x, 5 ) I e - c l c l E f ( t ) ~ L " ( [ -TI; -,

II If1II = I IIf I II x =

ik!

Ile-"l"'"f(t)

ll-,a.j=

Ilf(t)Il-.a--.ltl ItlST SUP

X , j ) }

,

.j

Lemma 2.3.2. Under the assumption (2.1.1) and f o r any a>O and B 2 0 , there is a constant Co>O such that for any K > O and with T = a / ( 2 r ) , we have, for all f,9€X, (2.3.6)

IIINo[f, sllll

co.-'lllflll

lllglll

.

Proof. Put w(t, s)= IIU(t-s)Q[f(s), g(s)]ll,,a-.lsl,~-r.Using Theorem 2.2.3 and then Lemma 2.3.1, we get w(t, ~ ~ ~ ~ u - K ~ lllglll s ~ l~l Cl fal/ l~ll l l JIllslll, lll for Is1 I T . Hence,

where

The last integral is majorized by

K - ~ ,

so (2.3.5) follows.

The fact that the last integral is bounded although Er/2+cc (IEl+oo)

for

y>O implies that the integration in t plays a role of a smoothing operator.

Now N is a bounded operator.

Further, it is a contraction on the space

Va={f€

x I lllflllla)

if a>O is chosen appropriately. Actually, we first choose d= 1- 4 C 0 K - ~ l l f o l l ~ , ~ , > s 0,

and then set a= (1- 42)/(2C,,K-') ,

which is the smaller root of the quadratic equation C o ~ - ' a ~ - a +Ilfollm.a,e=O

K

so large that

S. UKAI

50

We easily see that a<2/lfollm,m,j.By virtue of Theorem 2.2.3 and Lemma 2.3.2, we have, for any f , g c V,, (2.3.7)

III"fllI12 llfoll-,a,,9+ CoK-'a2=a II I"f1 -"sllII = IllNo[f+s, f-sll II 5 2CoK-'alIIS-Slll 9

7

which shows, since 2COr-'a=1--z/T< 1 , that N is a contraction on V,. Thus N has a unique fixed point f E V,. N o other fixed points exist in any V,, b#a. For, if g € V, be such a fixed point, we have, similar to (2.3.7),

I II f -

gl I I = I I I"f1

so we conclude Illf-glll=O, proved the

- "sll

choosing

K

II

CoK-Ya+b)l

I If-sll I

9

large enough. Summarizing, we have

Theorem 2.3.3. Suppose (2.1.1) and (2.2.5) be fulfilled. Then, for any f o € X z , p , n>O, @20,there are constants K , T>O such that (2.3.2) has a unique solution f € -LK,,d[--T,TI) satisfying Ilf(f)llm.n-I.,l~.B~211follmra,B

Remark 2.3.4. to (2.3.1).

If, in addition, a,,,fo

-

c X ; , B , then f is a classical solution

Remark 2.3.5. Thus the Boltzmann equation can be solved backwards as well as forwards, in t. However, f ( t ) is not in the same function class as f,, and in a worse class as It1 increases. This is the same situation as in the Cauchy-Kowalewski theorem. But, f happens to be within the same class as f , (see Theorem 2.4.3 below). 2.4.

Nonnegativity of solutions In this section, we always assume q, f o 2 0 . We shall prove thatf(t)>O for t 2 0 , a physically plausible property required for the Boltzmann equation because f is the density and q is the cross section. First, we write Q2 of (2.1.4) as

and (2.3.1) as

Given a function g, we define the operator V ( t ,s; g) by

Solutions of the Boltzmann Equation

51

Obviously, f= V(r, s; g)fosolves a linearized equation of (2.4.2), and similarly as before, we can write (2.4.2) as, instead of (2.3.2),

Recall (2.3.5) and set,

X + = X L , = l f € &,,.B([O,

TI) l f ( 4 2 0 ) .

Here, t is in [0, T I , not in [ - T , T I , and similarly for Then, @>O and hence,

111.111.

Suppose f,g €

X+.

for O < s < t < T = a / ( 2 r ) . The latter is proved noting that ~ [ f satisfies ] the linear version of the estimate (2.1.3) and proceeding as in the proof of Lemma 2.3.2. Repeat the proof of (2.3.7), using (2.4.4), to see that a slightly modified version of (2.3.7) holds for N , and for f, g € VanX+. Furthermore, (2.4.5)

N,[f]>O

whenever f 2 0 .

Consequently, N , is contractive on V a nX'. It is clear that the fixed point of N , coincides with that of N . Thus, we proved the Theorem 2.4.1. Let f = f ( t ) , t~ [ - T , TI be the solution given in Theorem 2.3.3. Thenf(t)>Ofor t € [ O ,TI i f q , f o 2 0 . Remark 2.4.2. This is not the case for t € [ - T , 0). For, then (2.4.4) no longer holds and N , is not contractive. Now, we can have a solution belonging to the same class as f,. Theorem 2.4.3. Let f be that in Theorem 2.3.3. Under the situation of Theorem 2.1.3, assume further that y < 1 and q, f o > O . Then, f € L"([O, T I ] ;X;,& with some TI < T . Proof. If y<1, then (2.1.8) shows that Q z is bounded, so by the help of (2.4.4), we can deduce from (2.4.3) that

II If1II'= I IINl[fllI / I < Ilf o II+ CT'II If1I

9

S. UKAI

52

II-II

where

and

111-111’

denote the norms of X ; , p and L”([O, TI]; X ; , , ) =

X,,,,,([O, T’])respectively. If T’ is small, we have lllflll’5~llfoll.

Remark 2.4.4. Theorem 2.4.1 is stated in [S, 24, 361 and Theorem 2.4.3 in [l], all as existence theorems. 3.

Local Solutions, 11. The case of non-cutoff potentials

Let us turn to the non-cutoff case, where Q behaves like a pseudodifferential operator: It is well-defined only for smooth functions and induces the loss of smoothness (see S 3.1). Following [40], we show the local existence in the scale of spaces consisting of functions which are analytic in x and of Gevrey class in E. Up to the present, this scale is the only one in which the Cauchy problem can be solved. The existence of less smooth solutions is a n open big problem, a contrast to the spatially homogeneous case [3]. Also, the convergence of the cutoff (1.2.1) will be shown. 3.1.

The operator Q in Gevrey class Throughout this section we assume that q(v, 19) is measurable and satisfies /q(u, ell I q , ( l + v - r + v r ) l

(3.1.1)

cos 01-r‘ ,

with

This involves the inverse power law potential (1.1.5) for s > 2 . Although we deal with smooth functions only, q need not be smooth. Since 7‘2 1, Q cannot be evaluated termwise like (2.1.5)because the integrals in o is divergent, but it can be if suitably combined. To see this, define, Q,,j = 1 , 2, by

QJf,gl=

(3.1.2)

1

RnxSn-1

q(u, e){f(s)-f(E)}s(v’)de’do

and similarly for Q,with {f(v)--f(E)}g(v’) replaced by {f(q’)--f(E’)}g(E).This definition is different from that of Q, in (2.1.4), but (2.1.5) remains to hold. Since 17°C’ at 8=lr/2 by (1.1.3), the quantities in the above vanish and may cancel out the singularity of q, iff, g are smooth enough. T o illustrate this, let p,=exp (az)and put u=p,f, w=p,g. From (1.1.3) we see that

v=e,

Qi

{ . a * }

[f,gl(E)=p,’(E)

Let llull=sup lu(E)I and let lula=sup Iu(p)-u(E)l/lv-Elg be the Holder quotient. We have

Solutions of the Boltzmann Equation

where

~=j

R n xS"-1

53

m, s)ip,-i(E')is-Eidd~'do .

From (1.1.3), 1~-El=IvcosOl, so we get by (3.1.1),

Z Ic

1

R"

(1 +v-'+Z)r)Dap,-'(E')dE'

1:

lcos 01-r'+dd0

The integral in 0 converges if --y'+6> - 1, whereas that in 5' can be evaluated using (2.1.9). The result is Zr+d. QZcan be estimated similarly. Thus Q,, and hence Q , are well-defined if 6 E (7'- 1, 11. For our purpose, however, this estimate is not convenient, and we shall deduce a n estimate in Gevrey classes re$;, defined as follows. Let pa be as above and define the norm

Definition 3.1.1. Let a , p, ,020 and f = f ( x , E ) E Cm(RnxR") satisfying

X,

v2l.

rg;:b is the set of functions

(3.1.4) where the sum is taken for all k, 1 E N". Remark 3.1.2. ( i ) Endowed with the norm (3.1.4), algebra. This is seen by Leibniz' rule and by

(3.1.5)

k!m!/(k+m)!
is a Banach

for all k, r n E N n ,

, is analytic in x in the strip R"+i{y€R* I l y l < p } , ( i i ) If ~ = 1 fE7Ei$ and similarly in E if Y = 1. (iii) If a'
(3.1.6)

ll~tflla.P,c,

P-0, Y

and similarly for 8, in place of 3,.

(3.1.7)

IC " p Y - l ~ - YIla. l l fp , r , p , u

9

This follows from (3.1.4) and

s,uy (l-ulp)*((s+l)"
where C U = ~ ~ p 1e-$(s+ 2 : a 1)".

S . UKAI

54

Theorem 3.1.3. Suppose (3.1.1) andlet a>O, 6 € ( f - l , 11 and ~ 2 1 . Then there is a constant C 2 0 and for any p, p>O, r > l , /3€(0, a ) , u c ( 0 , p ) and f , g € r:$‘,$, it holds that

IIQrf,

(3.1.8) where

II.II=ll.Ila,p,r,P,”

a)llfll llgll

~ l l I n - ~ . p . ~ , p - a . P, ~ ~ ~ ~ ~ ( P ~9

and a,(

p, a) =/3-

(7

+ +

+al 2{ /3F2 ( 1 p)pyd-lu-yd} .

Moreover, C=C, is decreasing in a , and C,+O (a--,co), -00

(a-0).

This is a modified version of the previous estimate given in terms of the Holder norm. For details of the proof, see [40]. Comparing with (3.1.6), we see that Q is a pseudo-differential operator in E of order 6 € ( f - 1, 11. Construction of solutions In solving the Cauchy problem (2.3.1) under the assumption (3.1.1), we shall restrict ourselves in the case a(x, E ) = O : Thus,

3.2.

fc+E*vzf=Q[fl f Ic=o=fo . 9

(3.2.1)

We rewrite this in the form of the integral equation (2.3.2), where, according to Remark 2.2.2, U ( t )is given by U(t)fo=fo(X--rE, E )

(3.2.2)

*

With this, we recall the definition of the operators N o and N of (2.3.3) and (2.3.4). In order to find a fixed point of N , we shall first study U ( t ) of (3.2.2) in rg;:p. In spite of its simple appearance, and its nice behavior in L p spaces (Theorem 2.2.3), it induces the loss of smoothness in x in our space, as is shown in the Proposition 3.2.1. we have, for I t ] I T ,

Let f o € rg;J:pand assume

V 2 K 2 1 .

II U ( M o l l a , p , . r , p , v i I l f 0 l l a . p . r . p . v

p’ = { p - ( pI t I)l’E}K

Then, with T = p / p ,

?

.

For the proof, see [40]. Similarly as in S 2 , we must set the Cauchy problem in a scale of Banach spaces. More precisely, we define the space X = X : y B . p . l , p . a by X 3 f - f = f ( t , x , E ) € C o ( [ - T , T ] C”(RnxRn)), ;

lllflll

Ilf(f)lla-~ltl.,~-llrl.r.P-~,rl.”<

O0

-

55

Solutions of the Boltzmann Equation

In the sequel, T is fixed as

.

T=min (a/2/3, p/21, p/2n)

The essential point is, just as in Lemma 2.3.2, that the integration in t plays a role of a smoothing operator and N o becomes a bounded operator in X . To see this, notice from the last assumption of (3.1.1) that (3-p-f)/2(+ 1) > 1. Choose Y such that 1 < Y < (3-r- f)/2(7’- 1)

(3.2.3) Then, r’-1<(2-7)/(1+2u),

so we can find 6 € ( f - l ,

J- 1 < 0“ < (2- r)/(1 +2Y)

(3.2.4) Lemma 3.2.2. above. Then.

. 11 satisfying

.

Let a, @, p, 1, p, o>O with 1>p.

Let ~ = and l let 8,

Y

be as

Since ~ = and l 1>p, and since O s s i t or t i s I O , it holds that p”ip-1ltlf Hence, on account of Remark 3.1.2 (iii) and then by Theorem 3.1.3. we see rhat

plt-sl
@(s, p*”)I@(s,

p-~lsl)iC,-gltl~o(pl~--sl, p-olsl,

The last two norms are majorized by choice of T , Ca-altlI C,,, and ao(pIt -4,p -Alsl,

Illflll,

~ l ~ - ~ l ) l l f ( ~ ) l l J l l.s ( ~ ) l l s

lllglll for Isl
01 t -sI) I(1 +2( 1 +p)pU8-l)al(p, a)aAIt- sl) ,

where a, is as above and a,(T)= T -

(7+28)/2fT-

(7+d+Zd/Z

.

In view of (3.2.4), (r+26)/2<(y+d+2~6)/2<1, so a, is integrable;

56

S. UKAI

Substituting these estimates into (3.2.6), we conclude the lemma. Apart from the definition of norms, the estimate (3.2.5) differs from (2.3.6) only by the factor l i - I replaced by a,(P, a). Further, a,(/$ o)+O as j?, a+w. Therefore, the proof of (2.3.7) applies to our N , and N becomes contractive for large /3, a. Thus, we get, Theorem 3.2.3. Let a, p, p > O and let Y be as in (3.2.3). Then, for every foE r2G;p,there exist constants T , /3, 1, a > 0 such that (3.2.1) has a unique classical solution f =f ( t )on [ - T , T I , satisfying ( i ) f J2,;,/fJ,P,u? ( i i ) lllf 11112llfOlln.p,l.p.Y. (iii) f ( t ) is analytic in x in the strip R " + i { y € R " I lyl
for O < s < t

and t < s < O .

UI

tl)l/"If -sl1/=}" Ip(s) ,

Rewrite this as

p(s)1/r- p( t)"'

2 (p- a1t I ) 1 / ~ 1I-

SI 11.

,

which shows that p(t)"' is monotone for t 2 0 and for t10, differentiable if li > 1, a contradiction.

but also nowhere

Convergence of cutoff approximation Suppose q(v, 8) satisfy (3.1.1) and consider its angular cutoff qe(v,8) given by (1.2.1). Obviously, qe also fulfill (3.1.1), so Theorem 3.2.3 applies. Furthermore, the constants T , /3, 1, a given there d o not change. Denote the solution thus obtained for qf by f ' . The original solution (i.e. for q) will always be denoted by f. 3.3.

Theorem 3.3.1. Under the situation of Theorem 3.2.3, let f and above, for the same initial f,. Then,

f E

be as

111 f-fq[I
E,

and 6 is as in (3.2.4).

For the proof, see [40]. This theorem establishes the convergence of the cutoff approximation in spaces of Gevrey class. Any similar results are not known for less smooth solutions.

57

Solutions of the Boltzmann Equation

As a simple application of the above theorem, we can prove the nonnegativity off of Theorem 3.2.3. Theorem 3.3.2.

Suppose q(v, B ) > O , in addition to (3.1.1).

Then, f =f (t )20

for t € [ O , TI i f f o 2 0 .

Proof. qf satisfies (2.1.1) so that, according to Theorem 2.4.1, f S ( t ) 2 O for t € [ O , T I ] . In general T , may tend to 0 with E , but for our smooth f,, f‘ satisfies Theorem 3.2.3 (ii) independently of E . Hence we can take T,=T, and going to the limit E+O in the previous theorem, we are done. For the non-cutoff case, we cannot write (3.2.1) in the form of the integral equation (2.4.3) which makes use of the nice splitting of Q . Therefore, without appealing to f‘, a direct proof of the above theorem seems difficult. 4.

Global Solutions for the Cauchy Problem

We now discuss the global existence for the Cauchy problem to (1.1.1). The restrictions are: potentials are angular cutoff, the external force a ( x , E) is absent and initial data are ‘small’. Otherwise, the global existence is still an open question. The smallness of initial data is a condition specific to our method of constructing global solutions described in S 4.1, which requires nice decay estimates as t+m of solutions of the linearized problem. A method of obtaining such estimates is given in S 4.2 and is used in S 4.4 assuming the angular cutoff hard potential and a ( x , E ) = O . Two types of global solutions are presented: one is near non-zero Maxwellian and one near zero. The latter is much simpler to construct but the method employed seems applicable only for the Cauchy problem. The scheme Here we present a method for constructing global solutions. Consider the following equation of evolution in a Banach space X . 4.1.

*=Bu(t)+T[u(t)]

(4.1.1)

dt

where B is a linear operator in X while (4.1.2)

(i)

,

t 2 0 ; u(O)=u, ,

r is nonlinear.

Suppose,

B generates a C,-semigroup erB on X which has the decay estimate, IletBu0II
t20 ,

with constants C,20 and a>O independent of t , u,. ( i i ) r[O]=O and, with some constants a,, C,, r 2 0 ,

Ilrtul -rtv1 II ICI( 1. + Ilv II) r Ilu - II 2,

holds for all u, v € X with norms bounded by a,.

7

S. UKAI

58

Of course II. I I denotes the norm of X . Note that a reptesents the decay rate of solutions of the linearized problem to (4.1.1) and y + l the degree of the nonlinearity of r. Write (4.1.1) as the integral equation,

s:

(4.1.3)

u(t)=etBuo+ e ( t - x ) B[&)Ids f

.

Theorem 4.1.1. Suppose, in addition, a > ( y + l ) - I . Then, there are constants a,, a, > O such that for any uoE X with j[uo[l
Ilu(t)II< d + t ) - "

.

Proof. Set III~III=S~Ptto (1+t)=llu(t)Il and V,=Iu€Co([O, m); X ) 1 I l l u l l l ~ 4 . Define the operator N by writing the right-hand side of (4.1.3) as N [ u ] ( t ) . We shall show that N is a contraction on V , with a suitable choice of a>O. For this, note that if a > (y+ l)-l,

5:

,

(l+t-s)-"(l+s)-u(rtl)dsiC,(l+r)-"

t20

,

holds, which, together with (4.1.2), yields, for all u, v € V,, a l a , , (4.1.4)

II]"I I II 5 coII uo I1+ C,Czar + I III"u1 -"vll I I I ClCz(2a)rII Iu- 41I 9

This implies that if a, a , are chosen small enough and if lluoll
An asymptotic behavior of a semigroup In this section H denotes a Hilbert space with the inner product ( , ) and norm [I 11, and A and K are linear operators in H subject to the condition, (4.2.1) ( i ) A generates a C,-semigroup etA with I[etAll
72ptd

llKR(y+ir, A)II-O

(lrlbm) ,

where 11 I( denotes the operator norm and R ( I , A)=(RZ-A)-' is the resolvent of A which exists for Re I > & due to (i) ([25]). Denote the domain of A by D ( A ) , and the resolvent set and spectrum of A by p ( A ) and u ( A ) , respectively. The set of discrete eigenvalues of A with finite multiplicity is denoted by ud(A) and the essential spectrum ([25, p. 2431) by a,(B). Finally, C,(p)= (A E C I Re ;I2/3}.

Solutions of the Boltzmann Equation

59

Define the operator B by D(B)=D(A).

B=A+K,

By (i) and (ii) in the above, B is a semigroup generator. Our aim is to deduce a n asymptotic behavior of e t B . To this end, we first discuss o(B).

(iii).

Theorem 4.2.1. ( i ) a ( B ) c C - ( p +llKIl), p(B)3C+(p+IlKll). (ii) a,(~)=o,(~)co(~)cC-(p). (iii) a(B)n C + ( P + S ) is in o,(B) and is a finite set, where 6 is that of (4.2.1)

Proof. (i) and (ii) follow from [25, Theorems IX. 2.1 and IV. 5.351 respectively. Set G(R)=KR(R, A ) and write the second resolvent equation, R(R, B)=R(R, A ) + R ( I , B)G(I) .

(4.2.2) Solve this as

R ( I , B)=R(R, A)(Z-G(R))-' ,

(4.2.3) and conclude that

(4.2.4)

R € p(A) and I-G(R) is boundedly invertible

3

R E p(B) .

Now, by [25, Theorem VII. 1.91, a(B)n C+(p)is contained in a,(B), with possible accumulation points only on the boundary of C+(p). Put ao=p+6. In view of (4.2.1) (iii), one can find a r,>O such that IlG(R)]I<1/2 for all RELY= (?+(a,) f l {R I IIm 21 >ro}. Therefore, by the Neumann series, Z-G(R) can be boundedly invertible with (4.2.5)

ll(Z-G(I))-1/l<2,

for all R E L Y .

From this and (4.2.4), a(B) does not contain Z. Therefore, by (i) of the present theorem, a,(B)nC+(a,) is a compact set, and thereby, necessarily a finite set. To proceed further, we need the

Lemma 4.2.2. For any r > p, we have,

'j* llR(r+ir, -m

~)ull2dr<~~K(r--~)-lllull2 ,

Proof. Recall ([21]) that (4.2.6)

R(I, A ) =

'j- e-lte 0

dt,

tA

REC+(p).

~

E

.H

S . UKAI

60

s".,

Let ~ + ( be t ) such that ~ + = for 1 t 2 0 and =O for t < O .

+

Then,

R( y ir, A ) = ( 2 ~ ) - " ~ e-""{ ( 2 7 ~ ) t( " t~)e-rcecA}dt ~ . This is a Fourier transform, and since Parseval' equality is valid for Hilbert space valued functions, one gets,

Using (4.2.1) (i), we conclude the lemma. According to Theorem 4.2.1 (iii), the eigenvalues R , of B in Ct(uO)are finite in number, so they can be enumerated as

-

Re R,2Re d , 2

2 R e R,2u0

.

Denote the multiplicity, eigenprojection and eigennilpotent of R j by m,,P, and Q,,respectively. Write

P= 2 P, . 7

j=1

The main result of this section is as follows. Theorem 4.2.3.

Suppose Re R,#uo=P+6.

(ii) IletB(l-P)II
Then,

.

Proof. The inverse Laplace transform of (4.2.6) for B is, [21],

with y sufficiently large. By (4.2.2) and (4.2.3), we have R ( I , B ) = R ( I , A ) + Z(& Z(d)=R(R,A)(Z-G(d))-lKR(R, A ) . Substituting this,

(4.2.7)

1 ecBu=etAu+s-lim-U,,,(t)u , a-m 2111

U,,,(t) =

r+ia

r--ta

eltZ(R)d;i .

The main ingredient is to shift the integration path to the line ReR=u,. Since Z(R) is meromorphic in Ct(uo), and with ro of (4.2.5), we have,

61

Solutions of the Boltzmann Equation 7

Ua,r(t)=2jri C Res e2CZ(I)+Z+Ua,,,,,(t), 1=1 2 = 2 j

for all a > r o , where Res means the residue and

First, since I j € C + ( , 8 ) c p ( A )and , by [25, p. 1811, we find Res eAcZ(I) = Res e"R(I, B ) =eijc(.. 2=2j

2=2j

"'l +2 -Q$). k! tk

k=l

Second, it readily follows from (4.2.1) (i) that (4.2.8)

IIR(I,A)II
Rel>B.

Combine this with (4.2.1) (iii) and (4.2.5) to deduce that llZ(rkiu)ll+O (@-.a) uniformly for y>uo, whence IlZll-0 (a-too). Finally, since I,#u, and by [25, Theorem VII. 1.91, ~ ~ ( Z - G ( u o + i r )

-a

~ ~ ~ ( u , + i r )llR(uo-ir, ull A*)vlldr ,

where A* is adjoint to A. Lemma 4.2.2 is valid also for A*, so by Schwarz' inequality, the last integral is found to be majorized by Con(ao-,8)-111~11 11011. This implies not only that Ua,o,,(t)converges as a+oo in the weak operator topology, but also that the limit Um,,,,,(f)satisfies

II Um,oo(t) II

Czeaot,

t€R

.

Combining these estimates in (4.2.7) proves the theorem. It is clear that eCB(Z--P)=eCn+(2~i)-~U,,,,,,( t). The above theorem has been proved in [37] in a concrete context of the Boltzmann equation.

4.3.

The linearized Boltzmann operator Our aim is to find a global solution to the Cauchy problem,

(4.3.1) We seek a solution near a n absolute Maxwellian g,#O. Here, go may be any Maxwellian, but can be normalized, without loss of generality, as

62

S. UKAI

go(E)=e-lel*/*.

(4.3.2) Putf=g,+g;% (4.3.3)

Then, (4.3.1) is reduced to u,=--E*V,u+Lu+f[u] ~I,=o=~o

,

7

where (4.3.4)

Lu=2g;'/*Q[g0, g;"~] , f [ u , ~ ] = g ; l / ~ Q [ g ~gA/2~] /~u, .

I'[u]=f"u, u] ,

The linearized Boltzmann operator B is the linear operator appearing in (4.3.3);

B= -f.V,+L

(4.3.5)

.

Then, (4.3.3) takes the form (4.1.1). Therefore, if B and f fulfill the assumption (4.1.2), Theorem 4.1.1 will have given global solutions. In particular, since our r is quadratic, i.e., r=1, it will suffice to deduce the estimate (4.1.2) (i) with a > 1 / 2 . This will be done in the next section. In our situation, however, r is unbounded, so it violates (4.1.2) (ii). In § 4.5, we will give a modified version of Theorem 4.1.1 which ensures the global existence for (4.3.3). In the rest of this section, we state some fundamental properties of L and r. We assume the angular cutoff hard potential proposed by Grad in [18], which includes (1.1.4) and angular cutoff of (1.1.5) for s 2 4 . In [18], the followings are proved. First, (4.3.6)

L=--Y(E)x + K ,

where -Y(E)=Y[~~](E) (see (2.4.1)) and satisfies (4.3.7)

O < Y , I I J ( ~ ~ Y ~ < E ,> ~

with some positive numbers

yo,

IJ,and

=(l+IEIz)'/*

~ 2 0 and ,

where K(E, E') is real measurable on R* x R" satisfying (4.3.9) ( i 1 W E , E')=K(E', E ) ,

,

63

Solutions of the Boltzmann Equation

with constants k,, k,>0 independent of E. Actually (iii) holds only for n=3, and should be modified for n f 3 . Here, we assume it for all n to simplify the argument. Denote by B(X, Y ) the set of all linear bounded operators from a Banach space X into another Banach space Y, and by C ( X , Y ) its subset consisting of compact operators. We write B ( X )=B(X, X ) and similarly for C ( X ) . Set, L2=L2(Rn),

L;={u(E) 1 b(E)

€

L"(R")} .

Proposition 4.3.1. ( i ) K € B(L2)n B(L;, L;+J, p € R. ( i i ) K € B ( L 2 , L;). (iii) K C C ( L z ) . Proof. (i) and (ii) follows from (4.3.9) (ii) and (iii) respectively. Let 1 for IE[ < R and =O for 161 > R . In view of (4.3.9) (ii),

X,(E) be such that X,=

~ ~ ( z - ~ R ) K ~ ~
holds in B(LZ),while, thanks to (4.3.9) (iii), X,K is of Hilbert-Schmit type, i.e., XRK€ C ( L z ) . Then, [25, Theorem 111. 4.71 completes the proof of (iii). Let L be defined in L2 with the maximal domain D ( L ) and similarly for the multiplication operator u(E)x. Then, both are self-adjoint, with D ( L ) = D ( v ( E ) x ) . As for the spectrum, we have the Proposition 4.3.2. ( i ) o,(L)=o,(-vy(E)X)c(--, ( i i ) a ( L ) c ( - - , 01. (iii) a ( L )n (--yo, OIco,(L). (iv) 0 E u,(L) with the eigenspace spanned by

(4.3.10)

$j=go1'21zj ,

j=O, 1,

-vO].

- - -,n + I ,

where h,'s are those of Proposition 1.1.1. Proof. (i) and (iii) can be proved similarly as Theorem 4.2.1, by the aid of (4.3.7) and the previous proposition. (ii) and (iv) come from Propostion 1.1.1. Denote the eigenprojection for 0 € UJL)by P o . For f , we see

Lemma 4.3.3. ( i ) P , r [ u , vI=O. ( i i ) With some constant C20, Ilu(C)-'f[u, v]II
j20.

Proof. (i) is Proposition 1.1.1 (i), and (ii) is roughly Theorem 2.1.1, see ~91.

S. UKAI

64

Decay estimates of etB Let us confirm (4.1.2)(i) with a > 1/2 for our B of (4.3.5). However, this is possible only if the norms for etBuoand uo are chosen differently, as will be explained later. To start, we shall use the Fourier transformation in x ; 4.4.

k € R n, i = d q

s z u = f i ( k , E ) = ( ~ R ) - ~ ' ~ e-ik"2u(x, 5)dx , SR.

.

The Fourier transform of (4.3.5) is,

Introduce the auxiliary operator

A ( k ) = -(ik.E+v(E))x

(4.4.2)

i

We look at the dual variable k of x as a parameter and at the operators A ( k ) and B(k) as those in L 2 = L 2 ( R ; )with the maximal domains. Then, from (4.3.6), (4.4.3)

B(k)=A(k)+K,

D ( B ( k ) ) = D ( A ( k ) ) = { uLz € 1 k - f u ,Y(E)u€L'},

the last equality of which comes readily because ik.6 is purely imaginary while is real. The first step is to apply Theorem 4.2.3 to B ( k ) . Thus we shall establish (4.2.1) for A ( k ) and K , taking H = L 2 . We note from (4.4.2) that

v(E)

et"(k'

=,-(ik.E+Y(E))

x

t

which, together with (4.3.7), gives (4.2.1)(i) for A = A ( k ) with j3= -vo, irrespective of k . And (4.2.1) (ii) is just Proposition 4.3.1, so it remains to prove (iii). It is important to show that (iii) holds uniformly for k € R n . Put G ( I , k ) =KR(R, A ( k ) ) . Lemma 4.4.1. For any 6>0, we have, IIG(I, k)ll-O(lkl-co) uniformly for I e C + ( - v o + 6 ) , ( i i ) IIG(o+ir, k ) / l - O ( l r l + ~ ) uniforrnzy for o>-vo+6 and k , I k l I r , f o r each fixed K,,> 0 . (i)

Proof. Let X R be as in the proof of Proposition 4.3.1 and put G,= KXRR(I,A ( k ) ) . By (4.4.2), we have R ( I , A(k))=(R+ik.E+v(E))-'x, so by the aid of (4.3.9) (iii) and Schwarz' inequality,

Solutions of the Boltzmann Equation

65

Denote the last integral by Z=Z(R, k , R ) . Set R=a+ir and

.ZC,={E€ R" I 151< R , Ir+k.El I e l k l } , where e > O .

H2={E€ R" I IEI < R)\Z,,

It is easy to see that mes El CeRn-' ,

mes E2I CR" ,

holds where mes means the measure in Rn and C 2 0 is independent of k, R, E , r. Then, for o 2 - u 0 + 6 ,

I=

L'

+

sz,


/ ~ ,find, Choose ~ = ( R / l k l ) ~and llGR[l
On the other hand, put GR'=K(Z-XX,)R(R,A ( k ) ) . Then, by (4.2.8) and (4.3.9), we get I[GR'[lIC(1+R)-'6-' ,

(4.4.4)

Re12-uo+G

.

Now, choosing R=lk12/(3n+"in these two estimates, we conclude (i) of the lemma. T o prove (ii), let Irl 2 2 ~ 8 .Then, Ir+k.El> lrl/2 whenever Ikl I r , and 151I R, so that for a > -u0+S, Z
Choose here and in (4.4.4), R = I T ~ ~ ' ( ~which + ~ ) , is possible for Irl22s0R= 2rolr12/(n+2' or for I r ~ > ( 2 ~ ~ ) ~Thus, + ~ / (ii) ~ . follows. The above lemma assures (4.2.1)(iii) for our A ( k ) and K , uniformly for k € R n and with any d>O. Now that all conditions of (42.1) are confirmed, Theorems 4.2.1 and 4.2.3 apply to B(k) for all k € R n , with ,B=-u0 and for any d>O. R,, P,, Q, possibly depend on k , and will be denoted as R,(k), etc. The next step is to discuss these quantities. The following theorem is due to [16]. Set S 1 [ r ] = { k E R n I I k l S r } , SZ[r]=Rn\S1[r]. Theorem 4.4.2.

--

There exist positive numbers -,n+ 1, such that ( i ) f o r any k € Sl[~,l,

p j ( r ) € Cm([-ro, r , ] ) ,j = O ,

lie,

uo( < y o ) , and functions

u , ( ~ ( kn ) )C + ( ~ o ) = { ~ j ( k ),~ , " = R +, (dW. = P ~ ( I,~ I ) /f,(") = i&)K - p$%2+0

with py)€ R and py)> 0 , and

(I 4 3 )

( llil-0)

,

66

S . UKAI

for j = O ,

-,n+ 1,

where PSo)are orthogonal projections with

c P:"(k) ,

n+l

Po=

j=0

Po being that in Lemma 4.3.3, and P y ) ( k )e B(L2)with uniformly bounded norms, and ( i i ) for any k c S z [ l i o ]u(B(k))nC+(-a,)=Q,. , From this and Theorem 4.3.3, we conclude the

Theorem 4.4.3. There is a constant C>O such that, ( i 1 for any k E S1[liOI, &B(k)

=

n+l

2 epj ( ' x' ) t P j ( k)U+( t ,k ) ,

j=O

[lU(t,k)IIICe-"ot,

r20,

( i i ) for any ~ € S , [ K - ,IlecB(k)/IICe-"oc, ], t20. In the above, the constant C can be taken independently of k because we have established (4.2.1) (iii) uniformly in k. Now we can deduce a decay estimate of ere. Let HL(R;) be the L2-Sobolev space, and define, Hl=L2(R;; H1(R;)) ,

Lg,2=L2(R;; Lq(R;))

.

Theorem 4.4.4. For any I E R and qe [1, 21, there is a constant C>_Osuch that, ( i ) IIecBull IC(1 t)-"ll u IIH l n Lq.2, (ii) IlecB(Z-Po)uII C(1+ t ) - a - 1 ' 2 1 [ u I I H ~ n ~ q 1 2 , with a=(n/2)(l/q- 1/2).

+

Proof. (4.4.5)

Referring to (4.4.3) and by Parseval's equality,

\

llefBuII~z= (ItIkl)2111etB(x)li(k, t ) l l h R ; ) d k. Rn

Split the integral over S , [ K o ] and S2[liO], and denote the respective integrals as ZI and Z,. By virtue of Theorem 4.4.3 (ii), I z ~ C e - z u ~ t I l u, l l ~ , whereas, by (i) and Theorem 4.4.2 (i) for P,(k), I,I

CCY I ~+ e, - ~2 ~ o ~ ~,~ u ~ ~ ~ c ) 1-0

Solutions of the Boltzmann Equation

61

so by the aid of Holder's inequality and then, thanks to the well-known interpolation inequality for the Fourier transformation,

where

which is majorized by C(l+r)-(nt*)'2if m 2 0 . These estimates prove (i) of the theorem. T o prove (ii), it suffices to take account of Theorem 4.4.2 (i) and put m=q' in the above. Remark 4.4.5. mum at q= 1.

Thus, a > 1/2 if q E [l, (l/n+l/2)-l) and a=n/4 is the maxi-

The nonlinear operator the space HL,a defined by (4.4.7)

K , p 3 24

-

r of (4.3.4)is not well-defined

L,ARf ;H1(R:)) II~II1,~--suP
in H,, but it is, in

7

E

E)llHhR;)

<

*

This will be shown in the next section. Hence, the preceding decay estimate should be translated into that in this space, to solve the nonlinear problem. This can be done by Grad's nice idea [19]. Set, (4.4.8)

(1+t)"llu(t)ll1,,E.

lll~llla*l,p=~~

Theorem4.4.6. Letq€[1,2],Z€R,B20,m=O, 1,andput a=(n/2)(l/q-11/2). Then, there is a constant C 2 0 and for any u,

.

IlletB(~-~o)m~IIIat,,z*L*pl~II~llxz,B"Hz"L~'~

Proof. Define the operator A = -E-V,-v(E) Remark 2.2.2 and (2.4.2) that

x . We readily see from

S. UKAI

68 et.4u=

e-u(E)t

U(X-tE,

5) ,

and hence, by (4.3.7), that (4.4.9)

in B(H,) and B(H,,,)

IletAll
.

On the other hand, since B = A + K , etBsolves uniquely the equation (4.4.10)

e(t-a).4KeoBdS.

elB=etA+\* 0

For any Banach space X,we define the norm,

a special case of which is (4.4.8), with X=H,,,. (ii) and (4.4.9), we get readily from (4.4.10), (4.4.1 1)

llletB411a,~< c(llullX+ IlletBullla,u),

Using Proposition 4.3.1 (i), a20,

for the pairs X = H , , , , Y=H,, and X=H,,,+,, Y = H , , , , 1 2 0 . The iterative use of this with respect to then shows that this is also valid for the pair X= H,,,, Y=H,, for any 1 2 0 . Now, Theorem 4.4.4 giving a n estimate of IletBu]Ia,xz completes the proof of the theorem. In the space H,,,, the operator r happens to be unbounded, like Q in S 2.1, but can be handled similarly. For this, define the operator Y by (4.4.12)

(?@f)(t)= e(t-s)B(Z-Po)Af(s)ds , Jo

where A = v ( E ) x . By (4.3.7), v ( 5 ) may tend to 00 with [ E l , so A is a n unbounded operator. Nevertheless, Y is bounded because the integration in t plays a role of a smoothing operator, as in S 2.3. Theorem 4.4.7. LetO0andO O for n=2. Then,

IIIYflIla, t , ,

2 C(lI If1I12a'.r,a+

III Afl IL',H,"

L1.Z)

.

Proof. On account of nice behaviors of q5j of (4.3.10), we see that Pod is bounded. Define

Y,,, f = \ ' e(~--")A(Z-Po)mAf(s)ds,m=O, 1 . 0

Similarly as in the proof of Lemma 2.3.2, with E ( x , E ) replaced by can easily prove for any a 2 0 ,

v(E),

we

Solutions of the Bobzmann Equation

69

Combine this with Theorem 4.4.4 for q= 1, to verify the theorem.

4.5. The global existence

Now we are ready to solve (4.3.3). Using V, we write the corresponding integral equation (4.1.3) as u(t)= etBuo+(VA-If [u, u])(t)= N [ u ] ( t ),

(4.5.1)

T o find a fixed point of N ,

where Lemma 4.3.3 (i) was taken into account. we need the

Lemma 4.5.1. Let a 2 0 , I>n/2 and ,8> (n+y)/2 where y is that of (4.3.7). Then, there is a constant C20 such that

II IA-lf[u,

vll I Iza, 1 , a+

IIIf[u, 4 I IIza, H p L 1 . 2 5 CIII4I l a ,

1,BI

I1vI lla.1.p

*

Proof. Notice the following three facts: H l , Bis a Banach algebra for I > 4 2 , HLis continuously imbedded in Hl,a if p>n/2, and if u, u € L z in x, then uv E L'. Consequently, the lemma can be substantiated as Lemma 4.3.3 (ii). Suppose q E [ l , min (n, 2)] and set a=a'=(n/2)(l/q-l/2) so that the condition of Theorem 4.4.7 is fulfilled. Combining Theorems 4.4.6 and 4.4.7 with Lemma 4.5.1, we see that N of (4.5.1) satisfies

I II"u1 II la, 1 , p I co IIuo IIH I , n L I , 2 + c,II IUIII", I"I] - "1 Ila, p CI(lII4I Iu. B+ II I4 I 1,

1,

1.

,b

Il,U,

7

@)II Iu-

vII lu.1.p

This gives (4.1.4) ( y = l ) , so N is contractive if uo is small. Thus, we have proved the

Theorem 4.5.2. Let qc [I, min (n, 2)], Z>n/2 and ,8>(n+y)/2. are positive constants a , and a , such that for any uo with I I ~ o l l i , p + IluollLq.2Ian 9

the equation (4.5.1) has a unique global solution uEL"([O,

H1.p)

9

satisfying, with a=(n/2)(l/q- 1/2),

-

III~lllu,l,p~alI1~ollH,,B"'~,~

Then, there

S. UKAI

70

It remains to discuss the regularity of u.

(4.5.2)

Hl,p={uOE

Hl.,9;

Define

IlxR(k, 8u0111,p40

(R-too)}

7

> R , and =O otherwise, k being the dual variable where xR(k,E)= 1 for Ikl+ of x . It can be seen that etAof (4.4.9), and hence elB, are C,-semigroups on Hl,,9,although not in H l , p ,with the domains

D ( A ) = D ( B ) ~ t j , b l , p +, r , r ' = m a x ( r , 1)

(4.5.3) for any I ,

(4.5.4)

/?. Also, A-lrl-,

-1

maps H t , p x H l , pinto H l , p

if

I>n/2, 8 2 0 .

For the proof, see [42]. Therefore, it follows that N[u] is in Co([O,a);H1,& if so is u and if u , € & [ , ~ . Using (4.5.3) with 1, /3 replaced by 1-1, /3-f, we then conclude the Theorem 4.5.3. then,

Let u , uo be as in Theorem 4.4.2. Zf, in addition, U ~ E H ~ , ~ ,

UE

co([o, m ) ; H1,,dncl([o, m); Hl-l,,9-r,),

and u is a classical solution to (4.3.3), and hence so is f=gO+g;% to (4.3.1). Since H l , , 9 c H l - t , p -for e any E > O , the above two theorems assert the

Theorem 4.5.4. Let u , uo be as in Theorem 4.5.2. Then, u is a classical solution to (4.3.3) such that

u€mto, for any

E

m);

K , ~n )co([o,a);L , ~ - A

n cl(to, m ) ; H ~ + ~ , ~ - ~,, - J

> 0.

Remark 4.5.5. ( i ) Thus f(t)+go ( t - a ) at the rate t-", provided f o is close to go, a physically natural conclusion. ( i i ) If n 2 2 , one can take q=2. Then, a=O, but one can show that u(t)-FO strongly in H l , p . Remark 4.5.6. It is in the form of Theorem 4.5.4 that the results of [37,38] were stated. Theorem 4.5.2 has been proved in [28, 32, 381 independently. The periodic case The previous arguments also provide the global existence for the Cauchy problem (4.3.1) associated with the periodic (in x) boundary condition. This is not a trivial result because the special initial boundary value problem for (4.3.1), where the domain Q is a parallelepiped with the specular reflection boundary condition (1.4.4) (ii), is reduced to the periodic case, [19]. 4.6.

71

Solutions of the Boltzmann Equation

In this case it is natural to use the Fourier series, instead of the Fourier transform. Thus, for example, (4.4.5) is replaced by

+

jlecRuII,,l=kzn(l Ikl)zLllecB(k)fi(k, -1 lliz

,

where ii is the Fourier coefficient and HL=L2(R;;H L ( T ; ) ) ,T" being the ndimensional torus. In applying Theorem 4.4.3 to this, we may assume O < K,< 1, from which readily follows the Theorem 4.6.1. There is a constanr oo>O, and for any l € R , ( i 1 IleWI 5 Cllull, ( i i ) ~lecB(Z-Po)ull n/2 and 9, > (n+r)/2. Then, there are positive numbers a,, a, such that for any uo with [luollr,a
eYu(t)llL,a
Remark 4.6.3. ( i ) Theorems 4.5.3-4 are also true, cf. [34]. ( i i ) If Pouo=O,f tends to go exponentially. (iii) The condition Pouo=O does not make a restriction. For, it can be always realized by choosing the Maxwellian go, i.e., its hydrodynamical quantities p, u, T o f (1.1.11) in such a manner that (4.6.1)

< j J ,gO>=
fO>

9

i=o, .

*

- 9

n+

9

<.,

hold, where h, are as in (1.1.8), f a the given initial in (4.3.1), and is the inner product of L z ( T ; x R ; ) . (4.6.1) is equivalent to P,u,=O, and owing to the conservation laws (1.3.1), P,u(t)=O. Notice that for the pure Cauchy problem such a go cannot be found because hJ, go are constant in x, so both sides of (4.6.1) diverge. Theorem 4.6.2 is due to [37], and is the first global existence theorem to (1.1.1). a>

Soft potentials If s < 4 in (1.1.5), the angular cutoff does not lead to (4.3.7), though (4.3.9) is still valid in case s>2. Instead, v ( E ) behaves like

4.7.

72

S. UKAI

(4.7.1) See [18]. For (1.1.5), we have 6=4/s-l. Whereas v ( E ) is bounded away from 0 in (4.3.7), it is not in the present case. A direct consequence is that the spectrum u(L) changes drastically; o,(L)=o,( - v ( E ) x ) = [ - v l , 01 (cf. Proposition 4.3.2). Hence, o e ( B ( k ) ) = { l € CI Re1€[-v1,0]},

(cf. Theorem 4.2.1), and Theorem 4.4.2 is no more valid. Nevertheless, it is possible to deduce decay estimates. First, note that is simply estimated as

where a>O is arbitrary and Cn=supt,o Two different methods which prove that etBtk)inherits this type of decay have been developed so far. In [9], the cutoff operator B,(k)= -ik-E+%,LX, has been introduced, where X R = 1 for /El < R and = O otherwise. Since X,v(E) behaves like (4.3.7), Theorem 4.4.3 holds, of course with modification. Another method, given in [42], is to extract pseudoeigenvalues of B ( k ) imbedded in u , ( B ( k ) ) which behave like p,(lkl) of Theorem 4.4.2. Again, a modified version of Theorem 4.4.3 1s obtained. The remaining step which leads to the global existence is the same. The one and two dimensional cases The estimates (4.3.7), (4.3.9) (and (4.7.1)), upon which the previous results all rely, are valid only for the case of the space dimension n 2 2 . On the other hand, the physically reasonable Boltzmann equation for n = 1 , 2 is not (1.1.1) itself, but is the one reduced from the case n=3 assumingf(t, x, E ) is constant in the variables x 2 , x3 or x , and keeping the variables E always 3-dimensional. Then, the Cauchy problem (4.3.1) is rewritten with the term E-V, replaced by C,"., E,a/dx,, where x € Rn(n= 1,2) and E E R3. The existence theorems 4.5.2-4 and 4.6.2 apply because the operator B ( k ) of (4.4.1) which is our starting point of the existence proof remains unaltered if restricted for k = ( k , , 0, 0) ( n = l ) and k = ( k , , k,, 0) (n=2).

4.8.

Global solutions near zero Theorem 4.5.2 which gives global solutions near go fails if go=O because the constant (1, given there becomes small with the magnitude p of go. Global solutions near 0 have been constructed in [20] and [23], by a method which also takes advantage of a decay of the linear part, although it is much simpler than, and seemingly quite different from, the preceding one. Here, we present a simplified proof of [201, [231. We consider the Cauchy problem (4.3.1) written in the integral equation (2.3.2), in which U ( t )is then (3.2.2); 4.9.

73

Solutions of the Boltzmann Equation

This is the solution of (4.3.1) linearized at f = O . The key point is that, in spite of its simple appearance, it has a decay if fo is nice. Indeed, define the norm

and write L">q=L"(R;; Lg(R;)). Then, if a, P > O , we get,

Illflll=lllflllu,g=~~P II~(-Mt)llu,p t t R

*

Evidently, we have

(4.9.2)

I f ( t , x,


lllflll

f

from which follows the same decay as (4.9.1). Recall the bilinear operator No of (2.3.3). The global existence follows from the

Lemma 4.9.1. Suppose the cutoff (2.2.1) with the restriction O O . Then, IIINo[f,

sllllrclllflll lllslll *

Proof. Let Q,,j = 1 , 2, be given by (2.1.4) and N o , , defined by (2.3.3) with Q replaced by Q,. Noting the group property U(t+s)=U(t)U(s), and then using (4.9.2), we get IU(-t)NO,U,

s l k x, E l l =

11'

U(-s)QJf(s), g(s)ldsl

lJlllflll where

lllglll

Y

74

S . UKAI

(4.9.4)

q

e-alxta(€-€')12~S

-m

Since I X + ~ ( E - E ' ) ~ ~ ~ ~ ~ ( ~ + ( E - E ' ) . X v/ =D ~IE-E'I, ) ~ , we get W I C v - ' , and hence, J I C e x p (-a1x12-P1512) by means of (2.1.7). Now, the lemma follows for N,,,, and similarly for No,2more straightforward. Noting (2.1.5) then proves the lemma for No. Using this and noting the fact IllU(t)folll=Ilfollu,p by definition, we have, for the operator N of (2.3.4),

III"f1lll Ilfollu.p+clllfll12 III"f1 -"91 II I C(lIIf1I I + I I 191I I )I I If-91 I I * 7

As before, this implies that N is a contraction if f,,is small, and thereby, proves the Theorem 4.9.2. Let q(v, 0)be as in Lemma 4.9.1 and a, P > O . Then, there are positive constants a,, a, such that for each f, with IIfollu,8
lllflll ~a~llfolla,p Remark 4.9.3. ( i ) It is astonishing that, in this situation, the global solution exists backwards as well as forwards in t , which is never the case in the situation of S 4.5. ( i i ) The above condition for q(v, 0) is fulfilled by (1.1.4) and the cutoff of (1.1.5) for s>4/3, (n=3). 5. The Initial Boundary Value Problem The local and global existence for the inital boundary value problem to (1.1.1) can be established in the same manner as for the Cauchy problem: First, the linearized problem is solved, and then, the nonlinear term is added as a perturbation. The latter can be handled exactly in the same manner as before, but the former is much more involved. First of all, we must establish a trace theorem which makes sense of the boundary condition (S 5.1). The existence of solutions for the linearized problem which is necessary to solve the nonlinear problem locally is much more delicate to prove than for the Cauchy problem (S 5.2), and the decay estimate which promises the global existence for the nonlinear problem is even more (S 5.4). 5.1.

Trace theorem Many authors have discussed trace theorems associated with the operator

Solutions of the Boltzmann Equation

(5.1.1)

75

( f ,X ,E ) € R X R X R " ,

A = ~ , + ~ . V , + U ( XE).Ve, ,

and related ones, see e.g. [ 5 , 8 , 39,44,45]. Here we follow the line of [39] and show that trace operators (1.4.2) are bounded. In the sequel, 9 is a domain in R", bounded or not, and the boundary aR is piecewise C'

(5.1.2)

.

With S' defined by (1.4.1) and with a T>O,we set, V=QxR", D=(O, T ) x V , I*=(O, T)xS' , V'={T*}X V ,

(5.1.3)

aD+=z'U

v'

,

(same signs)

T + = T , T-=O,

.

The assumption on a ( x , 5 ) is always (2.2.5), but x is within

a;

4x9 E ) = -V,b(x) + a , ( x , 5) ,

(5.1.4)

( i ) ~ E C ~ ( Gb)( ,x ) > l , a, E CL(V ) , ( - a , = v e . a l = 0 . According to Theorem 2.2.1, therefore, (2.2.2) has a unique solution S,(x, E)= ( X , 5 ) for any initial ( x , E ) € V as long as X stays in R. Denote the forward ( t > O ) stay time by t + ( x ,E ) and backward one by t - ( x , E). By definition, S,€V, -t- O . In any case, S , € F at t = f t * . It is now convenient to write y = ( f , x , E ) . The characteristic (integral) curve of A passing through y € D u i?D+u 3D- is given by ( ii

YO, y ) = ( t + s , SJX, 0)

9

-l-(Y)
where Z'(y)=min (T*t+t,P ( x , E)), T-=0, T + = T . Obviously, Y E D (-Z-
y E F (s=kZ+)

and

Z*=O

(y€aD*).

We claim that if f € L'(D), then f( Y ( . , y ) )E L1(-Z-(y), Z'(y)) for almost all y €a D* and (5.1.5)

holds, where dy=dtdxdE and du*=ln(x)-51dfdo,dE on

I*,=dxdE on V* ,

n(x) being the outward normal to aR and du, the measure on aR.

Set,

S . UKAI

76

These will be used to denote Lp-Lq pairs. Our trace theorem will be established between the spaces, wp= tf€W D ) I LP(D)}7 L$**=Lp(dD'; @do*), B = W = m i n (1, Z + ( Y ) + ~ - ( Y ) )

,

where Af is defined in the following distribution sense. Let f € L:,,(D) and suppose there exists a g € L:,,(D) such that

(f,4) = -(a 9)

(5.1.7)

holds for every 9 € C,l(D) (C1 and support compact in D). Then we put g= Af. Note from (5.1.4) that A * = - A (formally). The trace operators 7; are defined primarily on CA(fi)by (5.1.8)

~ ; f = f ~ a D *

f c c ~ ( o- )

Theorem 5.1.1. Let p E [l, a]. 7; have extensions belonging to B( W,, LZ-') and denoted again by 75. Thus, it holds that (5.1.9)

Ilr$fll,gn*

Cllfllwp=C(IIUIILp(D~ f IIAfllLpCD,)

*

Proof. Let f € W p and write f(s, y ) = f ( Y(s, y ) ) . Since &=@/as holds for $ € C A ( D ) , we can deduce from (5.1.5) and (5.1.7) that for almost all y € aD*, f(s, y ) is absolutely continuous in s and ( 5.1.10)

f(s, Y)'

fb', Y ) +

1'

( i i S ) ( r y)dr ,

8'

holds for any s, s'€ [--2-(y), Z+(y)]. Now we define j $ f ~ f ( l * ( y )y,) , yEi3D' which coincide with (5.1.8) if f C €A(@ (note that CA(D)C~W,). It remains to prove (5.1.9). In (5.1.10), we put s=l'(y)=O but retain s' This gives (5.1.9) forp=oo, while forp
Integrate both sides first with respect to s' on ( - l - ( y ) , l + ( y ) )and then with respect to y over d o * . In view of (5.1.5), we are done. Observe that since l ( y ) = l + ( y ) + Z - ( y ) < Tby definition, 0 and Z are equivalent as weight functions so long as T < 03. In the above, we cannot remove the weight function B if p < 03. For this reason, some authors have obtained Lg:-traces only ([8,451). The present theorem is given in [39] for the case a ( x , E)=O. The space Wp is a nice space to solve the linear problem (S 5.2) but Lg,'traces which are natural traces in W , are not adequate for the boundary con-

Solutions of the Boltzmann Equation

ditions of

77

S 1.4; we need LP.*-traces where, Lp**=Lp(eD+;do+).

Note that Lp,*=LP,*'forp=a but Lp.+$ZLP,.*i f p < a .

w, ={f E wp I 7; f E LP'*} c w,

wp=

Define,

.

Theorem 5.1.2. ( i ) If p = a , then W,. I f 7; f E Lp,+, then 7; f € L p , - and vice versa. f E W,, p E [l, a ) . Let (ii) I n this case, it holds that for any 1€ R, (5.1.11)

JPllf IIPLPCD, +Ilr;f

ll:P.+

= Ilrif IIZP.--P

Re

SD

IflP-'sgn (f)(A+Rlfdy ,

where sgn ( f ) = f / l f l gf(x)+O,= O i f f ( x ) = O . Proof. (i) is obvious. One can show that i f f € W,, p < a,then, IflpE W, in place o f f and set In (5.1.10), put Ifl" and Alflp=pRe Iflp-'sgn(f)Af. s=--I-(y), s'=l+(y), yEdD-. Then (5.1.11) follows by integration on dD- and by (5.1.5), proving (ii). The following Green's formula is essential in

S 5.2.

Theorem 5.1.3.

1, and 1 € R. Then,

(5.1.12)

Proof. 5.2.

Let f E

wp,g E

p-'+q-'=

( ( A + 4 f , s)+(f, (A-J)9)=+--

Put fg in place off in (5.1.10) and proceed as before.

The linear initial boundary value problem Let A be as in (5.1.1) and M be as in S 1.4.

(5.2.1)

-

We shall solve

in D , J E R , on I-, in V- .

(A+;l)f=O r-f=MrFf f(0)=fo

Evidently, 7'f and f(0) =f It=O should be understood as

r'f=r;firi

,

f(T')=rgflYi

.

If llM/l< 1, (5.2.1) can be simply solved by successive approximations

([8,39]), but the case llMll= 1 which involves the physically important examples (1.4.4) (ii)-(iv) is delicate. Three different methods have been developed so far, making use of Riesz' representation theorem [39], the limiting absorption principle [ 5 ] and the monotonicity [8],respectively. Here, following [39], we discuss weak solutions. Only L"-solutions are useful for the nonlinear prob-

S. UKAI

78

lem, a contrast to the transport equation which requires L'-solutions [8, 451. In the sequel, we assume (5.1.2) on 9, (5.1.4) on a ( x , E ) and the following on M . Set Yps+=Lp(S' I In(x).Elda,dE). (5.2.2)

with the norm

M E B( Ypp k, Yps-)

IlMll I 1

.

Denote the adjoint to M by M*.

Then, for p E [l, a), we have,

(5.2.3)

with

M*EB(Yqs-, Y " + )

llM*ll
For p = m , this is taken as a n additional assumption. See Remark 5.3.2. Note that since we are assuming that M does not act on t, Yp,' can be replaced by Lp,*l. = L * ( I ' ; In(x)-Eldtdu,d~). The weak solution is defined through Green's formula (5.1.12), with the space of test functions 2

W : = ( g E Wlq1 T + g = M * r - g , g ( T ) = O j ,

p-'+q-'=l

.

Suppose f E W p solves (5.2.1) (the strong solution). Recall (5.1.6). Then it follows from (5.1.12) that for any g E W';,

(f,( A-4g)

(5.2.4)

= - v -

.

Definition 5.2.1. Let f o E L p ( V ) . f € L p ( D ) is called a weak solution to (5.2.1) if (5.2.4) holds for every g E W z .

Theorem 5.2.2.

Suppose p E (1,

a],f,, € Lp(

Then a weak solution

V ) , R>O.

f~ Lp(D) exists. Proof. Apply (5.1.11) to g E W z replacing p , R by q, - A Note that q € [ 1 , m) if p E ( 1 , a]. Then by (5.2.3), we have,

(5.2.5) where 11

llgllq

llP is the

9

respectively.

~ ~ ~ " ' ~ - ' ' ~ l l l~l ( A~ -~~ )~ ! JI l ll q; ~

norm of Lq(D) and

9

I.11;

that of Lq( V ) . Define,

Z,={(A-R)g I g € W $ } C L S ( D ).

(5.2.5) shows that for each h c Z , , there exists a unique g € W$ such that h = ( A - R ) g . Therefore, F(h)= -,- is a well-defined functional on Z,. Further, F is bounded because by (5.2.5),

Consequently, F has a bounded extension P to Lq(D) (the Hahn-Banach theorem [15]) and P ( h ) = ( f , h) with some f ELp(D) for any h € L q ( D ) (Riesz' representation theorem [15]). Restricting h in 2, and putting h=(A-R)g, we see that this f is a desired weak solution.

79

Solutions of the Boltzmann Equation

R,emark 5.2.2. For the case p = 1, the above proof works only if llMll< 1, and gives a weak solution in Lm(D)*=ba(D) (the set of bounded additive set functions which vanish on sets of measure 0 [15])2L1(D). When l[M[l<1, the situation is fairly simple: Theorem 5.2.3. Suppose IIMII strong solution and has the estimate

for p = 00 and p

(1,

00)

< 1. Then f of Theorem 5.2.2 is a unique

respectively.

This follows readily from (5.1.11)once the following characterization of the weak solution is established. Theorem 5.2.4. Any weak solution f € L p ( D ) satisfies ( i 1 fe W p ,( A + J ) f = o , ( ii 1 f(0)=f,E Lp(V ) , , or weakly* (iii) r-fa-My+f6-0 ( € 4 0 ) weakly if p ~ ( 1 a) Lp,+(E+;du'), where f e = x e f and x E = l ( l ( y ) > ~ =O ), (l(y)<~).

if

p = a , in

Let us consider the case IIMII=l. Replacing M by KM with K E ( O , 1) in (5.2.1), and by Theorem 5.2.3, we have unique strong solutions f K satisfying (5.2.6) uniformly for K . Hence, passing to a subsequence,

p-.f

in

Lp(D) ,

p(T)+g

in

Lp(V)

as r-tl ,

both weakly (resp. weakly") for p € ( l , 00) (resp. p = 0 0 ) , with some limits f, 4. This f is a weak solution with f ( T ) = $ , and going to the limit in (5.2.6), we conclude the Theorem 5.2.5.

(5.2.7)

When IIM[I = 1, (5.2.1) has a weak soZutionfE Lp(D) such that Ilf(~)IILPCV,

IIlfoIlLPc",

*

The uniqueness is not known without additional conditions on M (see [5]). So far, we have assumed I>O. However, i f f is a solution to (5.2.1), so is eplfwith I replaced by A-p. In particular, (5.2.7) holds for 220. Since f ( T ) € L P ( V )by (5.2.7) and since T may be arbitrary (even negative), we can define the operator V ( t ) ,t = T € R, by

(5.2.8) Theorem 5.2.6.

~ ( t ) f o = f ( t* ) Z f p € (1,

a), V ( t )is

a C,-group on L p ( V ) .

This is not true when p = a , just a s in the case Q = R n

(S 2.2).

S . UKAI

80

5.3. The local existence We can now solve the initial boundary value problem to (1.1.1): (5.3.1) Let U ( t ) be as in (5.2.8) for 1=0. Then, (5.3.1) is reduced to the integral equation (2.3.2). With pa,Bof (2.2.10), we redefine the space X;,! of (2.2.11) on V . Clearly, Lemma 2.3.1 for Q is also valid in this space, so that if our U ( t )satisfies the estimate

(5.3.2)

I l ~ ( ~ ) f oI l l Ilfoll

in

x:,)4

7

for all a in some interval [a,, a,], a,>a,>O and for some p 2 0 , then we get (2.3.7) and conclude the local existence for (2.3.2). To prove (5.3.2), put f=p;,lPg in (5.2.1) and note that A P ~ , ~ = by O (5.1.4) (ii). We see that g also solves (5.2.1) with M replaced by a a ~ j = p a , f i M p i 9~ ~

and with the initial p;,lBfo=g,. Hence, if &fa,psatisfies (5.2.2-3) for p = w , then (5.2.7) applies to g; lIg(T)II I llgollin L”( V ) . This is nothing but (5.3.2). Thus, we have the same existence theorem as Theorem 2.3.1. Let X,,,,B([-T, T I ) be the space defined by (2.3.5) with our X:,B.

Theorem 5.3.1. Suppose (2.2.1) on q(v, B ) , (5.1.2) on aQ, (5.1.4) on a ( x , f) and (5.2.2-3) on & f a , B for p = w and f o r all a € [ a , , a,] with some aI>az>O and b 2 0 . Then, for any f o € X ; , B , a € [a,, a,], (5.3.1) (reduced lo (2.3.2)) has a unique solution f € [ - T , T I ) with some T , K > 0 satisfying

Ilf(0II-,a-z,t l ,,8 211 f o II-,a,B . Remark 5.3.2. If M is one of (1.4.4) (i)-(iii), then, a a , , = M and fulfills (5.2.2-3), for any a, B E R . If M is (1.4.4) (iv), G,,, also does, but only for a=(2TJ1, p=O. This does not suffice and we must assume that pw is a small parameter independent of T,. This is permitted in a convex linear combination of (1.4.4) (i)-(iv). Remark 5.3.3. If, in addition, M is nonnegativity preserving (i.e., like (2.4.5)), then Theorems 2.4.1 and 2.4.3 are also true for the present case. Thus we can get solutions belonging to the same function class as fa.

5.4. The global existence

Up to the present, the only method available to establish the global existence for (5.3.1) is the one described in 5 4 . 1 which promises a solution One might expect that the method of 5 4 . 9 is also in the form f=g,+gh’2u.

Solutions of the Boltzmann Equation

81

useful. However, because of its special choice of norms, the bounded domain cannot be dealt with, and for the case of the unbounded domain with boundary, it seems difficult to deduce a nice estimate like (4.9.4) because 0 of (4.9.3), modified by the boundary condition, has n o longer any simple properties. To apply the method of 4.1, we shall first rewrite (5.3.1) in the form (4.1.1) putting f = g O + g i ’ Z u . Then, the Maxwellian go should be such that Ag,=O and

(5.4.1) Otherwise, inhomogeneous terms appear both in the equation and the boundary condition. The case where (5.4.1) is violated will be discussed in the next section. Now, with such a go, we have,

(5.4.2)

in

D

y-u=l\)r+u,

on on

I-,

u=u,,

where L,

(5.4.3)

,

u,=--E.V,u--a.VEu+Lu+r[u],

r are as in (4.3.4) and l\)=g;‘f2MgAf2

V-,

.

Therefore the linearized Boltzmann operator €3 is,

(5.4.4)

B = --5.Vz--a.V~+L

,

(x,6)€

v,

associated with the boundary condition r - u = f i r + u . The global solution to (5.4.2) can be found by the method of S 4.5, provided elB has a nice decay. In S 4 , we have observed two different types of decay of elB. One is Theorem 4.4.4 for the Cauchy problem, i.e., for the case SZ=Rn, which can be taken as a typical case of the unbounded domain. The other is Theorem 4.6.1 for the case SZ=Tn, to which the initial boundary value problem for a parallelepiped, a special example of the bounded domain, with the specular reflection can be reduced. For the general case, we may infer from this that (a) if 8 is bounded, e t B ( I - P o )decays exponentially, and (b) if Q is unbounded, erBdecays like I r a , a>O. Several works have been done to confirm this, under the assumptions of a ( x , E)=O and Grad’s cutoff hard potential described in S 4.3. Taking (4.3.2) as go, (a) has been shown for the diffuse reflection in [17], for the specular one in [35], both assuming that SZ is convex, and for the general 8 and M in [6] with rather restrictive conditions on M including (5.2.2) and (5.4.1). The proof can be carried out appealing to Theorem 4.2.3. As for (b), the case of the exterior domain to a bounded convex obstacle was studied in [6, 411. This is a special case (c=O) of the result described in the next section. See also [22, 271.

S. UKAI

82

6. The Stationary Flow The existence and stability of the stationary flow having a prescribed velocity c at infinity and passing arround a n obstacle is one of the classical problems in fluid and gas dynamics and has been discussed extensively. However, most works start from the fluid equations such as the Euler and NavierStokes equations, and few from the Boltzmann equation. Here, following [44], we will show that if c is small, the Boltzmann equation has a stationary solution which is asymptotically stable in t . The case where c is large, especially the case where c is close to the Mach number 1, is a physically more interesting problem in connection with the transonic flow in which the shock appears, but remains unsolved. However, we should mention the works [12, 291 on the one-dimensional shock profile described by the traveling wave solution to the Boltzmann equation, for c near the Mach number 1. Its stability is still a n open question. We assume always Grad's cutoff hard potential and a(x, E)=O. Nothing is known in other situations.

The stationary problem Denote the obstacle by 0 ( c R n ) and its exterior by Q. Suppose that, at infinity, the gas is in equilibrium and moving with the velocity c € R". Then, our gas flow is described by, 6.1.

(6.1.1)

ft=- E * V J + r-f =M r 'f

Q[f I

in

D ,

on C - ,

f-g,(E)=exp(-lE-c12/2)

f It=o=fo

(Ixl-ta)

(t,E ) € R x R " , in V ,

where 9,is the Maxwellian with p, T normalized appropriately. we shall solve the corresponding stationary problem, (6.1.2)

-E-F,f+Q[fl=O r -f = M r 'f

in V , on S - ,

f->9,

for all E E R " ,

(IXl -~~)

First of all,

where f=f(x, 6). Note that g, is not, in general, a solution to (6.1.2) if c Z 0 , because it violates the boundary condition on S - as is seen for M of (1.4.4) (ii) (iii). However, since these M satisfy (5.4.1) with g0=g,=,,, we may expect that if c is small, (6.1.2) has a solution which differs slightly from 9,. T o show this, put f=ge+g;% and reduce (6.1.2) to (6.1.3)

--E.V,u+L,u+r[u]=O @y+u f h , u+o (Ixl400) T-U=

in V , on S - , for ail E € R " ,

Solutions of the Boltzmann Equation

83

where r is as in (4.3.4), ?I? in (5.4.3) and (6.1.4) One might expect that it is more convenient to set f=g,+gE% because L , then becomes selfadjoint in L Z ( R ; )for all c, whereas our L , is not if c f O . However, ?I? then becomes unbounded, for exapmle, for M of (1.4.4) (ii) (iii), which makes (6.1.3) ill-posed. Let B, be the linearized Boltzmann operator, (6.1.5)

B,=-E-V,+L,,

associated with the boundary condition r-u=?I?rtu, and suppose it have a n inverse B;l. Then, (6.1.2) can be reduced to (6.1.6) where

U+ $c

B,-'I'[u] -$c

is a solution to the linear stationary problem, -E.V,$+L.,$=O

(6.1.7)

=O ,

r-$=?I?r+$+h, $+O (1x1-00)

in V , on S - , for all e E R n .

Once the existence of B;' and $c is known, (6.1.6) can be solved by the implicit function theorem (S 6.3). A delicate problem is B;'. It will be seen in s6.2 that O € a ( B , ) and thus B;' does not exist, in L 2 ( V ) . However, the principle of limiting absorption which is familiar in the scattering theory and enables us to find the values of resolvents on the boundary of spectrum is applicable to our B,. Thus, B;' will be constructed as a limit of R ( I , B,) as 2-0. Denote the solution to (6.1.6) by u,. Then,

f , =9,+9:/2uc solves (6.1.2), and hence, is a stationary solution to (6.1.1). Now, set fc+g~'2v=g,+g~/2(u,+v) and rewrite (6.1.1) as

u,=-E.V,v+Lcv+2r[u,,

v]+I'[v]=O

in D , on 8 - , EER"

vl,=o=vo

f=

in

,

V.

By definition, the stationary solution f , is asymptotically stable if (6.1.8) has a global solution v which tends to 0 as f+w, whenever uo is small. This will

S. UKAI

84

be shown in

S 6.4 by

solving the integral equation

where we have put E,(t)=exp (tBJ

(6.1.10)

9

i.e., the semigroup generated by B,. As in S 4 , (6.1.9) will be solved by the help of a nice decay of E,(t). Note that the linear operator to (6.1.8) is not B, but Bc+2r[u,, -1 and that if the corresponding semigroup is used, the linear term of the right-hand side of (6.1.9) disappears. However, it seems difficult to deduce a decay for that semigroup. The extra linear term in (6.1.8) can be made small with u,, for small c. The limiting absorption principle To illustrate our method, we first discuss B, for the special case S2=Rn, i.e., the operator B," given by (6.1.5) but in V m = R ;x R ; . Then, as in S 4.4, it suffices to study B,m(k)=-ik.E+L,. Under Grad's cutoff hard potential, L, has the same properties as L = L , = , of (4.3.4) except that it is not selfadjoint unless c=O. In particular, 6.2.

(6.2.1)

Lc = -

Y A E ) + K

9

where v e ( E ) = v ( E - c ) and K , is a n integral operator to which Proposition 4.3.1 applies, continuously in c E R". Further, 0 € u,(L;) whose eigenspace is invariant in c, that is, if P, denotes the eigenprojection, then P,=P,, [44]. Using this and Theorem 4.4.2, we can prove the following theorem. Let L2, Lz are as in S 4.3, and lie, uo, p r ( r ) , S J r ] as in Theorem 4.4.2. Set, E(a, U ) = { R E C , ( - - U )

I -ReRla[ImA[2}.

) the maximal domain. Then, for Theorem 6.2.1. Define B; in L 2 ( V mwith any c o 2 0 , there is a positive number a , such that the followings hold for all C~ SJC O l.

( i ) P ( B 3 3 a a 0 , uo)\{O}, 0 € 0,"). ( i i ) R(R, B;)=C:f,2 V,(;C,c), for all ;Ce2'(ao,uo)\{O}, where, for O
(6.2.2)

V,(;C,c)=.Fz-lx(k)(R-;Cj(k,c))-'P,(k, C ) F Z , x(k)=l (kES,[rol) =o (keSl[rol) 9 4 =pJ( Ikl) i k . c Pj(k, C ) Bo(Si[rolx Si[col; W L 2 ,LF)) 1520 9

+

9

9

while for j=n+2,

85

Solutions of the Boltzmann Equation

(6.2.3)

U,+,(J,c) E Bo(,Wo,a,) x S1[cOl; B F 2 (v")).

Further, Uj'sare mutually orthogonal, and P,'s are mutually orthogonal projections 00 L2 with P,(k, O)=P,(k), C Pj(0, c)=P,=Po. According to (6.2.3), U,+,(O, c) is a bounded operator, whereas, since Aj(k, c)-l has a singularity at k=O as seen from the asymptotic expansion of ,u,(K) given in Theorem 4.4.2, U,(O, c), O l j l n f l , are unbounded, in L2(Vm). However, since this singularity is integrable, U,(17, c) can be made continuous at R=O, and hence, U,(O, c) become bounded, if the spaces of domain and range are chosen appropriately. This is the principle of limiting absorption. To state this more precisely, we set, (6.2.4)

L ~ , ~ = L $ , r ( V m ) = E{ )~I ~u€LI(RF; ~=~(~, Lp(R;))}

Theorem 6.2.2.

.

Let 1 < 4 < 2 < p < w , O € [ O , l), m=O, 1 with

(6.2.5)

4-1 -p-1>

(2-m)/(n+O).

Then, for O < j < n + l , lcleU,(17, c)(Z-P,(O, c))" € Bo(Z(a0,ao)xS,[c,];

B(L:B~,

.

Proof. It suffices to discuss the case m=O. By the interpolation for the Fourier transforms, and then, proceeding as in (4.4.6), IlUj(17, C)UIIL$'"ICII-iTZU~(R, C)UllL;;'.== Ic$llull'q.2

(p-'+p'-'=l)

Y

where, putting 7=4-l--p-l,

+=(

\slcco)

i l - q k , c)1-1/7dk>1.

After a lengthy calculation using the asymptotic expansion of ,u,(K), we see that

(D
In order to link BY to B,, it is necessary to solve

(6.2.6) for a given h € Yp,- (see (5.2.2)). Suppose, (6.2.7)

0 is a bounded convex domain and

W=aQ

Then, (6.2.6) can be easily (and explicitly!) solved.

is piecewise Cz.

Denote the solution by

86

S. UKAI

u=R,(l)h, R,(R) being the solution operator. Let e be the extension operator from V to V" by 0, and r the restriction operator from V - to I/. Further, set A?Z=r--&?y+ and (6.2.8)

T,(R)=A?ZrR(I, BT)eK,R,(R)

.

After some manupilations taking account of (6.2.6), we have an explicit formula of R(1, B J : (6.2.9)

,

R(2, B,) =rR(1, B:)e+S,(I)(Z- Tc(1))-lA?ZR(A, B,)e S,(R)=R,(l)+rR(R, B,")eK,R,(R)=(r-rR(I, B;)*e)*

.

Originally, this is derived as an equation in L2(V ) for 1 such that 1 € p(B:) n in other spaces as far as the right-hand side makes sense. A crucial point is the invertibility of Z-T,.(I).

p(B,) and l € p ( T c ( l ) )but , can be used to define

Proposition 6.2.3. Let n 2 3 , p~ [2, positive constants a,, c,, a1such that

001,

P>n(2-1-p-1).

Then, there are

(z-Tc(l))-l B o ( m l ,0 1 ) x S,[c,l; B( Y$*-Ni where Y;;,-={u 1 < t > @ uY~p 3 - J . We evaluate the right-hand side of (6.2.9) by the aid of Theorem 6.2.2 and this proposition. Besides, we need some estimates for & ( I ) and must appeal to Grad's argument used in the proofs of Theorems 4.4.6 and 4.4.7. Define L;S'=L;.~(V)by (6.2.4) with V" replaced by V . Set, (6.2.10)

X ; = L g y I p nL ; ' " ,

and set A , = Y , ( ~ ) x . Theorem 6.2.4.

(6.2.11)

zq=L2,2nL q J ,

Our result is, Let n 2 3 , 1 < 4 < 2 < p < m , ,B>n/2, O € [ O , l ) , m=O, 1 with

q-1-p-1>(2-mm)/(n+e)

,

p<1-2/(n+0)

Further, let a € [0, 11 andpur r=l+p-'--q-'. Also, with a,, c,, 6.2.3, set z = 2 ( a 1 , u , ) ~ S , [ c , ] . We have, ( i ) There is a constant C 2 0 andfor any (1, c) € 3,

. u,

of Proposition

+

IcPllR(1, B ~ ) ( Z - ~ ~ ) ~ A ~ u l C(ll~Ilx; l ~ $ ~ y / pl I ~~ 3 4 l z q ) ( i i ) Let e > O and 6>Or. U(C) E

then,

L "( s , [ ~ ,;l X $

If u=u(c) be such that

n B o ( S , [ ~ ,X$-J l; ,

A:,u(c)E Bo(S,[c,l;Zq),

Solutions of the Boltzmann Equation

87

IcldR(R, B,)(Z-P,)~A:U(c)€BO(Z';L;:;,p-E).

Compared with Theorem 6.2.2, the behavior of R ( I , B,) near c=O is worse than that of U,(;C,c). Put m=a=O and let u € X ; n Z q . Then, for c€S,[c,] fixed, R ( I , B,)u E Bo(Z(al,a,); L $ L ~ , ~ so - € that ) , B;'u= -R(O, B,)u E L;:;,p-a exists as a limit as k 0 . Using this inverse, we can solve (6.1.7) in the form, (6.2.12)

$c

Theorem 6.2.5. (6.2.13)

.

=R,(0)h,-B~lK,R,(O)h,

Let n 2 3 , p € [ 2 , 001, O € [ O , 1) with p-'< 1-2/(n+B)

.

Let /3 > n and suppose h, be such that

(6.2.14)

hcEBO(Si[cil; YF*-)t

llhcll=O(l~l) (c+O)

.

Then, $c solves (6.1.7) in Lp-sense and, with r=2-1/p, $,EBD(SJc,l;L;;*")

(6.2.15)

9

ll$cll=~(IcI1-o~).

So far, we have not mentioned the conditions to be imposed on M . Here, we only point out that all the arguments from Proposition 6.2.3 on are valid for M of (1.4.4) (i)-(iii), and for M of (iv) if IT,--T,I

(6.2.16)

Salcl

holds with some a 2 0 , where T,=l is T of our Maxwellian ge. The last condition comes from the second requirement in (6.2.14). The proofs of the statements in this section are all long, and we refer the interested readers to [44]. 6.3. Existence and stability From Theorem 6.2.4 and Lemma 4.3.3, we can see the Proposition 6.3.1. Let n 2 3 , [0,1) and ,!3>n/2+1. P € [2,41 n ((n+0)l(n+0-2), n+O)

(6.3.1) and put

(6.3.2)

r= 1+2/p.

Suppose,

,

There is a constant C20 such that, for c Sl[cI], IIB;'r[u, v]ll
in

X;

.

This and Theorem 6.2.5 enable us to apply the contraction mapping principle to solve the stationary problem (6.1.6). At the first glance, however, (6.3.2) does not seem nice because if we choose 0 f 0 , it diverges as c+O while

88

S. UKAI

in the physically important case n=3, the choice 8=0 is excluded by (6.3.1). It is the nice behavior of $, near c=O given in (6.2.15) that compensates for this defect. Let 8 E [0,2/7) and p 2 2 . Then, we can find (Y such that al = 8( 1+ p - ' ) < a < 1-8(2-p-')

(6.3.3)

= a,

.

Put u=Iclav and rewrite (6.1.6) as

By virtue of (6.2.15) and (6.3.2), it holds that

where II-II is the norm of X ; , C , and C, are positive constants independent of c, v, w, and u=a-a,, r=a,-a. Since u, r>O, G(., c ) becomes contractive for small c, which proves the Theorem 6.3.2. Let n 2 3 , 8 € [0,2/7), p>n/2+1, and suppose (6.3.1) and (6.3.3). Then, there is a positive number c, ( I c l ) such that f o r any c€S1[co], (6.1.6) has a unique solution u, in X ; satisfying

(6.3.4) that

IIu,IIX;
a+r=a,= 1-8(2-

l/p) ;

Further, the continuity properties in c stated in Theorems 6.2.4-5 prove u, € BO(S,[col;X;-J

,

E

>0 .

Also, it can be shown that u, E Wp(V ) and satisfies (6.1.3) in Lp-sense. With this u,, we now solve (6.1.9). Since the second term on its right-hand side is linear, Theorem 4.1.1 must be looked at with y=O, and hence E,(t) must decay faster than t - l . Taking the inverse Laplace transform of (6.2.9) gives a n explicit formula of E , ( t ) ; (6.3.5)

E,(t)=rE~(t)e+(y-rE,(t)*e)*

T D,(t) 9 n;lE:(t)e ,

where E,(t)=exp (tB,"),9 means the convolution in I and D,(t) is the inverse Laplace transform of (I-Tc(,2))-1,see (4.2.7). Knowing Theorem 6.2.1 and following the line of Theorem 4.4.6, we have, Theorem 6.3.2.

(6.3.6)

Let 1 < 9 < 2 < p < o o and m=O, 1. Then,

IIE,(t)(Z--P,)mull.$.-I

C(l+t)-~-~'~lluII,$~""Zs ,

with ~=(n/2)(1/q-l/p)and C 2 0 independent of c , t , u.

89

Solutions of the Boltzmann Equation

By this and Theorem 6.2.2, etc., we have,

Proposition 6.3.3. Let n 2 3 and l e t c , be that of Proposition6.2.3. each 8 € [0, l), there is a constant CTO such thar, ll(~c(t)-Z)ully;~- i Clcl -V +t)-rIIuII,;~-

(6.3.7)

holds for all c€S,[cJ, with r=(n-1+8)/2 is even.

Then, for

,

if n is odd and =(n-1)/2 is n

Substituting these into (6.3.5) yields a desired estimate. Write the righthand side of (6.1.9) as N [ v ] ( t ) . In order to evaluate the second (linear) term of "v], it is necessary that r > l in (6.3.6) ( m = l ) and (6.3.7), while for the third, it suffices that 7>1/2, according to Theorem 4.1.1. For the former, therefore, we should take 8> 0 in (6.3.7) when n=3. Otherwise, we can choose 8=0. If 8>0, a divergent factor IcI-8 appears, but this can be cancelled by (6.3.4). In any case, a careful choice of parameters is required. Write p , 8 of (6.3.1) as p o , 8, and impose the additional condition p ,
P [2,41 n ( ( 1 - 2 / n ~ , (112- U P ~ I - ,~ ) q € [ L 21n [ l , (l/p+l/n)-l) , Oc(0, a ) , p>n/2+1 , r=min ((n/2)(1/q-U~), (n/p,+l)/2, ( n / ~ + 1 ) / 2 ).

(6.3.8) Then,

r> 112.

Set

l l l ~ l l=y: l (1+ t)rllv ( t )11x5 .

We have,

III"v1lll I I I"4

+

IC(Ilvollx$nzg ( I C I -@a+Ill~lll)llI~lll) 7 I I C(IC I +a+ II1411+ I IIWII l)llIy- WI II

- "wll

9

where a=lluell in X $ , p=p,. By (6.3.4), IcI-@a+O as c+O, so N is contractive if v, is small as well as c. Thus, we proved,

Theorem 6.3.4. Let n 2 3 and suppose (6.3.8). Then, there are positive numbers a,, a,, co such that f o r any c € S,[c,] and if IIvoll
v = v ( t ) e B o ( [ O00); , X;)

,

llv(t)llia,(l+t)-T .

Now, the stability off, has been established.

7. The Euler Limit and the Initial Layer The justification of the Hilbert expansion has been discussed in [31].

90

S. UKAI

Here, we follow [43] which simplified the argument of [31] on the one hand and is suitable for the study of the initial layer on the other hand. The problem of unboundedness arises in establishing uniform estimates of solutions as E -> 0, and will be resolved by introducing a Banach scale again. We always assume a ( x , 5)=0 and Grad’s cutoff hard potential, and deal with the Cauchy problem only. Otherwise, all are open. In particular, the case SfR” where the boundary layer appears as well as the initial layer is a physically important open problem. Our result is local in t . A long time behavior which may involve the shock layer is also a n open problem. In this respect, however, [12] is suggestive, in which the Chapmann-Enskog approximation of the one-dimensional shock is discussed. The uniform existence of solutions We consider the Cauchy problem to (1.6.2) for all E > O , with a fixed initial data f,, and seek a limit of the solutions f’ as E+O. If such a limit exists and coincides with the first term f of (1.6.1), then the Hilbert expansion will have been justified to the 0-th order. For such a limit to exist, it is primarily necessary that f E exists on the time interval [0, T ] independent of E . According to Theorem 2.3.1, f’exists on [0, ~ € 1 but , T~ is easily checked to tend to 0 with E . O n the other hand, f’ exists globally in t if f, is near go, as shown in Theorem 4.5.2, but f, should approach g, as E + O , i.e., a,+O. The desired solutions exist i f f , is near go and analytic in x . To prove this, we shall make use of the result from S 4.4. Put f ’ = g o + g ~ ’ 2 ~where E go is as in (4.3.2). Then, (1.6.2) is reduced to

7.1.

(7.1.1)

U;=B%c+-

1 r [ u e ],

UfI,=o=u,

E

,

where

B.=-E.V,+L L E

.

As before, we shall investigate the integral equation, (7.1.2)

uE(r)=etBEuo+€

Roughly speaking, (7.1.3)

lt

.

e(c-s)Bar[~E(~)]d~

0

is uniformly bounded for E > O , while

elBE

1 E

e t B ( Z - P , ) = I V , l a + ~e-‘Ot’€b , E

with uniformly bounded operators a, b and a constant o,>O.

Thus, the un-

Solutions of the Boltzmann Equation

91

bounded factor a - l is replaced by the unbounded operator lVsl, a pseudodifferential operator with the symbol Ikl, The last term of (7.1.3) contributes to the initial layer. Now, our situation is much like that in S 3 in which Q is a pseudo-differential operator in E, and (7.1.2) can be solved by introducing a Banach scale to control the unboundedness of IV,1. Since its order is 1, the scale should be that of analytic functions in x. To be more precise, recall E ( k ) of (4.4.1) and set E e ( k ) = -ik-E+a-'L, the Fourier transform of EL. Since Ez(k)=a-lE(ak), Theorems 4.4.2-3 apply to Bc(k)with k , t replaced by ek, t / e respectively. We write the result as follows.

where L 2 = L 2 ( R ; ) . Now (7.1.3) is visible since (Z-P,)P:o)(i)=O by Theorem 4.4.2 (i). By Proposition 4.3.2 and proceeding as in the proof of Theorem 4.4.6, we can infer the

Y V )3 u=u(t)

-

III~III~=III~IIId,~*l,a,l=SUP llmIla-Tt,L,p< t€I

O0

9

where =(l+ly/2)1'2,y € R n , i=F& and Z c R is a n interval. sequel, we fix a, I , p such that, (7.1.5)

a>O,

l>n,

In the

/3>n/2+1.

Then, X is a Banach algebra, and u E X is analytic in x in the strip R"+ , P by i{lyl
Lemma 7.1.2. There is a constant C 2 0 depending only on a, I , and the following hold. ( i 1 IletBcuollO, t 2 0 .

B of

(7.1.5)

92

S. UKAI

( i i ) Let r > O andput r=a/y.

Then, writing ~ ~ ~ * ~ ~ ~ = ~ ~ ~ * ~ ~ ~ ~ , , ~ ,

l l l ~ ~ l l l ~ ~ ~ ~ ,+ r ~E >l O~.l l l ~ l l l Proof. By (7.1.4) and Lemma 7.1.1, (i) is immediate, and (ii) also, using thrice the inequality,

with 0, u ( E ) / c and u,/e as 5. Write the right hand side of (7.1.2) as N'[u'](t). Then, NE[u]=etBEuO+ P A - l r [ u ] . Since X is a Banach algebra, and by Lemma 4.4.3, IIIN"~1lIls c{IIklII

+( 1+ $)lIluIli~)

IIIN"ul-~"~IlllsC

9

~ l l l ~ l l l + l l l ~ l l l ~ l l l ~* - ~ l l l

This indicates that if uo is small, then N' is contractive, uniformly for E > O . Thus, we proved, Theorem 7.1.3. Suppose (7.1.5). Let r>O andputr=a/r. Then, there are positive constants a,, a, such that if IIuollO, (7.1.2) has a unique solution uEE Y ([0, T I ) with

lllUflll s a ~ l l ~ o* l l Consequently, u'(t) is analytic in x in the strip R"+i{jy]
X={ue

Let

I [ l x R ( k , E)u/140 (R400)} ~ = { u ~ I( e-Ttlfilue(t) t) E BO((O, ~ J )[O, ~ rIt; x X)}. 9

Then, we can infer that N e is a contraction in Z, and hence, [43], Theorem 7.1.4. Let ue, u, be those of Theorem 7.1.3. If, in addition, u, E 8, then u'EZand i s a classicalsolution to (7.1.1). 7.2.

Limit of solutions Define the space Y = Y ; ; : * l by P={u(t) I e - r t l k l u ( t ) ~ B ~ (T[I O ; 2)) , .

Evidently,

PC Y([O,r]). Going to the ,limit in (7.1.4), we can show,

Proposition 7.2.1. Let uo€ X and u(t) € P. Then, as e+O,

93

Solutions of the BoItzrnann Equation

etB'uo+E(t)uo,

W'u(t)-Fu(t) ,

strongly in Y ( [ 8 ,r ] )for any 6 > 0 with the limits,

C #,(t, k)P:O)(R)ii,

nfl

E(r)uo=.Fz-l

n+lst 3=0

F u ( t ) = X z - l .C 3=0

where

o

# j ( t ,k)llclP:l)(O, L)(Z-P,)Aa(s)ds-L-l(Z-Po)Au(t),

#3(r, k)=exp (i2y)Iklr). Further, E(t)uo,Fu(t)E p .

In the above, 8, p cannot be replaced by X , Y([O,71). From this, it follows that if u, E 8, N g maps w={u*(t) I e-rtl*luE(t) E Bo([o,11x 10, 7]\{(0,0)}; 8 ) },

into itself. Since W may be regarded as a subset of Z , then u' of Theorem 7.1.4 is in W . Consequently, us(t)-uo((t) in Y ( [ & T I ) for any 8>0. The limit uo((t)satisfies (7.2.1)

u0(t ) =E( t ) u 0 + F k 1 f[uo](t ) ,

on (0, r ] , but since this can be solved in Y by the contraction mapping principle, we can say that u o ( t ) E I ' and satisfies (7.2.1) on [0, 71. Recall that LPo=O, by which (Z-PO)P:2)(O,k)(Z-P,)=O follows. Hence, (7.2.1) gives (7.2.2) (7.2.3)

(Z-P,)uo(t)=-L-lf[uo(t)],

Luo(r)+f[uo(t)]=O ,

or

P,u"O) =Pouo.

Set f f = g o + g ~ ' 2 u Lfor € 2 0 . (7.2.2) is equivalent to Q [ f o ] = O , so f o is a local Maxwellian. In view of Theorem 1.1.1, (1.6.6) holds also for f',E > O . Going to the limit, we have, E-c=

-

s:


which is nothing but (1.6.7). Also, putting (7.2.4)

E.V,f"S)>e

t=O

e= E

ds

9

here gives 7

which is just (7.2.3). Summarizing, we have,

Theorem 7.2.1. Let ue be that of Theorem 7.1.4. Then, uc(t)+uo(t) strongly in ~([6, 71) for any 6>0, with uo(t)EF. (i ( i i ) f O ( t ) = g o + g ~ / 2 u o ( tis) a local Maxwellian whose fluid dynamical quantities p(t, x ) , u(t, x), T(r, x ) solve the compressible Euler equation (1.6.7) with the initial condition (7.2.4).

94

S. UKAI

In (i), 6=0 is not permitted, i.e., the convergence is not uniform near t=O. In fact, f ' ( 0 ) = f o is not in general a local Maxwellian, but f o ( 0 ) is. Physically, this non-uniform convergence is called the initial layer. However, if the initial f, is itself a local Maxwellian then the convergence becomes uniform and the initial layer disappears;

Theorem 7.2.2. If,in addition, uo=Pouo,then (i) of Theorem 7.2.1 holds good with 6=0. The proof is simple but is referred to [43]. Note that we have, at the same time, constructed a solution to the compressible Euler equation, although within a class of analytic functions. An interesting converse is [lo]: Suppose (1.6.7) has a smooth (Sobolev) solution and construct the local Maxwellian f 0 ( t )by (1.1.11) from the solution. Then, we can construct a solution f S ( t ) to (1.6.2) which tends to f o ( t ) , on some time interval [0, r ] . In this case, initial layers are absent because f c ( 0 ) = f o ( O )is a Maxwellian. The method does not apply to non-Maxwellian initials. So far, we have justified the Hilbert expansion to the 0-th order. Justification to higher orders is not known, but a slightly different expansion is possible ([7]); f"(t)=f"(a,

t)+fO(E,

t/E)+Ef1,*(E,

t) ,

where ( i ) f , ( ~ t,) is sufficiently smooth in [0, 11x [0, T], with f o ( O , t ) = f o ( t )of Theorem 7.2.1 (ii), ( i i ) ~ O ( Ea), is sufficiently smooth in { ( E , a)/€€ [0, 11, EU E [0, r ] } ,and behaves like e-aa with a>O, (iii) f l r * is uniformly bounded and ~f'-* is sufficiently smooth, in [0, 1 ] x [O,r]\{(O,O)}. Further, f l , * has the form, f"*(E,

t)=f'(E,

f)+J'(E,

t/E)+ES2.*(E,

4,

and similarly for f 2 , * , fs2* and so on, each having a property like (i) (ii) (iii). As for the Chapman-Enskog expansion, we must mention [26] which shows that the solution of (1.6.2) with a fixed E > O approaches, as t + a , that of the compressible Navier-Stokes equation with the viscosity and heat diffusion coefficients proportional to E . This is not, however, sufficient for the justification of the expansion to the first order.

References [ 1 ] T. Arai, (in preparation). [ 2 ] L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal., 45 (1972), 1-34. [ 3 ] -, Intermolecular forces of infinite range and the Boltzmann equation, Arch.

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95

Rational Mech. Anal., 77 (1981), 11-23. L. Arkeryd, Loeb solutions of the Boltzmann equation, Arch, Rational Mech. Anal., 86 (1984), 85-97. K. Asano, Local solutions to the initial and initial boundary value problems for the Boltzmann equation with an external force, I, J. Math. Kyoto Univ., 24 (1984), 225-238. -, On the initial boundary value problem of the nonlinear Boltzmann equation in an exterior domain, (in preparation). K. Asano and S. Ukai, On the fluid dynamical limit of the Boltzmann equation, Lecture Note in Numer. Appl. Anal., 6, North-Holland, 1983, 1-19. R. Beak and V. Protopopescu, Abstract time-dependent transport equations, (preprint). R. E. Caflisch, The Boltzmann equation with a soft potentials, Comm. Math. Phys., 74 (1980), 71-109. -, The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33 (1980), 651-666. -, Fluid dynamics and the Boltzmann equation, Nonequilibrium phenomena I, The Boltzmann Equation, (Eds. J. L. Lebowitz and E. W. Montroll), NorthHolland, 1983. R. E. Caflisch and B. Nicolaenko, Shock profile solutions of the Boltzmann equation, Comm. Math. Phys., 86 (1982), 161-194. T. Carleman, Probltme Mathtmatiques dans la Thtorie Cinttique des Gaz, Almquist et Wiksell, Uppsala, 1957. C. Cercignani, Theory and Application of the Boltzmann equation, Elsevier, Amsterdam, 1975. N. Dunford and J. Schwartz, Linear operators I, Interscience Publ., New York, 1957. R. S. Ellis and M. A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. Pures Appl., 54 (1972), 1825-1856. J. P. Giraud, An H-theorem for a gas of rigid spheres in a bounded domain, Colloq. Intern. CNRS, 1975, N236, 29-58. H. Grad, Asymptotic theory of the Boltzmann equation, Rarefied Gas Dynamics I, (Ed. J. A. Laurmann), Academic Press, New York, 1963. -, Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations, Proc. Symp. Appl. Math., (Ed. R. Finn), AMS, Providence, 1965. K. Hamdache, Existence in the large and asymptotic behavior for the Boltzmann equation, to appear in Japan J. Appl. Math. E. Hille and R. S. Phillips, Functional Analysis and Semigroups, AMS, Providence, 1957. A. G. Heintz, Solution of the boundary value problem for the nonlinear Boltzmann equation in a bounded domain (in Russian), Aerodyn. Rarefied Gases, 10 (1980), 16-24. [231 R. Illner and M. Shimbrot, The Boltzmann equation; Global existence for a rare gas in an infinite vacuum, (preprint). 1241 S. Kaniel and M. Shimbrot, The Boltzmann equation, Comm. Math. Phys., 58 (1978). 65-84. T. Kato, Perturbation Theory of Linear Operators, Springer, New York, 1966.

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[26] S. Kawashima, A. Matsumura and T. Nishida, On the fluid dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation, Comm. Math. Phys., 70 (1979), 97-124. N. B. Maslova, Stationary boundary value problems for the nonlinear Boltzmann equation (in Russian), Aerodyn. Rarefied Gases, 10 (1980), 5-15. N. B. Maslova and A. N. Frisov, Solution of the Cauchy problem for the Boltzmann equation (in Russian), Vestnik Leningrad Univ., 19 (1975), 83-85. B. Nicolaenko, A general class of nonlinear bifurcation problems from a point in the essential spectrum, application to shock wave solutions of kinetic equations, in: Application of Bifurcation Theory, Academic Press, New York, 1977. T. Nishida, A note on a theorem of Nirenberg, J. Differential Geometry, 112 (1977), 629-633. -, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Commun. Math. Phys., 61 (1978), 119-148. T. Nishida and K. Imai, Global solutions to the initial value problem for the nonlinear Boltzmann equation, Publ. R.I.M.S., Kyoto Univ., 12 (1976), 229-239. A. Parczewsky, Local existence theorem for the Boltzmann equation in L’, Arch. Mech., 33 (1981), 971-981. Y. Shizuta, On the classical solution of the Boltzmann equation, Comm. Pure Appl. Math., 36 (1983), 705-754. Y. Shizuta and K. Asano, Global solutions of the Boltzmann equation in a bounded convex domain, Proc. Japan Acad., 53A (1977), 3-5. C. Trusdell and R. G. Muncuster, Fundamentals of Maxwell’s Kinetic Theory of a Simple Monoatomic Gas, Pure Appl. Math., Vol. 83, Academic Press, New York, 1980. [37] S. Ukai, On the existence of global solutions of a mixed problem for the nonlinear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. Les solutions globales de 1’8quation de Boltzmann dans I’escape tout entier 1381 -, et dans le demi-espace, C. R. Acad. Sci., Paris, 282A (1976), 317-320. The Transport Equation, (in Japanese), Sangyo Tosho Publ., Tokyo, 1976. [391 -, Local solutions in Gevrey classes to the nonlinear Boltzmann equation with[401 -, out cutoff. Japan J. Appl. Math., 1 (1984), 141-156. S. Ukai and K. Asano, On the initial boundary value problem of the linearized Boltzmann equation in an exterior domain, Proc. Japan Acad., 56 (1980), 12-17. -, On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. R.I.M.S., Kyoto Univ., 18 (1982), 477-519. -, The Euler limit and initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J., 12 (1983), 303-324. -, Steady solutions of the Boltzmann equation for a gas flow past an obstacle. I, Existence, Arch. Rational Mech. Anal., 84 (1983),248-291,11, Stability (preprint). J. Voigt, Functional analytic treatment of the initial boundary value problem for collisionless Gases, Habilitationsschrift, Univ. Munchen, 1980. Department of Applied Physics Osaka City University Sugimoto 3, Sumiyoshi-ku Osaka 558, Japan