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Geometry and Discrete Velocity Approximations to the Boltzmann Equation*
DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) OElsevier Science Publishers B.V. (North-Holland), 1984
417
GEOMETRY AND DISCRETE VELOCITY APPROXIMATIONS TO THE BOLTZMANN EQUATION* Michael C. Reed Department of Mathematics Duke U n i v e r s i t y Durham, NC 27706
I would l i k e t o d e s c r i b e some c a l c u l a t i o n s which Reinhard I l l n e r and I have been making on t h e Carleman model, t h e s i m p l e s t d i s c r e t e v e l o c i t y model o f t h e Boltzmann e q u a t i o n . But, perhaps i t i s w o r t h w h i l e t o s t a r t by d e s c r i b i n g how these d i s c r e t e v e l o c i t y models a r i s e . The Boltzmann e q u a t i o n i s an i n t e g r o d i f f e r e n t i a l e q u a t i o n f o r t h e d e n s i t y i n c o n f i g u r a t i o n and momentum space, n!x,p,t), of a l a r g e number o f p a r t i c l e s ( f o r example, t h e molecules o f a gas). Since t h e Boltzmann e q u a t i o n i s n o t o r i o u s l y d i f f i c u l t t o s t u d y , one can i n s t e a d p e r m i t o n l y f i n i t e l y many f i x e d v e l o c i t i e s p1 ,. . . ,pk. One then wishes t o d e t e r mine t h e c o n f i g u r a t i o n space d e n s i t i e s
By analogy w i t h t h e pi(x,t) = n(x,pi,t). Boltzmann e q u a t i o n t h e s e s h o u l d s a t i s f y e q u a t i o n s o f t h e f o r m
The l e f t hand s i d e m e r e l y expresses t h e f a c t t h a t t h e p a r t i c l e s i n t h e d e n s i t y pi a l l have f i x e d v e l o c i t y pi. The r i g h t hand s i d e s a r e ( u s u a l l y ) taken t o be quadratic functions
o f the
pi
w h i c h mimic i n some way t h e i n t e r a c t i o n s i n t h e
Boltzmann e q u a t i o n ; i n p a r t i c u l a r , t h e q u a n t i t i e s a r e conserved.
qi
s h o u l d be such t h a t t h e expected
The Carleman model,
1 (Ut + u x ) = v2 - u -
u(x,O)
= uo(x)
20
1 (Vt -
v(x,O)
= v (x)
20
n
VP
-
v x ) = u2
-
v2
i s t h e s i m p l e s t d i s c r e t e v e l o c i t y model. p a r t i c l e s w i t h v e l o c i t y p l u s one; v ( x , t ) v e l o c i t y minus one. The t o t a l mass m =
/
0
u ( x , t ) i s the density a t time t i s the density o f p a r t i c l e s w i t h
of
m
u(x,t) + v(x,t)dx
-m
i s conserved. The i n t e r a c t i n g terms on t h e r i g h t guarantee t h a t i f t h e r e i s ( l o c a l l y ) an excess o f u ' s o v e r v ' s , t h e n some u p a r t i c l e s w i l l be t u r n e d i n t o v's. Some y e a r s ago, I l l n e r [ 7 ] showed t h a t t h e s o l u t i o n o f ( 1 ) i s g l o b a l i n time. 1 Suppose t h e i n i t i a l d a t a a r e C , nonThen, a s t a n d a r d c o n t r a c t i o n argument proves
Here i s t h e i d e a i n a s i m p l e case. n e g a t i v e , and have compact s u p p o r t .
*Research s u p p o r t e d b y NSF G r a n t #MCS-8201258
M.C.Reed
478
the existence f o r s h o r t time o f a so u t i o n o f Let B ( t ) = max I u ( x , t ) , v ( x , t
1 ) w i t h t h e same t h r e e p r o p e r t i e s .
}
X
and l e t
xo
be a p o i n t where
u(xo,t) = B ( t )
and
u(xo,t)
>
v(xo,t).
Then
has a maximum a t xo so u x ( x o , t ) = 0. Thus, f r o m t h e d i f f e r e n t i a l au < 0 so t h e maximum w i l l decrease. equation, ( x o , t ) = v(x0,t)' - u(x0,t)' A au s i m i l a r b u t somewhat more t e c h n i c a l p r o o f shows t h a t 5 (x,,t) 5 0 a t points u(x,t)
where b o t h
u(xo,t) = B ( t ) = v(xo,t).
Thus,
B(t)
i s n o n - i n c r e a s i n g so t h e
l o c a l s o l u t i o n o f (1) i s g l o b a l . Two y e a r s ago, I l l n e r and I g o t i n t e r e s t e d i n t h e a s y m p t o t i c p r o p e r t i e s o f t h e Carleman model and proved t h a t [8]
0 2 v(x,t) 5 ;
t 1.1
where c i s a c o n s t a n t depending o n l y on t h e mass. I t i s easy t o see t h a t t h i s t i m e decay i s t h e b e s t one c o u l d e x p e c t ( j u s t observe t h a t 1 - (ut + u x ) 2 -u2 becomes u ' ( s ) -u(s)' on r i g h t w a r d c h a r a c t e r i s t i c s ) . The
Q
f a c t t h a t c depends i n i t i a l data v e r i f i e s mechanics: l o n g t e r m i n i t i a l data b u t o n l y m w i l l disperse very o f t h e same mass m.
o n l y on t h e mass and n o t on any o t h e r p r o p e r t i e s o f t h e ( i n t h i s model) one o f t h e b a s i c i d e a s o f s t a t i s t i c a l b e h a v i o r s h o u l d n o t depend on d e t a i l e d p r o p e r t i e s o f t h e on a few o v e r a l l parameters. Very peaked d e n s i t i e s o f mass r a p i d l y and w i l l have t h e same a s y m p t o t i c s as f l a t d e n s i t i e s
The c a l c u l a t i o n which I want t o s k e t c h today i s f o r t h e Carleman model ( 1 ) i n box. The p a r t i c l e s a r e c o n s t r a i n e d t o s t a y on t h e f i n i t e i n t e r v a l [O,e] by r e f l e c t i n g w a l l s a t z e r o and e.
(2)
u(0,t)
= v(0,t)
u(e,t)
= V(Q,t)
These boundary c o n d i t i o n s m e r e l y a s s e r t t h a t p a r t i c l e s t h a t a r r i v e a t e w i t h speed p l u s one i m m e d i a t e l y l e a v e w i t h speed minus one and v i c e v e r s a a t zero. Q
u + v dx i s again conserved. L o c a l l y , excesses o f 0 produce v ' s and excesses o f v ' s produce u ' s so t h i s suggests t h a t u m. That i s , there should be s h o u l d approach t h e homogeneous s t a t e as t d as t m. O f course, d = c o n s t a n t d so t h a t u ( x , t ) + d, v ( x , t ) by c o n s e r v a t i o n o f mass. Here i s t h e r e s u l t .
m =
The t o t a l mass
-f
-f
-f
Theorem [9]-
Suppose t h a t
conditions (2).
uo(x)
Then t h e s o l u t i o n
and u, v
vo(x) of
are
C1
u's and v a m/2e
and s a t i s f y t h e boundary
(1) s a t i s f i e s
419
Approximations to the Boltzmann Equation
where
c
i s a constant depending o n l y on
I want t o sketch t h e main step i n completely elementary, i l l u s t r a t e s t h e t h e geometry o f t h e c h a r a c t e r i s t i c s i s on i n these d i s c r e t e v e l o c i t y models. 8(t) = a(t) =
max
m
and
e.
t h e p r o o f s i n c e t h e technique, though main p o i n t which I want t o make. Namely, fundamental t o understanding what i s going Set
iu(x,t),v(x,t)l
OLXZP" min { u ( x , t ) , v ( x , t ) }
ozxze
By conservation o f mass, a ( t ) < d < 8 ( t ) and a s i m i l a r a r ument t o t h e one i s non-decreasing. sketched above shows t h a t B ( t ) - is-non-increasing and act! I w i l l sketch t h e main step i n t h e proof t h a t B ( t ) - d 5 c / t . Since B ( t ) i s n = 1,2,... non-increasing i t i s s u f f i c i e n t t o show t h a t B(ne) - d 5 c/n, What we want t o do i s t o show t h a t B((n + lie) - d i s s t r i c t l y s m a l l e r than B(niL) - d. That i s , a f t e r a time s t e p o f l e n g t h e the maximum a c t u a l l y decreases (and o f course we need an e s t i m a t e on the decrease).
.
Define 6 = ( B ( n k ) + d)/2. Then, by conservation of mass t h e r e must be a at s e t o f reasonagly l a r g e measure so t h a t e i t h e r u o r v i s l e s s than 6, t = ne. More p r e c i s e l y ,
where
u
i s Lebesgue measure on [ O , e ] . Thus one o f t h e two terms on t h e l e f t m 7 , suppose i t i s t h e second. Then
must be g r e a t e r than
I n o t h e r words, we have a lower bound on t h e s i z e o f the s e t N a t t = n& on Set N- = N n [O,pG), N+ = N n [ p o t e l where po i s chosen so which v < 8.,
t h a t uW-} ' 7 ( B n - d ) , p{N+> 2~ (B,, - d ) . What we have i s t h a t v i s small ( l e s s than B n ) on a f a i r l y l a r g e s e t a t t = na. We want t o use t h i s t o C
show t h a t b o t h u and v a r e s t r i c t l y l e s s than ~ ( n e ) a t This i s where the geometry of t h e c h a r a c t e r i s t i c s comes in.
Figure 1
t = ( n + 1)e.
480
M.C. Reed
Let
p
be a p o i n t i n N+ and c o n s i d e r t h e l e f t - w a r d c h a r a c t e r i s t i c f r o m 1 S e t t i n g D- = - ( a t - ax) and l e t t i n g s denote a r c l e n g t h we have
6
and
Thus
v ( 0 ) < B,.
-
5 u2
(D-v)(s)
v(s)
v2 = ( u
*
2 w(s)
where
p.
v ) ( u - v ) 5 2B(ne)(3(ne) - v(s)) w(s)
solves
(D-w)(s) = Z a ( n e ) ( B ( n e ) - w) w(0) = Bn S o l v i n g t h e comparison problem e x p l i c i t l y y i e l d s
where
c
depends o n l y on
be a p o i n t on AP1 and c o n s i d e r t h e r i g h t w a r d c h a r a c t e r i s t i c , r. L e t M denote t h e p o i n t s on R where v s a t i s f i e s ( 4 ) . 1 utM} 2- c(Bn - d ) . We want t o f i n d a p o i n t on R where u i s
We know t h a t small.
~
n
Suppose B(nL)
f o r a l l p o i n t s on From ( l ) ,
u
2b2
We know t h a t
-
u2
b
B(l1k)
-
5 (o(ne)
I w i l l e x p l a i n why
R.
J,
u ( r ) = u(q) +
v2
d)
e.
and
m
Now l e t q from q t o
R,
-
v ( s ) 5 B(ne) - c(B(ne)
(4)
2 u(r),
M
ufq)
b
1 v2
v2 - u2
d)
b
If
B(ne),
i s very close t o
M
i s large.
possible positive contribution o f
But i f
1
v2
-
b
i s very close t o
u2
i s small s i n c e
B(ne)
b < a(ne)
-
-
i n any
When one makes
c(B(ne) - d)'
m
depends o n l y on
u(wo) 5 b ( n a )
the
v 5 B(nk)
MC case. Thus we g e t a c o n t r a d i c t i o n if b i s t o o c l o s e t o B ( n e ) . t h e e x p l i c i t e s t i m a t e s i n t h e above argument, one f i n d s t h a t
c
then
makes a s u b s t a n t i a l n e g a t i v e c o n t r i b u t i o n because o f ( 4 ) and t h e f a c t
M t h a t t h e measure of
where
~(ne).
c a n ' t be t o o c l o s e t o
- u2.
M
5 B(ni).
-
and
c ( 0 ( n e ) - d)'.
e . Thus t h e r e e x i s t s wo
on
R
such t h a t
By u s i n g t h e comparison argument which was used
above one concludes f r o m t h i s t h a t (5)
u ( r ) 5 B ( n i ) - c(B(nil) - d)
Notice t h a t t h i s holds f o r
r
in
2 ~
.
P2C.
We would l i k e t o show t h a t ( 5 ) h o l d s f o r
r
in
6 too.
Here i s t h e
geometric argument ( o m i t t i n g a l l t h e a n a l y s i s ) . v i s s m a l l on a l a r g e s e t i n ApO. T h e r e f o r e by t h e comparison argument, v i s s m a l l on a l a r g e s e t i n Apl. ~
By t h e boundary c o n d i t i o n s ,
t h e comparison argument
u
F.
u i s t h e r e f o r e small on a l a r g e s e t i n By is s m a l l on a l a r g e s e t o f r i g h t w a r d c h a r a c t e r i s t i c s
48 1
Approximations to the Boltzrnunn Equation
q.
from T h u s , every leftward c h a r a c t e r i s t i c from to must cross a large s e t of points where u i s small. By the argument above, v i s small a t a l l points of Thus, by the boundary conditions, u i s small a l l points of plB. Therefore, by the comparison argument, u i s small on Bp2. This i s how one concludes t h a t ( 5 ) holds f o r a l l r in BC. Similar methods show t h a t ( 5 ) holds i f u i s replaced by v. T h u s ,
p.
~
which imp1 i e s
Iterating t h i s inequality yields
which i s what we wanted t o prove. This gives a sketch of t h e main s t e p of the proof of the theorem. More There t h e theorem i s used t o prove, i n a d d i t i o n , d e t a i l s can be found in [9]. t h a t t h e decay t o d i s a c t u a l l y exponential in the L2 norm. The point of making these c a l c u l a t i o n s here i s t o emphasize t h a t t h e geometry o f the characteri s t i c s i s crucial to t h e Boltzmann-like properties of the Carleman model. This geometry becomes obscured i f one w r i t e s ( 1 ) in standard evolution equation form (by taking the x d e r i v a t i v e to the o t h e r s i d e ) :
Of course, A generates a nice l i n e a r semigroup etA and one can t r y to t r e a t the whole equation by using semigroup theory, Ouhamel's formula, and estimates on the nonlinear map F. B u t , t h i s seems t o me t o be the wrong approach because i t obscures t h e underlying geometry of the problem. This then i s the point o f my l e c t u r e . There has been a tremendous development of functional analysis over t h e l a s t 50 years. I t i s therefore tempting t o r e c a s t a l l i n i t i a l - v a l u e problems i n semi-group form and t o i n v e s t i g a t e properties of solutions by investigating prope r t i e s of the generator. I think i t i s a mistake in hyperbolic problems and other problems where the underlying geometry i s important.
I t i s appropriate t o end t h i s l e c t u r e by suggesting to you a nice unsolved problem. The next simplest d i s c r e t e velocity model i s the Broadwell model. 2
Vt
+ vx = z
Wt
- wX
=
22
Zt
=
2(vw -
- vw
- vw 2
2)
.
Suppose t h a t one s t a r t s with smooth i n i t i a l data (say of compact support) a t time t = 0. I t i s known from the work of Nishida [lo], Crandall-Tartar [ l l ] , t h a t the solution e x i s t s globally and will be smooth. What i s the asymptotic behavior of the solution as t m? Notice t h a t i f the i n i t i a l data i s zero outside of the interval [ - k , a ] , then z and v should be zero f o r x < - 2 and z a n d w should be zero f o r x > P. Thus the i n t e r a c t i o n should be confined t o the s t r i p t > 0. -QZ X 5 ,.P +
482
M.C. Reed
Figure 2 Thus v w i l l be c o n s t a n t on r i g h t w a r d c h a r a c t e r i s t i c s from p o i n t s p on t h e r i g h t edge o f t h e s t r i p and w w i l l be c o n s t a n t on l e f t w a r d c h a r a c t e r i s t i c s f r o m p o i n t s q on t h e l e f t s i d e o f t h e s t r i p . I n o t h e r words, v and w w i l l be t r a v e l l i n g waves t o t h e r i g h t and t h e l e f t of t h e s t r i p r e s p e c t i v e l y . The t o t a l mass,
m =
v + w + z dx,
leak out o f the s t r i p ?
w
i s conserved.
The q u e s t i o n i s , does a l l t h e mass
I b e l i e v e t h a t i t does and t h a t t h e r e a r e f u n c t i o n s
i,
such t h a t
(6)
-
v(x,t)
-+
i(X
t)
w(x,t)
+
i(X + t )
z(x,t)
-+
0
as t m. I a l s o b e l i e v e t h a t one should be a b l e t o c o n s t r u c t a p r o o f u s i n g o n l y c a l c u l u s and t h e geometry of t h e c h a r a c t e r i s t i c s . -f
I should m e n t i o n t h a t a s t u d e n t o f J o e l S m o l l e r (D. Chang) has some numeri c a l evidence f o r t h i s c o n j e c t u r e and t h a t Russ C a f l i s c h and I have a p r o o f i n t h e case where t h e d a t a i s s m a l l . For l a r g e smooth data, t h e problem i s c o m p l e t e l y open. I t i s n o t even known whether t h e s o l u t i o n i s bounded. References
[ll Broadwell, J . E.
Shock s t r u c t u r e i n a s i m p l e d i s c r e t e v e l o c i t y gas, Phys. o f F l u i d s 7 (1964), 1243-1247.
[2] [3]
Cabannes, H. S o l u t i o n g l o b a l e du p r o b l h e de Cauchy en t h i o r i e c i n i t i q u e d i s c r G t e , J. de M6c. 17 (1978), 1-22. Cabannes, H. The D i s c r e t e Boltzmann Equation, L e c t u r e Notes g i v e n a t U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , 1980.
[4]
Cabannes, H. P r o b l b e s mathbmatiques dans l a t h i o r i e c i n i t i q u e des gaz, Publ. s c i . de l ' i n s t . M i t t a g - L e f f l e r , Uppsala, 1957.
[5]
G a t i n g n o l , R. T h g o r i e c i n i t i q u e de qaz 2 r 6 p a r t i t i o n d i s c r G t e de v i t e s s e s , L e c t u r e Notes i n Phys. 36 ( S p r i n a e r - V e r l a g , New York, 1975).
Approximntions to the Boltzmann Equation
[6]
Godunov, K. and Sultangazin, U. M., On the d i s c r e t e models of k i n e t i c equation o f Boltzmann, Uspekhi Mat. Nauk 26 (1971), 3-51.
[7]
I l l n e r , R . , Global existence f o r two-velocity models o f the Boltzmann equation, Math. Meth. Appl. Sci. 1 (1979), 187-193.
[8]
I l l n e r , R . and Reed, M . The decay of solutions o f t h e Carleman Model, Math. Meth. Appl. Sci. 3 (1981), 121-127.
[9]
I l l n e r , R . and Reed, M. box, p r e p r i n t , 1983.
48 3
Decay t o equilibrium f o r t h e Carleman model in a
[lo] Nishida, T. and Mimura, M. On t h e Broadwell's Yodel f o r a Simple Discrete Velocity Gas, Proc. Japan Acad. 50 (1974), 812-817. [ l l ] T a r t a r , L. Existence globale pour u n systhrne hyperbolique semi-lin6aire de l a t h 6 o r i e c i n h a t i q u e des gaz, Ecole Polytechnique, Sgminaire GoulaouicSchwartz, 28 Octobre, 1975. [12] Temam,,R: S u r l a r6solution exacte e t approchse d ' u n probleme hyperbolique nonlineaire de T. Carleman, Arch. f o r Rational Mech. Anal. 35 (1969), 351 -362.
417
GEOMETRY AND DISCRETE VELOCITY APPROXIMATIONS TO THE BOLTZMANN EQUATION* Michael C. Reed Department of Mathematics Duke U n i v e r s i t y Durham, NC 27706
I would l i k e t o d e s c r i b e some c a l c u l a t i o n s which Reinhard I l l n e r and I have been making on t h e Carleman model, t h e s i m p l e s t d i s c r e t e v e l o c i t y model o f t h e Boltzmann e q u a t i o n . But, perhaps i t i s w o r t h w h i l e t o s t a r t by d e s c r i b i n g how these d i s c r e t e v e l o c i t y models a r i s e . The Boltzmann e q u a t i o n i s an i n t e g r o d i f f e r e n t i a l e q u a t i o n f o r t h e d e n s i t y i n c o n f i g u r a t i o n and momentum space, n!x,p,t), of a l a r g e number o f p a r t i c l e s ( f o r example, t h e molecules o f a gas). Since t h e Boltzmann e q u a t i o n i s n o t o r i o u s l y d i f f i c u l t t o s t u d y , one can i n s t e a d p e r m i t o n l y f i n i t e l y many f i x e d v e l o c i t i e s p1 ,. . . ,pk. One then wishes t o d e t e r mine t h e c o n f i g u r a t i o n space d e n s i t i e s
By analogy w i t h t h e pi(x,t) = n(x,pi,t). Boltzmann e q u a t i o n t h e s e s h o u l d s a t i s f y e q u a t i o n s o f t h e f o r m
The l e f t hand s i d e m e r e l y expresses t h e f a c t t h a t t h e p a r t i c l e s i n t h e d e n s i t y pi a l l have f i x e d v e l o c i t y pi. The r i g h t hand s i d e s a r e ( u s u a l l y ) taken t o be quadratic functions
o f the
pi
w h i c h mimic i n some way t h e i n t e r a c t i o n s i n t h e
Boltzmann e q u a t i o n ; i n p a r t i c u l a r , t h e q u a n t i t i e s a r e conserved.
qi
s h o u l d be such t h a t t h e expected
The Carleman model,
1 (Ut + u x ) = v2 - u -
u(x,O)
= uo(x)
20
1 (Vt -
v(x,O)
= v (x)
20
n
VP
-
v x ) = u2
-
v2
i s t h e s i m p l e s t d i s c r e t e v e l o c i t y model. p a r t i c l e s w i t h v e l o c i t y p l u s one; v ( x , t ) v e l o c i t y minus one. The t o t a l mass m =
/
0
u ( x , t ) i s the density a t time t i s the density o f p a r t i c l e s w i t h
of
m
u(x,t) + v(x,t)dx
-m
i s conserved. The i n t e r a c t i n g terms on t h e r i g h t guarantee t h a t i f t h e r e i s ( l o c a l l y ) an excess o f u ' s o v e r v ' s , t h e n some u p a r t i c l e s w i l l be t u r n e d i n t o v's. Some y e a r s ago, I l l n e r [ 7 ] showed t h a t t h e s o l u t i o n o f ( 1 ) i s g l o b a l i n time. 1 Suppose t h e i n i t i a l d a t a a r e C , nonThen, a s t a n d a r d c o n t r a c t i o n argument proves
Here i s t h e i d e a i n a s i m p l e case. n e g a t i v e , and have compact s u p p o r t .
*Research s u p p o r t e d b y NSF G r a n t #MCS-8201258
M.C.Reed
478
the existence f o r s h o r t time o f a so u t i o n o f Let B ( t ) = max I u ( x , t ) , v ( x , t
1 ) w i t h t h e same t h r e e p r o p e r t i e s .
}
X
and l e t
xo
be a p o i n t where
u(xo,t) = B ( t )
and
u(xo,t)
>
v(xo,t).
Then
has a maximum a t xo so u x ( x o , t ) = 0. Thus, f r o m t h e d i f f e r e n t i a l au < 0 so t h e maximum w i l l decrease. equation, ( x o , t ) = v(x0,t)' - u(x0,t)' A au s i m i l a r b u t somewhat more t e c h n i c a l p r o o f shows t h a t 5 (x,,t) 5 0 a t points u(x,t)
where b o t h
u(xo,t) = B ( t ) = v(xo,t).
Thus,
B(t)
i s n o n - i n c r e a s i n g so t h e
l o c a l s o l u t i o n o f (1) i s g l o b a l . Two y e a r s ago, I l l n e r and I g o t i n t e r e s t e d i n t h e a s y m p t o t i c p r o p e r t i e s o f t h e Carleman model and proved t h a t [8]
0 2 v(x,t) 5 ;
t 1.1
where c i s a c o n s t a n t depending o n l y on t h e mass. I t i s easy t o see t h a t t h i s t i m e decay i s t h e b e s t one c o u l d e x p e c t ( j u s t observe t h a t 1 - (ut + u x ) 2 -u2 becomes u ' ( s ) -u(s)' on r i g h t w a r d c h a r a c t e r i s t i c s ) . The
Q
f a c t t h a t c depends i n i t i a l data v e r i f i e s mechanics: l o n g t e r m i n i t i a l data b u t o n l y m w i l l disperse very o f t h e same mass m.
o n l y on t h e mass and n o t on any o t h e r p r o p e r t i e s o f t h e ( i n t h i s model) one o f t h e b a s i c i d e a s o f s t a t i s t i c a l b e h a v i o r s h o u l d n o t depend on d e t a i l e d p r o p e r t i e s o f t h e on a few o v e r a l l parameters. Very peaked d e n s i t i e s o f mass r a p i d l y and w i l l have t h e same a s y m p t o t i c s as f l a t d e n s i t i e s
The c a l c u l a t i o n which I want t o s k e t c h today i s f o r t h e Carleman model ( 1 ) i n box. The p a r t i c l e s a r e c o n s t r a i n e d t o s t a y on t h e f i n i t e i n t e r v a l [O,e] by r e f l e c t i n g w a l l s a t z e r o and e.
(2)
u(0,t)
= v(0,t)
u(e,t)
= V(Q,t)
These boundary c o n d i t i o n s m e r e l y a s s e r t t h a t p a r t i c l e s t h a t a r r i v e a t e w i t h speed p l u s one i m m e d i a t e l y l e a v e w i t h speed minus one and v i c e v e r s a a t zero. Q
u + v dx i s again conserved. L o c a l l y , excesses o f 0 produce v ' s and excesses o f v ' s produce u ' s so t h i s suggests t h a t u m. That i s , there should be s h o u l d approach t h e homogeneous s t a t e as t d as t m. O f course, d = c o n s t a n t d so t h a t u ( x , t ) + d, v ( x , t ) by c o n s e r v a t i o n o f mass. Here i s t h e r e s u l t .
m =
The t o t a l mass
-f
-f
-f
Theorem [9]-
Suppose t h a t
conditions (2).
uo(x)
Then t h e s o l u t i o n
and u, v
vo(x) of
are
C1
u's and v a m/2e
and s a t i s f y t h e boundary
(1) s a t i s f i e s
419
Approximations to the Boltzmann Equation
where
c
i s a constant depending o n l y on
I want t o sketch t h e main step i n completely elementary, i l l u s t r a t e s t h e t h e geometry o f t h e c h a r a c t e r i s t i c s i s on i n these d i s c r e t e v e l o c i t y models. 8(t) = a(t) =
max
m
and
e.
t h e p r o o f s i n c e t h e technique, though main p o i n t which I want t o make. Namely, fundamental t o understanding what i s going Set
iu(x,t),v(x,t)l
OLXZP" min { u ( x , t ) , v ( x , t ) }
ozxze
By conservation o f mass, a ( t ) < d < 8 ( t ) and a s i m i l a r a r ument t o t h e one i s non-decreasing. sketched above shows t h a t B ( t ) - is-non-increasing and act! I w i l l sketch t h e main step i n t h e proof t h a t B ( t ) - d 5 c / t . Since B ( t ) i s n = 1,2,... non-increasing i t i s s u f f i c i e n t t o show t h a t B(ne) - d 5 c/n, What we want t o do i s t o show t h a t B((n + lie) - d i s s t r i c t l y s m a l l e r than B(niL) - d. That i s , a f t e r a time s t e p o f l e n g t h e the maximum a c t u a l l y decreases (and o f course we need an e s t i m a t e on the decrease).
.
Define 6 = ( B ( n k ) + d)/2. Then, by conservation of mass t h e r e must be a at s e t o f reasonagly l a r g e measure so t h a t e i t h e r u o r v i s l e s s than 6, t = ne. More p r e c i s e l y ,
where
u
i s Lebesgue measure on [ O , e ] . Thus one o f t h e two terms on t h e l e f t m 7 , suppose i t i s t h e second. Then
must be g r e a t e r than
I n o t h e r words, we have a lower bound on t h e s i z e o f the s e t N a t t = n& on Set N- = N n [O,pG), N+ = N n [ p o t e l where po i s chosen so which v < 8.,
t h a t uW-} ' 7 ( B n - d ) , p{N+> 2~ (B,, - d ) . What we have i s t h a t v i s small ( l e s s than B n ) on a f a i r l y l a r g e s e t a t t = na. We want t o use t h i s t o C
show t h a t b o t h u and v a r e s t r i c t l y l e s s than ~ ( n e ) a t This i s where the geometry of t h e c h a r a c t e r i s t i c s comes in.
Figure 1
t = ( n + 1)e.
480
M.C. Reed
Let
p
be a p o i n t i n N+ and c o n s i d e r t h e l e f t - w a r d c h a r a c t e r i s t i c f r o m 1 S e t t i n g D- = - ( a t - ax) and l e t t i n g s denote a r c l e n g t h we have
6
and
Thus
v ( 0 ) < B,.
-
5 u2
(D-v)(s)
v(s)
v2 = ( u
*
2 w(s)
where
p.
v ) ( u - v ) 5 2B(ne)(3(ne) - v(s)) w(s)
solves
(D-w)(s) = Z a ( n e ) ( B ( n e ) - w) w(0) = Bn S o l v i n g t h e comparison problem e x p l i c i t l y y i e l d s
where
c
depends o n l y on
be a p o i n t on AP1 and c o n s i d e r t h e r i g h t w a r d c h a r a c t e r i s t i c , r. L e t M denote t h e p o i n t s on R where v s a t i s f i e s ( 4 ) . 1 utM} 2- c(Bn - d ) . We want t o f i n d a p o i n t on R where u i s
We know t h a t small.
~
n
Suppose B(nL)
f o r a l l p o i n t s on From ( l ) ,
u
2b2
We know t h a t
-
u2
b
B(l1k)
-
5 (o(ne)
I w i l l e x p l a i n why
R.
J,
u ( r ) = u(q) +
v2
d)
e.
and
m
Now l e t q from q t o
R,
-
v ( s ) 5 B(ne) - c(B(ne)
(4)
2 u(r),
M
ufq)
b
1 v2
v2 - u2
d)
b
If
B(ne),
i s very close t o
M
i s large.
possible positive contribution o f
But i f
1
v2
-
b
i s very close t o
u2
i s small s i n c e
B(ne)
b < a(ne)
-
-
i n any
When one makes
c(B(ne) - d)'
m
depends o n l y on
u(wo) 5 b ( n a )
the
v 5 B(nk)
MC case. Thus we g e t a c o n t r a d i c t i o n if b i s t o o c l o s e t o B ( n e ) . t h e e x p l i c i t e s t i m a t e s i n t h e above argument, one f i n d s t h a t
c
then
makes a s u b s t a n t i a l n e g a t i v e c o n t r i b u t i o n because o f ( 4 ) and t h e f a c t
M t h a t t h e measure of
where
~(ne).
c a n ' t be t o o c l o s e t o
- u2.
M
5 B(ni).
-
and
c ( 0 ( n e ) - d)'.
e . Thus t h e r e e x i s t s wo
on
R
such t h a t
By u s i n g t h e comparison argument which was used
above one concludes f r o m t h i s t h a t (5)
u ( r ) 5 B ( n i ) - c(B(nil) - d)
Notice t h a t t h i s holds f o r
r
in
2 ~
.
P2C.
We would l i k e t o show t h a t ( 5 ) h o l d s f o r
r
in
6 too.
Here i s t h e
geometric argument ( o m i t t i n g a l l t h e a n a l y s i s ) . v i s s m a l l on a l a r g e s e t i n ApO. T h e r e f o r e by t h e comparison argument, v i s s m a l l on a l a r g e s e t i n Apl. ~
By t h e boundary c o n d i t i o n s ,
t h e comparison argument
u
F.
u i s t h e r e f o r e small on a l a r g e s e t i n By is s m a l l on a l a r g e s e t o f r i g h t w a r d c h a r a c t e r i s t i c s
48 1
Approximations to the Boltzrnunn Equation
q.
from T h u s , every leftward c h a r a c t e r i s t i c from to must cross a large s e t of points where u i s small. By the argument above, v i s small a t a l l points of Thus, by the boundary conditions, u i s small a l l points of plB. Therefore, by the comparison argument, u i s small on Bp2. This i s how one concludes t h a t ( 5 ) holds f o r a l l r in BC. Similar methods show t h a t ( 5 ) holds i f u i s replaced by v. T h u s ,
p.
~
which imp1 i e s
Iterating t h i s inequality yields
which i s what we wanted t o prove. This gives a sketch of t h e main s t e p of the proof of the theorem. More There t h e theorem i s used t o prove, i n a d d i t i o n , d e t a i l s can be found in [9]. t h a t t h e decay t o d i s a c t u a l l y exponential in the L2 norm. The point of making these c a l c u l a t i o n s here i s t o emphasize t h a t t h e geometry o f the characteri s t i c s i s crucial to t h e Boltzmann-like properties of the Carleman model. This geometry becomes obscured i f one w r i t e s ( 1 ) in standard evolution equation form (by taking the x d e r i v a t i v e to the o t h e r s i d e ) :
Of course, A generates a nice l i n e a r semigroup etA and one can t r y to t r e a t the whole equation by using semigroup theory, Ouhamel's formula, and estimates on the nonlinear map F. B u t , t h i s seems t o me t o be the wrong approach because i t obscures t h e underlying geometry of the problem. This then i s the point o f my l e c t u r e . There has been a tremendous development of functional analysis over t h e l a s t 50 years. I t i s therefore tempting t o r e c a s t a l l i n i t i a l - v a l u e problems i n semi-group form and t o i n v e s t i g a t e properties of solutions by investigating prope r t i e s of the generator. I think i t i s a mistake in hyperbolic problems and other problems where the underlying geometry i s important.
I t i s appropriate t o end t h i s l e c t u r e by suggesting to you a nice unsolved problem. The next simplest d i s c r e t e velocity model i s the Broadwell model. 2
Vt
+ vx = z
Wt
- wX
=
22
Zt
=
2(vw -
- vw
- vw 2
2)
.
Suppose t h a t one s t a r t s with smooth i n i t i a l data (say of compact support) a t time t = 0. I t i s known from the work of Nishida [lo], Crandall-Tartar [ l l ] , t h a t the solution e x i s t s globally and will be smooth. What i s the asymptotic behavior of the solution as t m? Notice t h a t i f the i n i t i a l data i s zero outside of the interval [ - k , a ] , then z and v should be zero f o r x < - 2 and z a n d w should be zero f o r x > P. Thus the i n t e r a c t i o n should be confined t o the s t r i p t > 0. -QZ X 5 ,.P +
482
M.C. Reed
Figure 2 Thus v w i l l be c o n s t a n t on r i g h t w a r d c h a r a c t e r i s t i c s from p o i n t s p on t h e r i g h t edge o f t h e s t r i p and w w i l l be c o n s t a n t on l e f t w a r d c h a r a c t e r i s t i c s f r o m p o i n t s q on t h e l e f t s i d e o f t h e s t r i p . I n o t h e r words, v and w w i l l be t r a v e l l i n g waves t o t h e r i g h t and t h e l e f t of t h e s t r i p r e s p e c t i v e l y . The t o t a l mass,
m =
v + w + z dx,
leak out o f the s t r i p ?
w
i s conserved.
The q u e s t i o n i s , does a l l t h e mass
I b e l i e v e t h a t i t does and t h a t t h e r e a r e f u n c t i o n s
i,
such t h a t
(6)
-
v(x,t)
-+
i(X
t)
w(x,t)
+
i(X + t )
z(x,t)
-+
0
as t m. I a l s o b e l i e v e t h a t one should be a b l e t o c o n s t r u c t a p r o o f u s i n g o n l y c a l c u l u s and t h e geometry of t h e c h a r a c t e r i s t i c s . -f
I should m e n t i o n t h a t a s t u d e n t o f J o e l S m o l l e r (D. Chang) has some numeri c a l evidence f o r t h i s c o n j e c t u r e and t h a t Russ C a f l i s c h and I have a p r o o f i n t h e case where t h e d a t a i s s m a l l . For l a r g e smooth data, t h e problem i s c o m p l e t e l y open. I t i s n o t even known whether t h e s o l u t i o n i s bounded. References
[ll Broadwell, J . E.
Shock s t r u c t u r e i n a s i m p l e d i s c r e t e v e l o c i t y gas, Phys. o f F l u i d s 7 (1964), 1243-1247.
[2] [3]
Cabannes, H. S o l u t i o n g l o b a l e du p r o b l h e de Cauchy en t h i o r i e c i n i t i q u e d i s c r G t e , J. de M6c. 17 (1978), 1-22. Cabannes, H. The D i s c r e t e Boltzmann Equation, L e c t u r e Notes g i v e n a t U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , 1980.
[4]
Cabannes, H. P r o b l b e s mathbmatiques dans l a t h i o r i e c i n i t i q u e des gaz, Publ. s c i . de l ' i n s t . M i t t a g - L e f f l e r , Uppsala, 1957.
[5]
G a t i n g n o l , R. T h g o r i e c i n i t i q u e de qaz 2 r 6 p a r t i t i o n d i s c r G t e de v i t e s s e s , L e c t u r e Notes i n Phys. 36 ( S p r i n a e r - V e r l a g , New York, 1975).
Approximntions to the Boltzmann Equation
[6]
Godunov, K. and Sultangazin, U. M., On the d i s c r e t e models of k i n e t i c equation o f Boltzmann, Uspekhi Mat. Nauk 26 (1971), 3-51.
[7]
I l l n e r , R . , Global existence f o r two-velocity models o f the Boltzmann equation, Math. Meth. Appl. Sci. 1 (1979), 187-193.
[8]
I l l n e r , R . and Reed, M . The decay of solutions o f t h e Carleman Model, Math. Meth. Appl. Sci. 3 (1981), 121-127.
[9]
I l l n e r , R . and Reed, M. box, p r e p r i n t , 1983.
48 3
Decay t o equilibrium f o r t h e Carleman model in a
[lo] Nishida, T. and Mimura, M. On t h e Broadwell's Yodel f o r a Simple Discrete Velocity Gas, Proc. Japan Acad. 50 (1974), 812-817. [ l l ] T a r t a r , L. Existence globale pour u n systhrne hyperbolique semi-lin6aire de l a t h 6 o r i e c i n h a t i q u e des gaz, Ecole Polytechnique, Sgminaire GoulaouicSchwartz, 28 Octobre, 1975. [12] Temam,,R: S u r l a r6solution exacte e t approchse d ' u n probleme hyperbolique nonlineaire de T. Carleman, Arch. f o r Rational Mech. Anal. 35 (1969), 351 -362.