Recent Topics in Nonlinear PDE

236 Pages·1984·4.614 MB·iii-v, 1-239\236

by  Masayasu Mimura and Takaaki Nishida (Eds.)

NORTH-HOLLAND
MATHEMATICS STUDIES 98
Lecture Notes in Numerical and Applied Analysis Vol. 6
General Editors:
H. Fujita (University of Tokyo) and M. Yamaguti (Kyoto University)
Recent Topics in Nonlinear PDE
Edited by
MASAYASU MIMURA (Hiroshima University)
TAKAAKI NlSHlDA (Kyoto University)
1984
NORTH-HOLLAND PUBLISHING COMPANY
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KINOKUNIYA COMPANY LTD.
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@ 1984 by Publishing Committee of Lecture Notes in Numerical and Applied Analysis
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Distributed in Japan by KINOKUNIYA COMPANY LTD.
Lecture Notes in Numerical and Applied Analysis Vol. 6
General Editors
H. Fujita M. Yamaguti
University of Tokyo Kyoto Universtiy
Editional Board
H. Fujii, Kyoto Sangyo Universtiy
M. Mimura, Hiroshima University
T. Miyoshi, Kumamoto University
M. Mori, The University of Tsukuba
T. Nishida. Kyoto Universtiy
T. Nishida, Kyoto University
T. Taguti, Konan Universtiy
S. Ukai, Osaka City Universtiy
T. Ushijima. The Universtiy of Electro-Communications
PRINTED IN JAPAN
P R E F A C E
The meeting on the subject of nonlinear partial differential equations
was held at Hiroshima University in February, 1983. Leading and active
mathematicians were invited to talk on their current research interests in
nonlinear pdes occuring in the areas of fluid dynamics, free boundary
problems, population dynamics and mathematical physics. This volume
contains the theory of nonlinear pdes and the related topics which have
been recently developed in Japan.
Thanks are due to all participants for making the meeting so success-
ful.
Finally, we would like to thank the Grant-in-Aid for Scientific Re-
search from the Ministry of Education, Science and Culture of Japan for
the financial support.
M. MIMURA
T. NISHIDA
Lecture Notes in Num. Appl. Anal., 6, 1-19 (1983)
Recent Topics in Nonlinear PDE,H iroshima, 1983
On the Fluid Dynamical Limit of the
Boltzmann Equation
Kiyoshi ASANO* and Seiji UKAI**
*Institute of Mathematics, Yoshida College, Kyoto University
Kyoto 606, Japan
**Department of Applied Physics, Osaka City University
Osaka 558, Japan
1. Problem and Results
This paper is a continuation of our paper C161 concerned with the Euler
limit of the Boltzmann equation. In C161 we studied the behavior of the
density distribution f (t,x,S) of rarefied gas particles, when the mean free
path E(>O) tends to zero. More precisely, if the intial density distribution
f,,(x,S) i s sufficiently close to an absolute Maxwellian and 'satisfies some rather
restrictive conditions, then the solution f"(t,x,E) of the Boltzmann equation
with initial data fo exists in a time interval [O,T1 independent of E E (O,m),
F 0
and converges to a local Maxwellian f (t,x,E):
when E tends to zero. Moreover, the fluid dynamic quantities {p(t,x),v(t,x),
e(t.x)} (i.e, mass density, flow velocity and temparature) satisfy the com-
pressible Euler equation with initial data specified by fo(x,S). This limit-
ing process is the first approximation to the Hilbert expansion of the solution
of the Bol tzmann equation.
In this paper we make a more detailed treatment of the Hilbert expansion
and establish an asymptotic formula such as
1
2 Kiyoshi ASANO and Seiji UKAI
.
and behaves like exp(-oT) with u > 0 (j=O,l....) However, the general for-
-.
mula to calculate fJ and fJ is so complicated that we prove only the special
case
0 -0
(1.2) fE(t,x,c) = f (E,t,Xsc) f (E,t/EsX,E) Ef'**(E,t,X,c),
and suggest the method to prove the next step of the expansion.
The 1 imiting process from the Boltzmann equation to the compressible
Euler equation was described in detail in 1101 and C161, and we state only
the conclusion.
The Cauchy problem of the Boltzmann equation is described as
-aaft t c*v,f = 1 QCf,fl, t>O, (x.6) E Rn x Rn (nr3),
Here f = f(Est,X.c) is the density distribution of gas particles with the
position x and the velocity 5 at time t, E-V, = E,a/axl +.a*+ cna/axn
and QCf,hl is the symetrized collision integral which is a quadratic operator
acting on the variable 5. The scattering potential is assumed to be the cut-
off hard type of Grad C51. E>O is the mean free path.
Since we consider (1.3) near an absolute Maxwellian , we put
2
g(E) = p(2re)-n/2 e -161 /(2e) , p > 0, e > 0,
(1.4) f(E,t,X,c) = + g1/2~(c,t,X,~) ,
.
fo(x,e) = g + g1l2 u0~x.c)
Then we obtain the equation for the unknown u :
-au- - -s.vxu t 1 LU t 1 r[u,ui ,
at
Fluid Dynamical Limit of the Boltzmann Equation 3
(1.5)
where
Denoting by Q(k,S) = Fxu(.,S) the Fourier transform of u,
O(k,S) = (2a)-"' u(x,E)dx,
we convert (1.5) to the folt owing
3a t = - iS.kQ + 1 LO + 1 r^CO ,01, i = fl ,
(1.6)
filt,o = Oo(k,S),
where
.
(1.7) F[u,vl(k,S) = (21r)-"' rCu(k-k',*),v(k',*)I(E)dk'
The equation (1.6) is actually solved in this paper (see also I: 61).
According to C31~ the collision integral Qtf,fl has (n+2 invariants
{h.(E) ; O<jsn+ll = {l,El,***,Sny 151 2 /21, i.e,
J
(1.8) QCf,fl(5)hj(S)dS = 0 , j = 0,1,***. n+l,
By the following formula we define fluid dynamic quantities associated with
the density f(t,x,S) of gas particles, i.e, the mass density p(E,t,x), fluid
flow velocity v(c,t,x), internal energy e(E.t,x). temparature e ( ,~t,x ), stress
tensor P(~,t,x) = (P. .(~,t,x)), heat flow vector q(E,t,x) and pressure
1J
p(E,t,x) (see C101):
P(E,t,x) = f'(t,x,S) hO(S) dS ,
R"
sf"
P (E ,t ,x)v (E ,t ,x) = (t,x ,5) h (5 Id5 (1" jsn ,
5
P(E,t,x){e(E,t,x) + 2 /v(E.~,x)/~=) fE(t,x,E)h,+l (E)dS,
4 Kiyoshi ASANO and Seiji UKAI
(1.9)
The last is the ideal gas condition.
Combining (1.3) and (1.81, we obtain
na
P+ VX'(PV) = 0,
Since P = P(II = the identity matrix) and q =O for the local Maxwellian f,
the equation (1.10) reduces to the compressible Euler equation for E = 0
(and t > 0).
According to Proposition 3.1 of C121, we have
P1. J.( E,*) - P(E,*)Gij = -2E!.l(R){-( 2i -axai vJ. + a Vi) - -n1v x*v I + O(E 2 ),
j
a 2
qJ.( E,') = - EK(e) ?&-e + o(E ) I
j
Thus the fluid dynamic quatitites {p,v,el obtained from the density
f(E,t,x,5) given in Theorem 1.2 will satisfy the compressible Navier-Stokes
equation with the error of O(E~). A more delicate treatment will be given
elsewhere.
To solve the equation (1.6) we use several function spaces and norms
(cf. C161).
We introduce these spaces. All functions are measurable or continuous.
(1.11) 3 u(k,O +===c.
.
I'Ia,~,~ = sk,ucp Rn e'(" l k' (1+1 kl f(l+[5[ )B\u(k,E)I <
Fluid Dynamical Limit of the Boltzmann Equation
Here x(l kl+lSI>R) is the characteristic function of the set {(kyS)cR$Rn;
0
(k(+lS(>Rl. With a Banach space X, B (D;X) denotes the space of X-valued,
bounded and continuous functions defined on D. We put RT = CO,lIxCO,Tl,
RT* = RT \ i(O,O)> and RT = {(E,T); (E,ET)ER~~F. or m = (mlym2)y ml 2 0,
m2 2 0 (resp. m 2 0 ),
(1.14) Z!$;i'* :B myy(RT* ;X"R ,B ) is defined similarly.
Theorem 1 .l. Let g be an absolute Mmellian and let
a > 0, 9. > n+l, B 5 1.
6 Kiyoshi ASANO and Seiji UKAI
Then there exist positive numbers a1,b0,b0 and bQ such that for each initial
data fo = g + g1/2uo satisfying
the following statements hold with constants Y > 0, T > 0 (a-yTzO) and u > 0.
lil For each E E (0,1],(1.3) (resp.(l.6)) has a unique solution
f(E,t,X,c) (resp. Q(E,t,k,S)) on the time interval [O,T], and there hold
f = g + g1/2u,
U(E,t) = u0 (E,t) + $(E,t/E) + E“l’*(E&)
1/2 0
liil For t E (O,T], f(O,t,x,S) = g(5) +g(E) u (O,t,x,S) is a ZocaZ
Mamellian whose fluid dynamical quantities Cp,v,Ol are the soZution of
the compressible Euler equation (1.10) with P = PI and q = 0.
liiil Moreover, there hold
A“0 ‘ ‘k,lBP,T ,Y ’ 11 A0 IIl,a,y,L,B,T ’ bb IQOla,L,B1
Theorem 1.2. Let g be an absolite Maxuellian and let
a > 0, II > n+3, B 2 2.
Then there exist positive numbers a2,bj, ’bj (j=O,l), bt, yj, aj
-
(j=O,l, a. = a, a,-y OT 2 a,, al ylT 2 0) ando such that for each initial
data fo g + g1/2uo satisfying
3
Fluid Dynamical Limit of the Boltzmann Equation 7
the solution f(E,t.x,S) of (1.3) (resp. O(E,t,k,C) of (1.6)) is described in
the following formula
U(E,t) = u0 (E,t) + “U(E.t/E) .t EU1 (E,t) .t Eij1 (E, t/E) + E2U2’*(E,t),
€a-aEa 2,* (€,t
We note that if 0 c X;,B , then u(x,E) is analytic in x E Rn + Bi,,
Ba = {y E Rn ; IyI < a 3, and uniformly bounded on Rn + iE6, 0 < 6 < CL .
According to the results of Theorem 1. l,w e put
f0(€,t,X,5) = g(5) + g(E)1’2Uo(E,tsX,5)
9
(1.16) P0 (E,t,/E,x,S) = g(5) 1/2 -u0 ( E,t/&.x,S) ,
.
fl’*(E.t,X,S) = g(5) 1/2u19* (E ,t ,x ,5 )
Then we have the desired formula (1.2).
Similar expansion formula can be estableshed using the results of Theorem 1.2.
i0
Considering that fo, and fl’* are analytic in x E Rn .t iBa,yt for
0 < t < T, our existence theorem is of Cauchy-Kowalewski type ([8],[91). We
hope to find more natural existence theorems.
2. Some estimates
Denoting the unknown by u(k,S) instead of ^u(k,S), we write (1.6) as