Applied Mathematics, 2010, 1, 504-509
doi:10.4236/am.2010.16066 Published Online December 2010 (http://www.SciRP.org/journal/am)
C 0Approximation on the Spatially Homogeneous
Boltzmann Equation for Maxwellian Molecules*
Minling Zheng
School of Science, Huzhou Teacher College, Huzhou, China
Received August 25, 2010; revised October 15, 2010; accepted October 19, 2010
Abstract
In this paper we study the viscosity analysis of the spatially homogeneous Boltzmann equation for Maxwel-
lian molecules. We first show that the global existence in time of the mild solution of the viscosity equation
(,)
tv
f
Qf ff

 

 . We then study the asymptotic behaviour of the mild solution as the coefficients
0


, and an estimate on 0
f
f

 is derived.
Keywords: Viscosity Boltzmann Equation, Mild Solution, Viscosity Approximation, Collision Kernel
1. Introduction
In this paper we shall investigate the asymptotic
properties of the solution of the viscosity
Boltzmann equation for Maxwellian molecules

,
tv
f
Qf ff

 
   in
 
3
0, R (1)
as the viscosity coefficients 0


. Here, (,)Qf f
is the Boltzmann collision operator for Maxwellian
molecules defined by its quadratic form
 

3**
0
*
,2
cos sin
R
Qfgfg fg
bddv



 

where function b is nonnegative and continuous,
and
 

1
cos sin0,bL

 . Here the shorthand

,'
f
ftv
,

**
,
f
ftv

 are used; *
',vv
 are the
post-collisional velocities corresponding to the
pre-collisional velocities * ,vv respectively, which
submit to the elastic collision law


*
**
',
',
vvvv
vvvv




 



 (2)
where (,) denotes the scalar product. 2 S   the 2-D
unit sphere and (')/|'|vv vv

 .

is the angle
between * vv and

,


0,

 . On physically, Q
satisfies the symmetrization and translation invariance.
For Maxwellian potential Q can be split into Q

and
_
Q:




,,,QffQ ffQ ff


 
32 **
,cos
RS
Qffffb ddv 







 
32 **
,,
cos
RS
Qff fLf
Lffb ddv 




The problem of viscosity approximation of the spa-
tially homogeneous Boltzmann equation, namely wheth-
er the solution of (1) converges to the solution of the
equation


,
t
f
Qff in
 
3
0, R (3)
as 0


, is very interested for mathematical theory of
Boltzmann equation as well as practical applications. We
know that the energy of the solution of (1) is increasing
with the time t due to the diffusion effect. We cannot
expect that the so lu tion o f (1) approaches to the Maxwel-
lian equilibrium in large time. This observation has re-
cently been shown by Li-Matsumura [1]. In early work
of the authors an explicit estimate of
f
f

 in 1
k
L
was derived which indicates also the dependence of time
[2]. It must be stressed this result excludes the case of
Maxwellian molecules. Actually, the produce of mo-
ments for cutoff potential is not valid for Maxwellian
molecules. In this paper we shall study the viscosity ap-
proximation for Maxwellian molecules. Our goal is to
*This work was supported by Huzhou Natural Science Foundation
(2008YZ06) and Innovation Team Foundation of Department of Educa-
tion of Zhe
j
ian
g
Province
(
T200924
)
.

M. L. ZHENG
505
study the existence and uniqueness of the global solution
of the viscosity equation (1) in time, and to estimate
f
f

 explicitly in 0 C-norm. The new tool is the
Gagliardo-Nirenberg inequality.
Let us mention some works about the spatially homo-
geneous Boltzmann equation with cutoff potential, see
[3-11] for example. For the Maxwellian molecules Mor-
genstern first deduced the existence and uniqueness of
the solution in 1
L space [12]. We also remark that the
approximation with diffusion term in velocity variable
was present in the work of DiPerna-Lions [13].
Now we complement the equation (1) and (3 ) with the
same initial condition:
3
00
||(),
tt
f
fvvR



 . (4)
In the sequel we always assume that
13
0() ()vLR   . (5)
It must be emphasized that the nonnegative hypothesis
of ()v

is not necessary in present paper.
In the following we denote the m Cnorm by ||
m

, and




3
,
0||
max maxsup
mjj
jm jvR
f
fandf Df

  

Here

is the multi-index.
This paper is organized as follows. We introduce a
mild solution to the Cauchy problem (1) and (4) in Sec-
tion 2. We prove local existence of the mild solution by
the contracted mapping principle. In Section 3, we pro-
pose the global existence of the mild solution. Our main
tool is the interpolation inequalities. Finally, we study
the 2,
p
Westimate of
f

in Section 4 and deduce the
following asymptotic expression
0
kt
f
fAe

  .
2. The Local Existence
In this section we shall study the local existence of the
solution of the C auchy problem (1), (4).
Definition 1. Given 0

. We call
f

is the mild
solution to the Cauchy problem (1) and (4), if





1, 3
0, ;1
p
fCWRp

  and satisfies



 
33
0
,0
,, ,
,, 0
t
RR
vt
ftvGtvdGtsv
Qf fsdsdt




 





 





(6)
where


2
3/2
1||
,exp,0
4
4
tv
GGtv t
t
t








The following is the local existence theorem.
Theorem 1. Given 0

. Let 1, 3 ()
p
WR  ,
1p

 and satisfy (5). Then there exists 0T such
that the Cauchy problem (1) and (4) has a unique mild
solution ,0
f
tT


 .
In order to prove Theorem 1, let us recall a
well-known result which is often called convolution
property.
Proposition 2 ([10,14]). For any 1p, if
13 3
(), ()
p
f
LRgL R then there exist constant 0C
dependent on bonly, such that

 1
,p
p
L
L
L
QgfCg f
The proof of Theorem 1 Consider the following space






1, 3
0, ;p fCTW R


T is determined. Defined the mapping:
f
f


 by



  
0
,0
,**,,0
t
tv tsv
vt
ftvGvGQffvdst











(7)
where *v denotes the convolution in variable v.
By


13
LR   and the definition of the mild solu-
tion (6), we have
 1 11
0,
t
L LL
f
Qf fds

 

Making use of the Prop. 2 and Gronwall’s lemma, we
obtain the estimate of 1
L
f

.
In terms of (7), we denote

,0ftvt

 by 1
I
and
2
I
. Obviously,
1, 1,
1
p
p
WW
I  
By Prop. 2 and Young’s inequality, noting that
11
tsL
G

, one obtain
 11
20,
pp
p
t
ts
LL LL
L
I
GQffdsCfft
  


Here the nonnegative constant C depends on b
only. In what following, we denote C for various non-
negative constants independent of

unless special
statements. On the other hand,

1
20,
pp
t
ts
LL
L
I
GQffds
 



1
11
22
0p
t
L
L
Ctsffds





1
11
22
p
LL
Cfft



. (8)
Therefore,

M. L. ZHENG
506
1, 1,
1
2
2p
p
WW
ICttf


 




This follows the mapping  is closed. Let
12
,ff, by the bilinearity of Q
  
112 2112
,, ,Qf fQffQf ff


 


1221 12122
,Qfff fLffffLf

Thus
 
112 212
,,
p
p
L
L
Qf fQffCff
 
1
2112 2
,,
p
L
Ctt QffQf f 
 
 



1,
1
212
p
W
CTTff 
 
 


 (9)
So, one deduces that the mapping  is locally Lip-
schitz continuous. By choosing 0T suitably, such
that 0tT , the Cauchy problem (1) and (4) exists a
unique mild solution.
3. The Global Existence of the Wild Solution
In order to prove the global existence of the mild
solution above, it suffices to show that
1, ,0 p
W
ft

 
First, let us recall the N dimensional Gagliar-
do-Nirenberg’s inequality: let 1,qr, j, m
are integer s and 0jm . Suppose that ,1
j
am






,
(1a if /mjNr is a nonnegative integer).
Then there exists a constant C dependent on q, r, j,
m, a, N such that for any u

D ()
N
R,
1
q
pr
a
a
L
LL
jm
Du CDuu









where 111jma
a
pm rNq


 


Lemma 3. Given 0

. Let 1, 3 ()
p
WR  , 1p


and satisfy (5). Then


0, 0,Tt T , the solution of
Cauchy problem (1) and ( 4) satisfies

,p
pLL
fCT

 
Proof From (1), we have
3
1
1sgn
p
pp
t
LR
d
f
ff fdv
pdt




 
3
1
sgn ,
p
R
f
ffQffdv
 





3
22
1p
Rpffdv




 


3
1
12
sgn ,
p
R
f
fQffdvII
 

 (10)
Obviously 10I

. By Prop. 2 and Holder’s inequality
 
3
11
,,
p
p
p
p
L
L
RfQffdvQff f
 




1
p
p
p
p
L
LL
Cf fCf
 
 (11)
By these estimates above and Gronwall’s lemma, we
derive

,p
pLL
fCT

 
This finishes the proof of the lemma.
Lemma 4. Given 0

. Let

1, 3p
WR  , 1p


and satisfy (5). Then


0, 0,Tt T , the solution of
Cauchy problem (1) and ( 4) satisfies

1,
,pp WL fCT

  .
Proof In terms of (6), for any 0t
 
0
,* *,.
t
tts
f
tvGGQ ffds





Therefore,

0
**,
t
tts
f
GGQffds



 
 (12)
By Young’s inequality

10,
p
pq
r
t
tts
LLLL L
f
GGQffds



 
 (13)
where
111
1pqr

, 1,,pqr.
Next we estimate q
ts
L
G and

,
L
r
Qf f

re-
spectively. Noting that
213
2
2,
2
z
zeez R

.
Thus
2( )
ts ts
v
GG
ts 


 

2
31 1
22 2
z
ts ze 
 

 
13111
22222
2
2etsCts

 
 (14)
Therefore

1
11 1
12
qqq
tsts ts
LLL
GGGCts




  (15)

M. L. ZHENG
507
By Gagliardo-Nirenberg’s inequality,
  

,, ,
rp p
a
LL L
QffQf fQf f
 


1
,
p
a
L
Qf f

 (16)
where
1111
,0 1
3
a
aa
rp p


 


By the translation invariance of Q, it is easily to show
that

,,,Qf fQf fQff



So making us of Prop. 2 again



11
,.
pp
pLLL L
L
Qf fCffff


(17)
By (12), it gives
 1 1111
0,
t
tts
LLLLL
f
GGQffds



 

11
0
t
LL
Cfds

  
 (18)
that is

1
1,LL
fCT

  (19)
Plugging (19 ) i nt o ( 17 ), gi ves



,pp
pLL
L
Qf fCff
 
 . (20)
By (20) and (16),



1
,
p
pp
r
aa
L
LL
L
Qf fCfff
 



1
p
pp
aaa
L
LL
Cfff


 (21)
Combining (15), (21) and (13), we deduce



11
2
0
p pppp
taaa
LLLLL
f
Ctsfffds


 


(22)
According to the Lemma 3, one has
 
11
22
00 .
p p p
tt
a
L LL
f
Cts dsCtsfds
 




(23)
By Gronwall type inequality we obtain the desired re-
sult.
Next using the basic theory of parabolic equation and
the a priori estimate above, we have the following theo-
rem.
Theorem 5. Gi ven 0

. Let

1, 3p
WR  , 1p


and satisfy (5). Then for any 0T the Cauchy
problem (1) and (4) exists a unique mild solution
f

such that







1, 33
0, ;0,
p
f
CTWR CTR



4. 2,
p
W Estimate and 0 C Approximation
In this section we shall make 2,
p
Westimate on
f

and
deduce the explicit estimate on the viscosity approxima-
tion.
Theorem 6. Given 0

, 2p . Then for any


2, 3p
WR   satisfying (5) the mild solution
f

of
the Cauchy problem (1) and (4) belongs to






2,3
0, ;p CWR.
Proof By the equation of (1), one has
 
,
f
fQff
t




. (24)
Thus
 
3
1
1sgn
p
pp
t
LR
d
f
ff fdv
pdt


 

  


3
1
sgn ,
p
R
f
ffQffdv
 





12
I
I

 (25)
Next we estimate 1
I
and 2
I
respectively.
 
3
1
1sgn p
R
I
ff fdv





 
3
2
2
1p
Rpf fdv




 (26)
and
 
3
1
2sgn ,
p
R
I
ff Qffdv
 



  
3
2
1,
p
Rpf fQffdv


 

  
3
22
22
1,
pp
Rpf fQfffdv


 

(27)
By Young’s inequality, for any 0

,
 
3
2
2
21p
R
I
pf fdv






3
22 1,
4
p
R
pQf ffdv
 



 
 (28)
Employing Young’s inequality again, the second term
of the above formulation can be estimated by


 
33
12
1,
24
pp
RR
pp
pQf fdvfdv
pp
 


 

(29)
Taking 4



, and plugging (28), (29) and (26) into (25), it
gives

M. L. ZHENG
508


3
2
2
31
1
4
p
pp
LR
p
d
f
ffdv
pdt




 








33
211 2
,pp
RR
ppp
Qf fdvfdv
pp
 


 

(30)
By (20) and Gr onwall’s lemma, and the Schauder the-
ory, we conclude the desired result.
Now, we consider whether the mild solution of the
Cauchy problem (1) and (4) converges to the solution of
(3) and (4) in 0 C-norm as 0 

. The following
theorem is our main result.
Theorem 7. Let 22,3 ()
p
CWR   and satisfying
(5), 2p. For any 0Tand 0

, set
f

is
the mild solution of the viscosity equation
 

3
3
0
,0,
|
t
t
f
fQffinTR
finR
 




 



 (31)
and
f
is the solution of
 

3
3
0
,0,
|
t
t
f
QffinT R
finR


 



 (32)
Then,
 
0
,, kt
f
tft Ae

   (33)
where
A
and k are constants independent of

.
Furthermore, for any 0



0
,ft f


 (34)
if
1
0minlog ,tT
kA




 



Proof By the theorem abov e and the result of spatially
homogenous Boltzmann equation, we know that


2, 3
,, p ftvfWR

.
Let wf f

, then
 
,,wfQffQff
t


 

Therefore,
 
  
||sgn sgn
,,.
wwf w
t
Qf fQff






(35)
Noting that
 
0
,,Qf fQff



0
,,Qff fQf ff
 



1 1
0LL
Cf fff

 (36)
By the estimate on 1
L
f

in Section 2, we know the
estimate (36) is uniform in

. Therefore




0
,,Qf fQffCw

 (37)
Next, we estimate 2
f

. Indeed, by (6) for any multi-
indices

, 2 

, we have
 
3
00
,
R
DfGt Dvd



 

  
3
00
,,,
t
RGtsDQ ffsvdsd



  

12
I
I

 (38)
noting that

3,1
RGtvdv

, 12
I

. Making use of
Leibniz form ul a, for 2 



||2
,,DQ ffQD fDf











 (39)
Thus


11,1
2
22
0
t
LW
I
Cfffds
 

. (40)
Noting that
 1 11
0,
t
L LL
fQffds

 

So,
2 22
0
t
f
CfdsCT

  
 (41)
By Gronwall’s inequality we can deduce the bound of
2
f

and the bound is independent of

.
Together these estimate with (35) we have
 
020
3,,wfQffQff
t


 

0
CCw   (42)
Therefore,


0
kt
wt Ae  
This finishes the pr o of o f T h eor em 7.
5. Acknowledgements
I am grateful to the reviewer for the meaningful sugges-
tions.
6. References
[1] H. L. Li and A. Matsumura, “Behaviour of the Fokker-
Planck-Boltzmann Equation near a Maxwellian,” Archive
for Rational Mechanics and Analysis, Vol. 189, No. 1,
2008, pp. 1-44.
[2] M. Zheng and X. P. Yang, “Viscosity Analysis on the

M. L. ZHENG
509
Spatially Homogeneous Botzmann Equation,” Asymptotic
Analysis, Vol. 53, 2007, pp. 13-28.
[3] L. Arkeryd, “On the Boltzmann Equation,” Archive for
Rational Mechanics and Analysis, Vol. 45, 1972, pp.
1-34.
[4] L. Arkeryd, “Stability in L1 for the Spatially Homogene-
ous Boltzmann Equation,” Archive for Rational Mechan-
ics and Analysis, Vol. 103, 1988, pp. 151-167.
[5] T. Elmroth, “Global Boundedness of Moments of Solu-
tion of the Boltzmann Equation for Forces of Infinite
Range,” Archive for Rational Mechanics and Analysis,
Vol. 82, 1983, pp. 1-12.
[6] L. Desvillettes, “Some Applications of the Method of
Moments for the Homogeneous Boltzmann Equation and
Kac Equation,” Archive for Rational Mechanics and
Analysis, Vol. 123, 1993, pp. 387-395.
[7] S. Mischler and B. Wennberg, “On the Spatially Homo-
geneous Boltzmann Equaion,” Annales de l’Institut Henri
Poincar´e - Analyse non lin´eaire, Vol. 16, No. 4, 1999,
pp. 467-501.
[8] B. Wennberg, “Entropy Dissipation and Moment Prod-
uction for the Boltzmann Equation,” Journal of Statistical
Physics, Vol. 86, 1997, pp. 1053-1066.
[9] B. Wennberg, “Stability and Exponential Convergence in
Lp for the Spatially Homogeneous Boltzmann Equation,”
Nonlinear Analysis, Theory, Methods & Applications,
Vol. 20, 1993, pp. 935-964.
[10] C. Mouhot and C. Villani, “Regularity Theory for the
Spatially Homogeneous Boltzmann Equation with Cut-
off,” Archive for Rational Mechanics and Analysis, Vol.
173, No. 2, 2004, pp. 169-212.
[11] C. Mouhot, “Rate of Convergence to Equalibrium for the
Spatially Homogeneous Boltzmann Equation with Hard
Potentials,” Communications in Mathematical Physics,
Vol. 261, 2006, pp. 629-672.
[12] D. Morgenstern, “General Existence and Uniqueness
Proof for Spatial Homogeneous Solution of the Maxwell-
Boltzm-ann Equation in the Case of Maxwellian Mole-
cules,” Proceedings of the National Academy of Sciences,
Vol. 40, 1954, pp. 719-721.
[13] R. J. DiPerna and P. L. Lions, “On the Fokker-Planck-
Boltzmann Equation,” Communications in Mathematical
Physics, Vol. 120, No. 1, 1988, pp. 1-23.
[14] I. M. Gamba, V. Panferno and C. Villani, “On the Boltz-
mann Equation for Diffusively Excited Granular Media,”
Communications in Mathematical Physics, Vol. 246, No.
3, 2004, pp. 503-541.