Mathematical Methods in the Applied Sciences
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
MOS subject classification: 35 Q 99, 82 D 10, 85 xx; 76 X 05
Long-time Behaviour for Solutions of the
Vlasov–Poisson–Fokker–Planck Equation
Ana Carpio*
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain
Communicated by Y. Shibata
We study the long-time behaviour of solutions of the Vlasov—Poisson—Fokker—Planck equation for initial
data small enough and satisfying some suitable integrability conditions. Our analysis relies on the study of
the linearized problems with bounded potentials decaying fast enough for large times. We obtain global
bounds in time for the fundamental solutions of such problems and their derivatives. This allows to get
sharp bounds for the decay of the difference between the solutions of the Vlasov—Poisson—Fokker—Planck
equation and the solution of the free equation with the same initial data. Thanks to these bounds, we get an
explicit form for the second term in the asymptotic expansion of the solutions for large times. ( 1998 B. G.
Teubner Stuttgart—John Wiley & Sons, Ltd.
KEY WORDS: Vlasov—Poisson—Fokker—Planck; long-time behaviour; fundamental solutions
0. Introduction
The simplest mathematical description of the state of a stellar system or a rarefied
plasma is based on collisionless kinetic models, the Liouille—Newton system or the
Vlasov—Poisson system in case the induced magnetic fields vary slowly.
The model of collisionless plasmas, specially in controlled fusion or laser fusion, is
too idealized and collisional effects need to be incorporated. A way to do that is to
model the motion of an individual particle as Brownian motion caused by collisions
with the background. The resulting system of mathematical equations is the stochastic
Langevin system
dx"v dt,
dv"(E(x, t)!bv) dt#J2p db,
where E is the electrostatic or gravitational field, b is a viscosity parameter, p a thermal diffusion parameter and b denotes the standard N-dimensional Brownian motion.
*Correspondence to: A. Carpio, Departamento de Matemática Aplicada, Universidad Complutense de
Madrid, Madrid 28040, Spain
Contract grant sponsor: DGICYT; contract grant number: PB93-1203
CCC 0170—4214/98/110985—30$17.50
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Received 9 December 1996
986
A. Carpio
The Vlasov—Poisson—Fokker—Planck equations result when one incorporates the
above Langevin system into the Vlasov equation in order to determine the dynamic
behaviour of the expected distribution of particles with respect to position and
momentum. Letting f (x, v, t) denote this distribution at time t we get the Vlasov—
Poisson—Fokker—Planck system
G
f !p* f#v+ f#(E( f )!bv)+ f!NBf"0, in RN]RN]R`,
t
v
x
v
f (x, v, 0)"f (x, v),
in RN]RN,
0
where E(o( f ))"e(K * o( f )) with o( f )":RN f (x, v, t) dv, and K(x)"x/S DxDN, S
N
N
x
being the area of the sphere in RN. We shall be concerned with the case N"3, b"0.
The parameter e"$1 depending on whether the interaction between the particles is
electrostatic or gravitational.
Some existence results for (VPFP) are known. Let us review the main ones. Degond
proved in Reference 9 existence of smooth global solutions in dimensions one and
two. Later, Triolo [16] proved global existence of smooth solutions for small initial
data in the class f 3¸1W¸=, 0)f )Ah(x)g(v) in dimension N*3. For small f he
0
0
0
proved global existence and some decay estimates
(VPFP)
C
DDE( f )DD¸=)
,
(1#t)N~1
C
DDo( f )DD¸=)
.
(1#t)N
The proof uses an iterative method and relies on the decay given by the dispersive part
of the equation to get uniform bounds.
Victory and O’Dwyer [18] proved global existence of classical solutions when
N)2 and local existence when N*3 for initial data
f 3C1W¸1 , (1#DvD2)c@2( f #D+ f D #D+ f D )3¸= , c'3.
0
b
xv
0
x 02
v 02
xv
For such data E( f )3¸= , +E( f )3¸= . The key point in their proof is the construction
xt
xt
of a fundamental solution for the linear, degenerate, parabolic like problem. In
Reference 17, Victory proved the existence of global weak solutions when n)3 and
f 3¸1W¸=,
0
P
(DxD2#DvD2) f (R.
0
Weckler and Rein [15] gave the sufficient conditions for the solutions constructed in
Reference 18 to be global: assuming the field E( f ) associated to f to decay fast
0
enough (like (1#t)~a with a'1), if for f we have a global solution and f !g is
2
0
0
0
‘small’ enough in an appropriate norm, then we get for g a global solution with
0
E decaying like (1#t)~1. The decay rate for E is therefore improved. The proof relies
on an estimate for large times of the fundamental solution in terms of the fundamental
solution of the linear problem with E"0 and a perturbation technique.
Bouchut [2] constructed global solutions when N"3 and
f 3¸1W¸=,
0
P
DvDm f dv dx(R, m'6.
0
Moreover, E( f )3¸= . This gives conditions for the solutions in Reference 18 to be
xt
global. In Reference 3 some further results on the smoothness of solutions are given.
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
987
Carrillo and Soler [5, 6] proved the existence of global weak solutions for initial
data in ¸1W¸p and, when N"3, for small data in Morrey spaces. For initial data f in
0
¸1, f *0 with S ( f )"sup
DDo DD (R and Max(sup
DDo DD , DD f DD )
0
9@4 0
j;0 0,j L9@4
j;0 0,j L9@4 0 1
small enough they get a unique global weak solution satisfying
(H) S ( f (t)))C,
DDE(t)DD¸=)Ct~1@2.
x
9@4
Carillo, Soler and Vazquez were the first to determine the asymptotic profile of the
solutions f to (VPFP) for large times. In Reference 8 they prove that the long-time
behaviour of weak solutions f satisfying condition (H) for large times is given by the
fundamental solution of the linear problem with E"0 with mass : f dx dv provided
0
that
f (1#DxD2#DvD2#Dlog f D)3¸1(R6),
E( f )3¸2(R3)3
0
0
0
and f 3¸9@7 in the case of gravitational forces, i.e. e"!1. Therefore, the first term in
0 xv
the asymptotic expansion of f for large times is determined by the free equation and
the mass of the inital data. Their study relies on scaling techniques.
We shall study here the long-time behaviour of solutions to (VPVP) when N"3
from another point of view. Our analysis is based on the obtainment of global
estimates on the fundamental solutions of linearized problems and its derivatives in
which the fundamental solutions and its derivatives are bounded in terms of the
fundamental solutions of the linear problem where the potential E"0. Once these
estimates are obtained we can get optimal decay rates on the remainder r(t) in the
expansion f (t)"G( f )(t)#r(t), where f (t) is the solution of (VPFP) with datum
0
f and G( f ) is the solution of the free equation with datum f , provided that f satisfies
0
0
0
0
some integrability conditions. Then, we can find explicit expressions for the second
term in the expansion for large times. In this term the influence of the non-linearity
appears. This result clarifies the deviation of ‘strong’ solutions of (VPFP) from the
solutions of the free equation.
The paper is organized as follows. In section 1 we get the decay estimates for the
solutions of the lineaerized (VPFP) problem with E"0. In section 2 we get the decay
estimates for the solutions of the linearized (VPFP) problem with a potential
E3¼1,=. In section 3 we study the long-time behaviour of solutions to (VPFP) for
initial data small enough and satisfying suitable integrability conditions. For such
data we improve the results obtained in Reference 8. Our study relies mainly in the use
of the estimates on fundamental solutions obtained in section 2.2. In the following we
shall take p"1.
1. Linear problem with E50
It is useful to think about VPFP as a perturbation of the linear problem with E"0,
that is,
(P )
0
G
g !* g#v+ g"0 in R3]R3]R`,
t
v
x
g(x, v, 0)"g (x, v)
in R3]R3,
0
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
988
A. Carpio
This equation is hyperbolic in the x variable and parabolic in the v variable. An
explicit fundamental solution is known (see References 13 and 14)
G(x, v, t; m, l, q)"G(x!m, v, l, t!q)
G
H
33@2
3Dx!m![(t!q)/2] (v#l)D2 Dv!lD2
"
exp !
!
(2n)3(t!q)6
(t!q)3
4(t!q)
with x, v, m, l3R3 and t'q*0.
1.1. Some known properties of the fundamental solution
Lemma 1. (i) For some positive constants m , m , M , M
1 2 1 2
M
x!m!(t!q)lD2
1 exp !
G(x, v, t; m, l, q) dv"
,
(t!q)9@2
m (t!q)3
R3
1
M
Dv!lD2
2 exp !
G(x, v, t; m, l, q) dx"
,
(t!q)3@2
m
(t!q)
R3
2
P
P
P
P
R6
R6
P
G(x, v, t; m, l, q) dx dv"
R6
G
G
H
H
G(x, v, t; m, l, q) dm dv"1,
G(x, v, t; m@, l@, q@)G(m@, l@, q@; m, l, q) dm@ dl@"G(x, v, t; m, l, q)
with x, v, m, l, m@, l@3R3 and t'q@'q*0.
(ii) ¹he following estimates hold for some positive constant C:
D+ G(x, v, t; m, l, q)D)C
v
G(x/2, v/2, t; m/2, l/2, q)
,
(t!q)1@2
D+ G(x, v, t; m, l, q)D)C
l
G(x/2, v/2, t; m/2, l/2, q)
,
(t!q)1@2
D+ G(x, v, t; m, l, q)D)C
x
G(x/2, v/2, t; m/2, l/2, q)
,
(t!q)3@2
D+ G(x, v, t; m, l, q)D)C
m
G(x/2, v/2, t; m/2, l/2, q)
,
(t!q)3@2
D* G(x, v, t; m, l, q)D)C
v
G(x/2, v/2, t; m/2, l/2, q)
,
(t!q)
D+ + G(x, v, t; m, l, q)D)C
l v
G(x/2, v/2, t; m/2, l/2, q)
,
(t!q)
D+ + G(x, v, t; m, l, q)D)C
x v
G(x/2, v/2, t; m/2, l/2, q)
(t!q)2
and so on.
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
989
Proof. Part (i) is obtained by integrating the explicit expression for G. Part (ii) follows
estimating the explicit expressions for the derivatives.
1.2. Decay of solutions
Once the fundamental solution is known, for any solution of the linear problem
with initial datum g 3¸1(R3]R3)
0
g !* g#v+ g"0 in R3]R3]R`,
t
v
x
(P )
0 g(x, v, 0)"g (x, v)
in R3]R3,
0
we have the integral expression
G
PP
g(x, v, t)"
G(x, v, t; m, l, 0)g (m, l) dm dl"G(g ).
0
0
We are interested in positive solutions. Since G*0, we have a maximum principle.
Thus, for positive initial data we get positive solutions. The decay estimates can be
extended to negative data changing signs.
Lemma 2. ¹he following decay estimates hold:
(E1) DDG(g )DD¸p (¸1))Ct~(3@2)(1@r~1@p) DDg DD¸p (¸1) , p*r,
0 v x
0 v x
C
DDg DD¸a (¸r ) , p*a, q*r,
(E2) DDG(g )DD¸p (¸q ))
0 v x
t(3@2)(~3@q`3@r~1@p`1@a) 0 v x
(E3) DDG(g )DD¸p )DDg DD¸p , 1)p)R,
0 xv
0 xv
DDo DD
(E4) DDG(g )DD¸p (¸1))Ct~(9@2)(1@r~1@p) Sup
j;0 0,j ¸rx
0 x v
)Ct~(9@2)(1@r~1@p) DDg DD¸1 (¸r ) , p*r.
0 v x
Proof. The estimates follow from the integral formula taking into account
Lemma 1(i):
(E1):
P
PPAP
G(g ) (x, v, t) dx"
0
PP A
B
A B AP
Dv!lD2
exp !
g (m, l) dm dl
0
m t
2
M
D ) D2
" 2 exp !
g (m, ) ) dm ,
0
t3@2
m t *v
2
M
" 2
t3@2
B
G(x, v, t, m, l, 0) dx g (m, l) dm dl
0
B
where 1/r#1/r@"1#1/p. Therefore,
K
A
BKK
C
D ) D2
DDG(g )DD¸p (¸1))
exp !
DDg DD¸r (¸1)
0 v x
t3@2
m t ¸r{ 0 v x
v
2
)Ct~(3@2)(1@r~1@p) DDg DD¸r (¸1) , p*r,
0 v x
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
990
A. Carpio
(E2):
KP
DDG(g )DD¸p (¸q ))
0 v x
KP
K
)
dl DDG( ) , v, l, t) * g ( ) , l)DD¸q
x
x 0
dv DDG( ) , v, l, t)DD¸r{ DDg ( ) , l)DD¸r
x
x
0
G
H
K
K
¸p
v
¸p
v
K
C
Dv!lD2r@
exp !
*v DDg0 ( ) , l)DD¸rx
t(3@2)(4~3@r{)
m t
¸p
v
2
C
)
DDg DD¸a (¸r ) , p*a, q*r.
t(3@2)(~3@q`3@r~1@p`1@a) 0 v x
)
Therefore,
C
DDg DD¸a (¸r ) , p*a, q*r.
DDG(g ) DD¸p (¸q ))
0 v x
t(3@2)(~3@q`3@r~1@p`1@a) 0 v x
(E3): Since
P
R6
P
G(x, v, t; m, l, q) dx dv"
R6
G(x, v, t; m, l, q) dm dv"1,
we have
DDG(g )DD¸1 )DDg DD¸1 ,
0 xv
0 xv
Interpolating
DDG(g )DD¸=)DDg DD¸= .
0 xv
0 xv
DDG(g )DD¸p )DDg DD¸p ,
0 xv
0 xv
(E4):
P K
P K
G
G
H
H P
K
M
D !tlD2
1 exp ! )
DDG(g )DD¸p (¸1)) dl
*x g0 ( ) , l)
0 x v
t9@2
m t3
¸p
x
1
C
D ) D2
) dl
exp !
g ( ) !tl, l) dl
t9@2
m t3 *x 0
1
)Ct~(9@2)(1@r~1@p)
KP
g ( ) !tl, l) dl
0
K
K
¸p
x
, p*r.
¸r
x
In view of this, we introduce
Sup
j;0
KP
g ( ) !jl, l) dl
0
K
"Sup
DDo DD ,
j;0 0,j ¸rx
¸r
x
where o ": g ( ) !tl, l) dl, so that
0
0,j
DDG(g )DD¸p (¸1))Ct~(9@2)(1@r~1@p) Sup
DDo DD
0 x v
j;0 0,j ¸rx
)Ct~(9@2)(1@r~1@p) DDg DD¸1 (¸r ) , p*r.
0 v x
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
991
Remark 1. Using estimate (E4) we get for g *0
0
DDo(G(g )) (t)DD¸p)C(DDg DD¸1(¸r ) )t~9@2(1@r~1@p), p*r.
x
0
0 v x
Thanks to this estimate and the properties of kernel K (see Reference 12 for the first
inequality) we get
DDE(o(G(g ))) (t)DD¸p)CDDo(G(g )) (t)DD2@3
¸= DDo(G(g ))) (t)DD1@3
¸1
x
x
x
0
0
0
)C(DDg DD¸1 , DDg DD¸1(¸r ) )t~3@r, 1)r)R.
0 xv
0 v x
As we can see, the field E decays very fast, it may reach the decay rate t~3 for initial
data in ¸1. When g 3¸1 (¸r ) the decay rate for large times is worse but the singularity
0 v x
at zero is smaller. For g satisfying more integrability conditions we can reduce the
0
singularity at time zero.
Remark 2. Differentiating the integral expression for G(g ) we get the following
0
integral expression for the derivatives:
PP
­ G(x, v, t; m, l, 0)g (m, l) dm dl"­ G(g ).
xi
0
xi
0
Using the estimates on the derivatives of G (Lemma 1(ii)) we obtain
P
P
D­ G(g )D) D­ G(x, v, t; m, l, 0)D Dg (m, l)D dm dl
xi
0
xi
0
)
G(x/2, v/2, t; m/2, l/2, 0)
Dg (m, l)D dm dl.
0
t3@2
Thus, we get the same decay estimates as for DDG(g )DD with an extra decay factor t~3@2.
0
For ­ G(g ) we get an extra decay factor of order t~1@2, and so on.
v
0
2. Linear problem with EO0
We deal here with the problem
G
g !* g#v+ g#E(x, t)+ g"0 in R3]R3]R`,
t
v
x
v
g(x, v, 0)"g (x, v)
in R3]R3
0
with g 3¸1(R3]R3).
0
(P )
E
2.1. E bounded and E ¸ipschitz
When DDEDD¸=(R3][0, ¹ ]))C and DD+ E DD¸=(R3][0, ¹ ]))C a classical fundamental soluxt
xt
x
tion ! (x, v, t; m, l, q) to problem P is known to exist [18]. This function is defined to
E
E
be the only function such that
(1) For fixed (m, l, q)3R3]R3]R`, ! as a function of (x, v, t) satisfies the equation
E
g !* g#v+ g#E(x, t)+ g"0 in R3]R3](q, ¹ ].
t
v
x
v
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
992
A. Carpio
(2) For every continuous and bounded function g (x, v)
0
lim
t?q
PP
! (x, v, t; m, l, q) f (m, l) dm dl"f (x, v).
E
0
0
We then have the solution to our problem P given by
E
PP
(I1) g(x, v, t)"
! (x, v, t; m, l, 0)g (m, l) dm dl
E
0
in [0, ¹ ]. Let us recall some known results on the behaviour of the fundamental
solution.
Lemma 3. ¸et us assume DDEDD¸=(R3][0, ¹ ]))C and DD+ E DD¸=(R3][0, ¹ ]))C. ¹hen
xt
xt
x
(i) ¼e have
A
B
x v m l
(FS1) 0)! (x, v, t; m, l, q))C(DDEDD¸= , ¹ )G , , t; , , q ,
xt
E
2 2 2 2
(FS2) D­ ! (x, v, t; m, l, q)D)C(DDEDD¸= , ¹ )
xt
v E
G(x/2, v/2, t; m/2, l/2, q)
,
(t!q)1@2
(FS3) D­ ! (x, v, t; m, l, q)D)C(DDEDD¸= , DD+EDD¸= , ¹ )
xt
xt
x E
G(x/2, v/2, t; m/2, l/2, q)
,
(t!q)3@2
(FS4) D* ! (x, v, t; m, l, q)D)C(DDEDD¸= , DD+EDD¸= , ¹ )
xt
xt
v E
G(x/2, v/2, t; m/2, l/2, q)
,
(t!q)
for t!q'0, t, q3[0, ¹ ].
(ii) It holds
P
P
P
R6
! (x, v, t; m, l, q) dx dv"
E
R6
! (x, v, t; m@, l@, q@)! (m@, l@, q@; m, l, q) dm@ dl@"! (x, v, t; m, l, q)
E
E
E
R6
! (x, v, t; m, l, q) dm dl"1,
E
with x, v, m, l, m@, l@3R3 and t'q@'q*0.
(iii) DDg(t)DD¸p )DDg DD¸p , 1)p)R.
xv
0
(iv) ¼e have an integral expression
(I2) ! (x, v, t; m, l, q)"G(x, v, t; m, l, q)
E
t
#
­l@G(x, v, t; m@, l@, s)E(s, m@)! (m@, l@, s; m, l, q) dm@ dl@ ds
E
q
t
"G(x, v, t; m, l, q)!
G(x, v, t; m@, l,@ s)E(s, m@)­l@! (m@, l@, s; m, l, q) dm@ dl@ ds.
E
q
P PP
P PP
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
993
Proof. Statements (i), (ii) and (iv) were proved in Reference 18. From (ii) it follows that
DDg(t)DD¸1 )DDg DD¸1,
xv
0
DDg(t)DD¸=)DDg DD¸= .
xv
0
Interpolating we get (iii) for every p.
Remark 3. The second identity in (ii) states that the value of the fundamental solution
at time t corresponding to an initial time q is the same as the value of the solution of
(P ) at time t taking the value of the fundamental solution with singularity at q as the
E
initial datum at time q@3(q, t).
Remark 4. In view of the estimates on ! , we could think of getting for f (t) the same
E
kind of decay estimates we got for G( f ) (t). However, the fact that the constant
0
C(DDEDD¸= , ¹ ), depends on ¹ makes it impossible at first sight.
xt
Remark 5. The condition DD+ EDD¸=(R3][0, ¹ ]))C is used in Reference 18 to ensure that
xt
x
the function ! , which is the only solution of the integral equation (iv), is a classical
E
solution of the partial differential equation (it allows to justify the differentiation
inside the integral term).
Corollary 1. ¸et us assume DDEDD¸=(R3][0, ¹ ]))C. ¹hen, a generalized fundamental soluxt
tion ! (x, v, t; m, l, q) exists. ¹his solution satisfies estimates (FS1), (FS2) and parts
E
(ii)—(iv) in ¸emma 3. ¹he integral expression (I1) for solutions of P also holds.
E
Proof. A function !(x, v, t; m, l, q) solving the integral equation in Lemma 3(iv) is
known to exist (see Reference 18). This function satisfies (FS1) and (FS2). We must
relate the integral equation to the partial differential equation.
Let us see that solutions f of (P ) with bounded potentials E are limits of solutions
E
f of mollified problems with potentials E 3¸= with + E in ¸= . We take
xt
x d
d
d xt
E "E * g where g is a mollifying family. We then have DDE DD¸=)DDEDD¸= and E PE
xt
d
d
d
d xt
d
in ¸= weak *.
xt
For each f we have the corresponding fundamental solution ! satisfying the
d
d
estimates in Lemma 3. In particular, we have the bounds
A
B
x v m l
(FS1 ) D! (x, v, t; m, l, q)D)C(DDE DD¸= , ¹)G , , t; , , q ,
d
d
d xt
2 2 2 2
(FS2 ) D­ ! (x, v, t; m, l, q)D)C(DDE DD¸= , ¹)
d
v d
d xt
G(x/2, v/2, t; m/2, l/2, q)
(t!q)1@2
and the identities
PP
(I1 ) f (x, v, t)"
d d
! (x, v, t; m, l, 0) f (m, l) dm dl,
d
0
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
994
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(I2 ) ! (x, v, t; m, l, q)"G(x, v, t; m, l, q)
d d
P PP
#
t
q
­l@G(x, v, t; m@, l@, s)E(s, m@)! (m@, l@, s; m, l, q) dm@ dl@ ds
d
P PP
"G(x, v, t; m, l, q)!
t
q
G(x, v, t; m@, l,@ s)E(s, m@)­ ! (m@, l@, s; m, l, q) dm@ dl@ ds.
l{ d
We assume that f is smooth with compact support. Then, the formulae can be
0
extended to integrable f by density.
0
Since ! is bounded (locally in t) in any ¸p space a subsequence denoted again ! ,
xvt
d
d
converges weakly (locally in t) in any ¸p (weakly * if p"R) to a function ! and we
xvt
E
can pass to the limit in the right-hand side. In the distribution sense, the derivatives of
! with respect to v converge weakly to the derivatives of ! .
d
E
On the other hand, if a sequence g converges to g in the sense of distributions and
n
g )h then g)h in the sense of distributions. Therefore, we can pass to the limit in
n
the bounds (FS1) and (FS2) for ! and get them for ! in the sense of distributions. By
d
E
density we can extend the inequalities to test functions decaying at infinity.
It is clear from the estimate (FS1 ) and the integral expression (I1 ) that f is
d
d
d
uniformly bounded in any space ¸p with respect to d and locally in t. Therefore, f
xvt
d
converges weakly (locally in t) in any ¸p space to a function f and their derivatives
xvt
also converge in the sense of distributions.
Now, multiplying the equation satisfied for f by f we get a uniform ¸2 bound on
xvt
d
d
+ f . If we multiply the equation by DvD2 we get a uniform ¸1 bound on DvD2f .
v d
d
We can pass to the limit in the equation satisfied by f except in the term E + f .
d
d v d
Let us take a test function a(x)b(v)c(t). Then, we must pass to the limit in
PP
T
0
P
dx dt E (x, t)a(x)c(t) dv + b(v) f (x, v, t).
d
v
d
We split the integral in two I #I , where I is the integral when DvD(R and I the
1
2
1
2
integral when DvD*R, R'0. I goes to zero as R tends to R uniformly with respect
2
to d thanks to the estimate on DvD2f . On the other hand, for a.e. fixed (x, t) the function
d
F (x, t)":DvD(R du + b(v) f (x, v, t) tends to F(x, t)":DvD(R dv + b(v) f (x, v, t) thanks
d
v
d
v
to the bounds on + f . We have also bound
v d
DF (x, t)D)C
d
P
DvD(R
dv D + b(v)D
v
PP A
G
B
x v m l
, , t; , , 0 D f (m, l)D dm dl"H(x, t)
0
2 2 2 2
with H(x, t)3¸1-0#. Therefore, F (x, t)PF(x, t) in ¸1 . Since E PE in ¸= weak
xt
xt,-0#
d
x,t
d
* we can pass to the limit.
Passing to the limit in (I1 ) and (I2 ) we see that ! (x, v, t; m, l, q) is such that the
d
d
E
solution of (P ) is given by
E
PP
(I1) f (x, v, t)"
! (x, v, t; m, l, 0) f (m, l) dm dl
E
0
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
995
and ! is a solution to the integral equation
E
(I2) ! (x, v, t; m, l, q)"G(x, v, t; m, l, q)
E
P PP
#
t
q
­ G(x, v, t; m@, l@, s)E(s, m@)! (m@, l@, s; m, l, q) dm@ dl@ ds
l{
E
P PP
"G(x, v, t; m, l, q)!
t
q
G(x, v, t; m@, l@, s)E(s, m@)­ ! (m@, l@, s; m, l, q) dm@ dl@ ds.
l{ E
In the same way, (ii) and (iii) in Lemma 3 remain true in the limit. We know (see
Reference 18) that the solution of (I2) satisfies (FS1) and (FS2).
2.2. E(x, t) decaying at infinity
2.2.1. Estimate on the fundamental solution. For E as in section 2.1 we ignore the
possibility of getting the decay for the solutions. The only bounds we have are
DDg(t)DD¸p )DDg DD¸p , 1)p)R.
xv
0 xv
However, if we assume some decay on E as t grows, we can bound the fundamental
solution ! by means of the fundamental solution of the problem with E"0. Thanks
E
to this bound, the decay estimates are immediate. More precisely, we have the
following theorem:
Theorem 1. ¸et us assume that the field E is such that
(i) E3¸= ;
x,t
(ii) DDE(t)DD¸=)C /((1#t)(1@2)`a) with a*0 and C small enough if a"0.
x
a
a
¹hen, the fundamental solution ! for the problem P satisfies a global in time
E
E
estimate of the format
A
x v m l
0)! (x, v, t; m, l, q))C(DDEDD¸= )G , , t; , , q
xt
E
2 2 2 2
B
for 0(t!q(R, q*0, x, v, m, l3R3.
Remark 6. Condition (i) guarantees the existence of a fundamental solution satisfying
the estimates and identifies in Corollary 1. Condition (ii) allows to get the global in
time version of (FS1) removing the time dependence of the constant.
Proof. Step 1: C small. The procedure to construct the fundamental solution was as
a
follows [17, 15]:
=
!(x, v, t; m, l, q)"G(x, v, t; m, l, q)# + (! !! ) (x, v, t; m, l, q)
n`1
n
n/0
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
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with
! (x, v, t; m, l, q)"G(x, v, t; m, l, q),
0
! (x, v, t; m, l, q)"G(x, v, t; m, l, q)
n`1
t
­ G(x, v, t; m@, l@, s)E(s, m@)! (m@, l@, s; m, l, q) dm@ dl@ ds.
#
l{
n
q
We have the following estimates:
P PP
A
B
x v m l
D(! !! ) (x, v, t; m, l, q)D)(M 26I C )n`1 G , , t; , , q
n`1
n
l 1 a
2 2 2 2
and for a3(0, 1 )
2
A
B
x v m l
D(! !! ) (x, v, t; m, l, q)D)(M 26C )n`1In I t~aG , , t; , , q ,
12
n`1
n
l
a
2 2 2 2
where the constants M , I , I , C are given by
l 1 2 a
x v m l
D­ G(x, v, t; m, l, q)D)M (t!q)~1@2 G , , t; , , q ,
l
l
2 2 2 2
P
P
t
0
t
0
A
B
(t!s)~1@2s~1@2 ds"I ,
1
(t!s)~1@2s~a~1@2 ds"I t~a, a3(0, 1 ),
2
2
C
a
.
DDE(t)DD¸=)
x
(1#t)1@2`a
The estimates are proved by induction. For n"0 we have
D(! !! ) (x, v, t; m, l, q)D
1
0
t
"
­ G(x, v, t; m@, l@, s)E(s, m@)G(m@, l@, s; m, l, q) dm@ dl@ ds
l{
q
t
(t!s)~1@2 (1#s)~(a`1@2) ds
)M C
l a
q
x v m@ l@
]
G , , t; , , s G(m@, l@, s; m, l, q) dm@ dl@
2 2
2 2
KP PP
P
PP A
P
)M C 26
l a
t
B
(t!s)~1@2 (1#s)~(a`1@2) ds G
q
We can bound this last quantity by either
A
x v m l
M C 26I G , , t; , , q
l a 1
2 2 2 2
B
A
K
B
x v m l
, , t; , , q .
2 2 2 2
A
B
x v m l
or M 26C I t~a G , , t; , , q .
l
a2
2 2 2 2
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
997
Assuming the bounds to hold for n!1, let us see that they hold for n. We have
D(! !! ) (x, v, t; m, l, q)D
n`1
n
t
)
­ G(x, v, t; m@, l@, s)E(s, m@) (! !! )(m@, l@, s; m, l, q) dm@ dl@ ds
l{
n
n~1
q
t
)M C (M 26I C )n (t!s)~1@2 (1#s)~(a`1@2) ds
l a l 1 a
q
x v m@ l@
m@ l@ m l
]
G , , t; , , s G
, , t; , l dm@ dl@
2 2
2 2
2 2
2 2
KP PP
K
P
PP A
BA
)(M 26C )n`1In
1
l
a
P
t
B
(t!s)~1@2 (1#s)~(a`1@2) ds G
q
We can bound this last quantity by either
A
x v m l
(M 26I C )n`1G , , t; , , q
l 1 a
2 2 2 2
B
A
B
x v m l
, , t; , , q .
2 2 2 2
A
B
x v m l
or (M 26C )n`1In I t~a G , , t; , , q .
1 2
l
a
2 2 2 2
The series + = (M 26I C )n`1 converges if M 26I C (1. This can be achieved if
l 1 a
n/0 l 1 a
C is small enough. In conclusion
a
x v m l
=
D!(x, v, t; m, l, q)D) M# + (M 26I C )n`1 G , , t; , , q
l 1 a
2 2 2 2
n/0
when C is small, where M is such that
a
x v m l
! (x, v, t; m, l, q))MG , , t; , , q .
E
2 2 2 2
A
BA
A
B
B
Step 2: a'0. When a'0 we can write for t*¹
C
C
a
a@2
)
DDE(x, t)DD¸=)
x
(1#t)a`1@2 (1#t)(a@2)`1@2
with C "C /((1#¹ )a`1@2)(1 when ¹ is large enough. Going back to the estia@2
a
mates in step 1, we can use this estimate on E in the integrals above when q'¹, so
that we get
A
B A
x v m l
=
D!(x, v, t; m, l, q)D) M# + (M 26I C )n`1 G , , t; , , q
l 1 a@2
2 2 2 2
n/0
when t'q'¹.
For 0)q(t)3¹ we have
D!(x, v, t; m, l, q)D)C(DDEDD¸=(R3][0, 2¹ ]) ) G
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
A
B
B
x v m l
, , t; , , q .
2 2 2 2
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
998
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When q)¹ and t*3¹ we may use the expression
P
R6
! (x, v, t; m@, l@, q@)! (m@, l@, q@; m, l, q) dm@ dl@"! (x, v, t; m, l, q)
E
E
E
with x, v, m, l, m@, l@3R3 and t'q@'q*0. Taking q@"2¹ we get
A
B
=
! (x, v, t; m, l, q)) M# + (M 26I C )n`1 C(DDEDD¸=(R3][0, 2¹ ]) )
l 1 a@2
E
n/0
]
PP A
G
B A
B
B
x v m@ l@
m@ l@
m l
, , t; , , q@ G
, , q@; , , q dm@ dl@
2 2
2 2
2 2
2 2
A
A
B
x v m l
=
) M# + (M 26I C )n`1 C(DDEDD¸=(R3][0, 2¹ ]) ) G , , t; , , q ,
l 1 a@2
2 2 2 2
n/0
where
A
B
x v m l
! (x, v, t; m, l, q))MG , , t; , , q .
E
2 2 2 2
Therefore, we get a global estimate
D!(x, v, t; m, l, q)D)C(DDEDD¸= ) G
xt
A
B
x v m l
, , t; , , q
2 2 2 2
for 0)q(t, when a'0.
2.2.2. Asymptotic behaviour of solutions. In view of section 2.2.1 we have the solution
of P
E
P
g(x, v, t)" ! (x, v, t; m, l, 0)g (m, l) dm dl
E
0
satisfies the following decay estimates
DDg(t)DD¸p )DDg DD¸p ,
xv
0 xv
DDg(t)DD¸p )Ct~6(1@r~1@p)DDg DD¸r ,
xv
0 xv
p*r,
DDo(g)(t)DD¸p)Ct~(9@2)(1@r~1@p)DDg DD¸1 (¸r ) , p*r.
x
0 v x
Expanding g (see Reference 10) we have
0
(!1)DaD#DbD
g " +
0 DaD)n (DaD#DbD) !
DbD)m
AP
B
xavbg ­a ­bd# + ­a ­bF
0 x v
x v a,b
DaD"n
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
DbD"m
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
999
with F 3¸1 , if g (1#DxDn`1DvDm`1)3¸1 . For such g , we get an expansion
xv
0
0
a,b xv
(!1)DaD#DbD
g(x, v, t)" +
! (x, v, t; m, l, 0)
xavbg ­a­b d dm dl
E
0 m l
DaD)n (DaD#DbD) !
P
DbD)m
+
#
DaD"n#1
DbD"m#1
AP
BP
xavbg
0
1
DaD)n (DaD#DbD) !
" +
DbD)m
#
AP
AP
B
! (x, v, t; m, l, 0) ­a­b F dm dl
m l a,b
E
B
xavbg ­a­b ! (x, v, t; 0, 0, 0)
0 m l E
P
(!1)DaD#DbD ­a­b ! (x, v, t; m, l, 0)F dm dl
a,b
m l E
DaD"n#1
+
DbD"m#1
for g (1#DxDn`1DvDm`1)3¸1 . When E"0, ! "G and we can estimate the decay
0
xv
E
rate of the different terms
DD­a­b G(x, v, t; 0, 0, 0)DD¸p )C
xv
m l
DDG(x/2, v/2, t; 0, 0, 0)DD¸p
xv
t(3 D a D/2)#D b D/2
)Ct!6 (1!1/p)!(3 D a D/2)#D b D/2.
Using the estimates obtained in Lemma 1, the decay norm of the remaining term,
which we denote R , is estimated as follows:
mn
DD : G(x/2, v/2, t, m/2, l/2, q)F (m, l) dm dl DD¸p
xv
a,b
DDR DD¸p )C +
mn xv
t3(n`1)@2`(m`1)@2
DaD"n
DbD"m
)Ct~6(1~1@p)~(3(n`1))@2~(m`1)@2.
When EO0 is Lipschitz, we can use estimates (FS3) and (FS4) to bound the rest and
we get
1
(DaD#DbD)
!
D aD)0
g(x, v, t)" +
DbD)1
AP
B
xavbg ­a­b ! (x, v, t; 0, 0, 0)#R
01
0 m l E
with
DDR DD¸p )Ct~6(1~1@p)~3@2~2@2.
01 xv
If E is only bounded we cannot estimate the rest in this way. Therefore, we have
proved
K
Theorem 2. (i) For a solution g of (P ) we have the expansion
0
1
g(x, v, t)" +
xavbg ­a­b ! (x, v, t; 0, 0, 0)#R ,
nm
0 m l E
DaD)n (DaD#DbD) !
DbD)m
AP
B
DDR DD¸p )Ct~6(1~1@p)~*3(n`1)+@2~(m`1)@2
mn xv
when g (1#DxDn`1DvDm`1)3¸1 .
xv
0
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
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(ii) For a solution g of (P ) with a ¸ipschitz potential E we have the expansion
E
1
g(x, v, t)" +
xavbg ­a­b ! (x, v, t; 0, 0, 0)#R ,
01
0 m l E
(DaD#DbD)
!
D aD)0
AP
DbD)1
B
DDR DD¸p )Ct~6(1~(1@p))~(3@2)~2@2
mn xv
when g (1#DxD1DvD2)3¸1 .
xv
0
Remark 7. The expansion in Reference 10 is proved when g (1#D(x, v)Dk`1)3¸1
xv
0
replacing the sums + DaD)n with + DaD#DbD)k and + DaD"n#1 with + DaD#DbD)k#1 .
DbD)m
However, the result can be extended to cover our case.
DbD"m#1
2.2.3. Derivatives. We would like to get an estimate global in time for the derivatives
of the fundamental solution. We have the following theorem:
Theorem 2. ¸et us assume that the field E satisfies
(i) E3¸= ;
x,t
(ii) DDE(t)DD¸=)C /((1#t)(1@2)`a) with a*0 and C small enough if a"0.
x
a
a
¹hen, the fundamental solution ! for the problem P satisfies an estimate global in
E
E
time of the form
A
x v m l
C(DDEDD¸= )
xt G
D+ ! (x, v, t; m, l, q)D)
, , t; , , q
v E
(t!q)1@2
2 2 2 2
B
for 0(t!q(R, q*0, x, v, m, l3R3.
Proof. The derivatives of ! with respect to v solve
E
+ ! (x, v, t; m, l, q)"+ G(x, v, t; m, l, q)
v E
v
t
+ G(x, v, t; m@, l@, s)E(s, m@)+ ! (m@, l@, s; m, l, q) dm@ dl@ ds.
!
v
l{ E
q
On the other hand, we have
P PP
=
+ ! (x, v, t; m, l, q)"+ G(x, v, t; m, l, q)# + (+ ! !+ ! ) (x, v, t; m, l, q).
v E
v
v n`1
v n
n/0
We shall prove that
D(+ ! !+ ! ) (x, v, t; m, l, q)D
v n`1
v n
A
B
x v m l
)(M@@26C I@)n`1(t!q)~1@2 G , , t; , , q ,
a
2 2 2 2
where M@@"max(M2 , M M@) with
l
vl
A
x v m l
D+ + G(x, v, t; m, l, s)D)M (t!s)~1G , , t; , , s
v l
vl
2 2 2 2
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
B
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
1001
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
and M as in section 2.2.1. The constants I@ and M@ will be fixed below. For n"0 we
l
have
D(+ ! !+ ! ) (x, v, t; m, l, q)D
v 1
v 0
)C
(t`q)@2
a
AP PP
P
#
q
t
(q`t)@2
Therefore,
(1#(s!q))~(a`1@2) D+ + G(x, v, t; m@, l@, s)DG(m@, l@, s; m, l, q) ds dm@ dl@
v l{
A
x v m l
(1#(s!q))~(a`1@2) M2 (s!q)~1@2(t!s)~1@2 ds26 G , , t; , , q
l
2 2 2 2
A
x v m l
D(+ ! !+ ! ) (x, v, t; m, l, q)D)M 26C I(t!q)~1@2G , , t; , , q
v 1
v 0
0
a
2 2 2 2
BB
.
B
with M "max(M , M , 1)2 and I"21@2 : 1 s1@2 ds"23@2/3. Let us assume that
0
l vl
0
x v m l
D(+ ! !+ ! ) (x, v, t; m, l, q)D)(26C I@)nM (t!q)~1@2G , , t; , , q
v n
v n~1
a
n~1
2 2 2 2
A
B
for M "(max(M , M , 1))n`1. Then, we have
n~1
vl l
D(+ ! !+ ! ) (x, v, t; m, l, q)D
v n`1
v n
(q`t)@2
(1#(s!q))~(a`1@2) D+ + G(x, v, t; m@, l@, s)D
)C
v l{
a
q
]D! !! D (m@, l@, s; m, l, q) ds dm@ dl@
n
n~1
t
x v m@ l@
#
M (1#(s!q))~(a`1@2) (s!q)~1@2(t!s)~1@2 G , , t; , , s
l
2 2
2 2
(q`t)@2
]+ D ! !! D (m@, l@, s; m, l, q) ds dm@ dl@).
l{ n
n~1
Using for D ! !! D the bounds in section 2.2.1 and for D + (! !! )D the inducn
n~1
v n
n~1
tion hypotheses, we get
AP PP
P PP
A
B
A
x v m l
D+ ! !+ ! ) (x, v, t; m, l, q)D)C 26M@ I(t!q)~1@2 G , , t; , , q
v n`1
v n
a
n
2 2 2 2
B
with M@ "max(M (M 26I C )n, M (26C I)nM ). We take I@"max(I, I ) and
n
vl l 1 a
l
a
n~1
1
M "(max(M , M , 1))n`2. Then
n
vl l
x v m l
D(+ ! !+ ! ) (x, v, t; m, l, q)D)(C 26I@)n`1 M I(t!q)~1@2 G , , t; , , q
v n`1
v n
a
n
2 2 2 2
A
B
Therefore,
D+ !(x, v, t; m, l, q)D
v
A
B
x v m l
=
)(M (t!q)~1@2# + (C 26I@)nM (t!q)~1@2) G , , t; , , q .
v
a
n
2 2 2 2
n/0
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
1002
A. Carpio
The series above converges if C is small enough. We denote M@ "M #
v
v
a
+ = (C 26I@)n.
n/0 a
When a'0 we can write for t*¹
C
C
a
a@2
)
DDE(x, t)DD¸=)
x
(1#t)a`1@2 (1#t)a@2`1@2
with C "C /(1#¹ )a`1@2(1 when ¹ is large enough. Going back to the previous
a@2
a
estimates, we can use this estimate for E in the integrals above when q*¹, so that we
get
A
B
x v m l
D+ !(x, v, t; m, l, q)D)M@ (t!q)~1@2 G , , t; , , q
v
v
2 2 2 2
when t'q*¹.
For 0)q(t)3¹ we have
A
B
x v m l
D+ !(x, v, t; m, l, q)D)C(DDEDD¸=(R3][0, 2¹ ]) ) (t!q)~1@2G , , t; , , q .
v
2 2 2 2
When q(¹ and t*3¹ we may use the expression
P
R6
+ ! (x, v, t; m@, l@, q@)! (m@, l@, q@; m, l, q) dm@ dl@"+ ! (x, v, t; m, l, q)
v E
E
v E
with x, v, m, l, m@, l@3R3 and t'q@'q*0. We get
D+ ! (x, v, t; m@, l, q)D)M@ (t!q@)~1@2 C(DDEDD¸=(R3][0, q@]) )
v
v E
x v m@ l@
m@ l@
m l
G , , t; , , q@ G
, , q@, , , q dm@ dl@
]
2 2
2 2
2 2
2 2
PP A
B A
B
A
B
x v m l
)M@ (t!q@)~1@2 C(DDEDD¸=(R3][0, r@]) )G , , t; , , q .
v
2 2 2 2
In the range of t we are dealing with (t!q@)~1@2)(t!q)~1@2, so that we get a global
estimate
A
B
x v m l
D+ !(x, v, t; m, l, q)D)C(DDEDD¸= ) (t!q)~1@2G , , t; , , q
xt
v
2 2 2 2
for 0)q(t, when a'0.
2.2.4. Asymptotic behaviour of the derivatives. In view of section 2.2.3 we have that the
derivative of the solution of P with respect to v
E
P
+ g(x, v, t)" + ! (x, v, t; m, l, 0)g (m, l) dm dl
v
v E
0
satisfies the following decay estimates:
DD+ g(t)DD¸p )Ct~6(1@r~1@p)~1@2DD g DD¸r , p*r.
xv
v
0 xv
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
1003
In the same way as in section 2.2.2 we get an expansion
(!1)Da D#DbD
+ g(x, v, t)" +
v
DaD)n (DaD#DbD) !
DbD)m
#
+
DaD"n#1
DbD"m#1
AP
DbD)m
#
+
DaD"n#1
DbD"m#1
B
xavbg + ! (x, v, t; m, l, 0)­a ­b d dm dl
0 v E
m l
B
xavbg + ! (x, v, t; m, l, 0)­a ­b F dm dl
m l a,b
0 v E
1
(DaD#DbD)
!
DaD)n
" +
AP
AP
B
xavbg ­a ­b + ! (x, v, t; 0, 0, 0)
0 m l v E
P
(!1)DaD#DbD ­a ­b + ! (x, v, t; m, l, 0)F dm dl
m l v E
a,b
for g (1#DxDn`1DvDm`1)3¸1 , with F 3¸1 . When E"0, ! "G and we can
E
xv
a,b xv
0
estimate the decay rate of the different terms
DD­a ­b + G(x, v, t; 0, 0, 0)DD¸p )C
xv
m l v
DDG(x , v , t; 0, 0, 0)DD¸p
xv
2 2
t3Da D/2#(DbD#1)/2
)Ct~6(1~1@p)~3Da D/2!(DbD#1)/2.
The decay norm of the remaining term, which we denote R , is estimated as follows:
mn
DDR DD¸p )C +
mn xv
DaD"n
DD : G (x/2, v/2, t, m/2, l/2, q)F (m, l) dm dlDD¸p
xv
a,b
t(3(n`1))@2`(m`2)@2
DbD"m
)Ct!6 (1!(1/p))!3(n#1)/2!(m#2)/2.
When EO0 is Lipschitz we get this expansion but with n"m"0.
3. Non-linear problem
3.1. ¸ong-time behaviour of the solutions
Let f be a global solution of VPFP
f !* f#v+ f#E( f )+ f"0, in R3]R3]R`,
t
v
x
v
f (x, v, 0)"f (x, v),
0
in R3]R3,
where E(o( f ))"e(K * o( f )) with o( f )":R3 f (x, v, t) dv, e"$1 (depending on
x
whether the interaction between the particles is electrostatic or gravitational) and
K(x)"x/S DxD3, S being the area of the sphere in R3. We know that for initial data
3
3
small enough and satisfying some suitable integrability conditions, global solutions
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
1004
A. Carpio
are known to exist and to satisfy the associated integral equations
P
(I1) f (x, v, t)" G(x, v, t; m, l, 0) f (m, l) dm dl
0
P PP
#
t
0
PP
(I2) f (x, v, t)"
ds
+ G(x, v, t; m, l, s)E( f )(m, s) f (m, l, s) dm dl,
l
! (x, v, t; m, l, 0) f (m, l) dm dl,
E
0
where G is the fundamental solution of the problem with E"0, and ! is the
E
fundamental solution of the problem with potential E( f ).
Remark 8. For ease of notation we denote E(o( f ))"E( f )"E.
We shall be interested in applying the results of the fundamental solutions obtained
in section 2 to fields E(o) with o"o( f ) and f a solution of (VPFP) corresponding to
an initial datum f . Therefore, we must know for which data we get solutions of
0
(VPFP) with Lipschitz or bounded potentials.
A first result in this direction is the following one: If
(RW) f *0, (1#v2)c@2f 3(¸1W¸=)(R6), (1#v2)c@2+
f 3(¸1W¸=)(R6), c'3
0
0
(x,v) 0
and we assume E(o( f ))3¸= , then + E(o( f )) belongs to ¸= (see Reference 15).
xt
xt
x
Therefore, when we have a global solution of (VPFP) and we know that the associated
potential E( f ) is bounded, we can always guarantee that the potential is also
Lipschitz, imposing more restrictions to the initial datum.
Let us recall some results of the global existence of solutions with bounded
potentials.
In Reference 16 solutions to (VPFP) with E(o( f ))3¸= are constructed for small
xt
initial data in the class
f 3¸1W¸=, 0)f )Ah(x)g(v).
0
0
In Reference 2 solutions to (VPFP) with E(o( f ))3¸= are constructed for initial data
xt
in the class
f *0, f 3(¸1W¸=)(R6), DvDmf 3¸1(R6) for some m'6.
0
0
0
By Reference 6 we can take m'15. By Reference 3 we can take m"2 if o( f )3¸=.
4
x
0
In these cases the potential E( f ) is Lipschitz when we impose the condition (RW)
on the initial data. All the results stated in Lemma 3 on the fundamental solutions for
linearized (VPFP) problems with Lipschitz potentials E hold.
If we do not add restrictions on the data, the potential is bounded and Corollary 1
applies: the existence of fundamental solutions and the estimates (FS1) and (FS2) on
the fundamental solution and its derivative with respect to v, as well as (ii) and (iii) in
Lemma 3 are guaranteed without adding conditions on the data.
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
1005
We have the following theorem:
Theorem 4. ¸et us assume that
(i) f is the unique solution to (VPFP) taking f as initial data satisfying (I1);
0
(ii) the fundamental solution ! corresponding to E"E( f ) exists and satisfies
E
x v m l
0)! (x, v, t; m, l, q))C(DDEDD¸= ) G , , t; , , q
xt
E
2 2 2 2
A
B
for 0(t!q(R, x, v, m, l3R3
(iii) f 3¸1W¸=(R6), f 3¸1 (¸= ).
0
0 v x
¹hen, f behaves for large times like the solution of the problem P taking the initial
0
data f , in the sense that
0
t6(1~1@p) DDG( f ) (t)!f (t)DD¸p P0 as tPR, 1)p)R.
xv
0
Moreover, we have the decay rate
DDG( f ) (t)!f (t)DD¸p )Ct~6(1~1@p)~1@2, t'0, 1)p)R.
xv
0
Corollary 2. ¹heorem 4 applies to the solutions for (VPFP) constructed in
(1) Reference 16 for small initial data in the class
f 3¸1W¸=, 0)f )Ah(x)g(v), h, g3¸1W¸=.
0
0
(2) Reference 2 for initial data in the class
f *0, f 3(¸1W¸=) (R6), f 3¸1 (¸=) DvDm f 3¸1(R6) for some m'6
0
0
0 v x
0
provided that the data are small enough. By Reference 6 we can take m'15 . By
4
Reference 3 we can take m"2 if o( f )3¸=.
x
0
Proof. We know by Theorem 1 that if E3¸= and
xv
C
a
DDE(t)DD¸=)
x
(1#t)a`1@2
for a*0, with C small enough if a"0, then (ii) holds. Therefore, it suffices to apply
a
Theorems 1 and 4 with E"E( f ). For the solutions constructed by Triolo, a"3'0.
2
For the solutions constructed by Soler and Carrillo, a"0.
Remark 8. In view of the results in section 2.2.2, it folows from this theorem that
DDMG(t)!f (t)DD¸p )Ct~6(1~1@p)~1@2, t'0, 1)p)R,
xv
where M is the mass of the initial data, provided that f (DxD#DvD2)3¸1 . For f only
xv
0
0
satisfying (ii), we can only say that
t6(1~1@p) DDMG(t)!f (t)DD¸p P0 as tPR, 1)p)R
xv
holds.
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
1006
A. Carpio
Proof of ¹heorem 4. From (i) and (ii) f satisfies (I2) with
A
x v m l
0)! (x, v, t; m, l, q))C(DDEDD¸= ) G , , t; , , q
xt
E
2 2 2 2
B
for 0(t!q(R, x, v, m, l3R3.
In those conditions, if f 3¸1W¸= it follows that, for p*r
0
DD f (t)DD¸p )Ct~6(1@r~1@p)DD f DD¸r ,
xv
0 xv
DDo(t)DD¸p)Ct~9@2(1@r~1@p) DD f DD¸1(¸r ) ,
x
0 v x
DD f DD2@3 ,
DDE(t)DD¸=)Ct~3@r DD f DD1@3
x
0 ¸1xv 0 ¸1v (¸rx)
DD+ f (t)DD¸p )Ct~6(1@r~1@p)~1@2DD f DD¸r ,
xv
0 xv
v
thanks to the results for the linear equations with potential E obtained in section 2.
Using the integral equation
P PP
P PP
f (x, v, t)!G( f ) (t)"
0
t@2
0
!
­ G(x, v, t; m, l, s)E( f ) (m, s) f (m, l, s) dm dl ds
l
t
t@2
G(x, v, t; m, l, s)E( f ) (m, s)­ f (m, l, s) dm dl ds,
l
so that
DD f (x, v, t)!G( f ) (t)DD¸p
xv
0
t@2
x v m l
)C
(t!s)~1@2 G , , t; , , s (1#s)~3 f (m, l, s) dm dl ¸p ds
xv
2 2 2 2
0
t
x v m l
G , , t; , , s (1#s)~3­ f (m, l, s) dm dl ¸p ds
#
l
xv
2 2 2 2
t@2
t@2
(t!s)~1@2~6(1@r~1@p)(1#s)~3(1#s)~6(1~1@r) ds
)C
0
t
(t!s)~6(1@r~1@p)(1#s)~3s~1@2~6(1~1@r) .
#
t@2
Taking r"1 in (0, t/2) and r"p in (t/2, t) we get
AP K PP
P K PP A
AP
P
A
B
B
K B
K
B
DD f (x, v, t)!G( f ) (t)DD¸p
xv
0
P
)C(t~1@2~6(1~1@p) (1#s)~3 ds#(1#t)~3t~1@2~6(1~1@p)t),
that is,
DD f (x, v, t)!G( f ) (t)DD¸p )Ct~(1@2)~6(1~1@p).
xv
0
Remark 10. The decay rate is the same as that obtained in Theorem 3 when
n"m"0.
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
1007
3.2. ¸ong-time behaviour of the derivatives
We have the following theorem:
Theorem 5. ¸et us assume that
(i) f is the unique solution to (VPFP) taking f as initial data satisfying (I1);
0
(ii) the fundamental solution ! corresponding to E"E( f ) exists and satisfies
E
A
A
B
B
x v m l
0)! (x, v, t; m, l, q))C(DDEDD¸=) G , , t; , , q ,
xt
E
2 2 2 2
x v m l
D+ ! (x, v, t; m, l, q))C(DDEDD¸=) G , , t; , , q
xt
l E
2 2 2 2
for 0(t!q(R, x, v, m, l3R3;
(iii) f 3¸1W¸=(R6), f 3¸1 (¸= ).
0
0 v x
¹hen
t6(1~(1@p))`1@2DD+ G( f ) (t)!+ f (t)DD¸p P0 as tPR, 1)p)R.
xv
v
0
v
Moreover, we have the decay rate
DD+ G( f ) (t)!+ f (t)DD¸p )Ct~6(1~(1@p))~1, t'0, 1)p)R.
xv
v
0
v
Corollary 3. ¹heorem 5 applies to the solutions for (VPFP) constructed in
(1) Reference 16 for small initial data in the class
f 3¸1W¸=, 0)f )Ah(x)g(v), h, g3¸13¸=;
0
0
(2) Reference 2 for initial data in the class
f *0,
0
f 3(¸1W¸=)(R6), f 3¸1 (¸= ), DvDm f 3¸1(R6) for some m'6
0
0
0 v x
provided that the data are small enough. By Reference 6 we can take m'15. By
4
Reference 3 we can take m"2 if o( f )3¸= .
x
0
Proof. The same as in Corollary 2 but using Theorem 2.
Remark 11. In view of the results in section 2.2.2, it follows from this theorem that
DDM+ G(t)!+ f (t)DD¸p )Ct~6(1~1@p)~1, t'0, 1)p)R,
xv
v
v
where M is the mass of the initial data, provided that f (DxD#DvD2)3¸1 . For f only
xv
0
0
satisfying (ii), we can only say that
t6(1~1@p)`1DDM+ G(t)!+ f (t)DD¸p P0 as tPR, 1)p)R
xv
v
v
holds.
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
1008
A. Carpio
Proof. From (i) and (ii) f satisfies (I2) with
A
A
B
B
x v m l
0)! (x, v, t; m, l, q))C(DDEDD¸=) G , , t; , , q ,
xt
E
2 2 2 2
x v m l
D+ ! (x, v, t; m, l, q)D)C(DDEDD¸=) G , , t; , , q .
xt
l E
2 2 2 2
Under those conditions, if f 3¸1W¸= it follows that
0
DD+ f (t)DD¸p )Ct~6(1@r~1@p)~1@2DD f DD¸r , p*r,
xv
v
0 xv
thanks to the results for the linear equations with potential E obtained in section 2.
Using the integral equation
D+ f (x, v, t)!+ G( f ) (t)D)C
v
v
0
P PP
t@2
D+ f (m, v, s)D
] l
ds
(1#s)3
0
KP PP
t
t@2
A
x v m l
(t!s)~1@2+ G , , t; , , s
v
2 2 2 2
A
B
B
K
x v m l
D f (m, v, s)D
(t!s)~1@2+ + G , , t; , , s
ds ,
v l
2 2 2 2
(1#s)3
so that
DD+ f (x, v, t)!+ G( f ) (t)DD¸p
xv
v
v
0
t@2
(t!s)~1~6(1@r~1@p)(1#s)~3(1#s)~6(1~1@r) ds
)C
0
t
#C
(t!s)~(1@2)`6((1@r)~1@p)(1#s)~3s~6(1~(1@r))~1@2DD f DD¸1 ds.
0 xv
t@2
Choosing r"p in the interval (t/2, t) and r"1 in (0, t/2) we obtain
P
P
DD+ f (x, v, t)!+ G( f ) (t)DD¸p )Ct~1~6(1~1@p).
xv
v
v
0
3.3. Second term
We know that the first term in the asymptotic development of a solution f of
(VPFP) with small data f 3¸1W¸=(R6) is MG(x, v, t; 0, 0, 0), M": : f (x, v) dx dv.
0
0
In order to find a more precise development we are going to study each of the terms
appearing in the integral equation
P
(I1) f (x, v, t)" G(x, v, t; m, l, 0) f (m, l) dm dl
0
P PP
t
ds
+ G(x, v, t; m, l, s)E( f ) (m, s) f (m, l, s) dm dl.
l
0
As far as the first term is concerned, we know (section 2.2.2) that
#
P
G(x, v, t; m, l, 0) f (m, l) dx dl"G( f )
0
0
"MG(x, v, t; 0, 0, 0)#m ­ G(x, v, t; 0, 0, 0)#R(x, v, t),
i li
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
1009
where m ": : v f (x, v) dx dv and the remaining term R satisfies
i
i 0
DDRDD¸p )Ct~1~6(1~1@p)
xv
when v2f 3¸1 , xf 3¸1 .
0 xv 0 xv
To study the asymptotic behaviour of the second integral, denoted w(x, v, t), that is,
P PP
w(x, v, t)"
t
ds
+ G(x, v, t; m, l, s)E( f ) (m, s) f (m, l, s) dm dl,
l
0
we are going to use a scaling technique. By rescaling, we see that the functions
w (x, v, t)"j12w(j3x, jv, j2t) satisfy
j
t
w (x, v, t)"j~5
­ G(x, v, t; m, l, s)E (m, s) f (m, l, s) dm dl ds
j
m
j
j
0
with E (m, s)"j6E(j3m, j2s), f (m, l, s)"j12f (j3m, jl, j2s), that is,
j
j
j2
m l s
­ G x, v, 1; , ,
E(m, s) f (m, l, s) dm dl ds.
w (x, v, 1)"j~1
m
j
j3 j j2
0
We remark that
P PP
P PP A
B
DDw (1)DD¸p "j12(1~1@p) DDw(j2)DD¸p .
xv
xv
j
Thus, if we want to make precise the asymptotic behaviour of w when tPR it suffices
to find a function g such that
DDw (1)!g (1)DD¸p d(j)P0 as jPR
xv
j
j
for some d(j) tending to R as jPR, where g (x. v, t)"j12g(j3x, jv, j2t). That
j
implies
DDw(t)!g(t)DD¸p t1~1@pd(t1@2)P0 as tPR.
xv
It is easy to prove that DDw DD¸p is bounded by Cj~1. Therefore, the same kind of
j xv
bound must hold for g but the difference DDg (t)!w (t)DD must go to zero faster.
j
j
j q
Under certain conditions it is possible to take g(x, v, t)"K ­ G(x, v, t; 0, 0, 0) with K
i li
i
to be precised below and d(t)"t1@2. More precisely, we prove the following:
Proposition 1. ¸et us assume that
(i) f is the unique solution to (VPFP) taking f as initial data satisfying (I1):
0
(ii) the fundamental solution ! corresponding to E"E( f ) exists and satisfies
E
x v m l
0)! (x, v, t; m, l, q))C(DDEDD¸=) G , , t; , , q ,
xt
E
2 2 2 2
A
A
x v m l
D+ ! (x, v, t; m, l, q)D)C(DDEDD¸=) G , , t; , , q
xt
l E
2 2 2 2
B
B
for 0(t!q(R, x, v, m, l3R3;
(iii) f 3¸1W¸=(R6), f 3¸1 (¸= ).
0
0 v x
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
1010
A. Carpio
¹hen, the integral w satisfies
K
AP P
=
t1`6(1~(1@p)) w(t)!
0
R6
B
E( f ) f dx dv ds + G(x, v, t; 0, 0, 0)
m
K
P0
¸p
xv
as tPR.
Proof. In view of the identity
w (x, v, 1)"j~1
j
P PP A
j2
B
m l s
­ G x, v, 1; , ,
E(m, s) f (m, l, s) dm dl ds,
m
j3 j j2
0
we expect
P PP A
j(w (1)!g (1))"
j
j
AP PP
=
j2
0
B
m l s
­ G x, v, 1; , ,
E(m, s) f (m, l, s) dm dl ds
m
j3 j j2
B
E(m, s) f (m, l, s) dm dl ds­ G(x, v, 1; 0, 0, 0)
m
0
to tend to 0 as jPR. Taking into account that
!
AP PP
P PP
j2
E(m, s) f (m, l, s) dm dl ds
0
B
=
E(m, s) f (m, l, s) dm dl ds ­ G(x, v, 1; 0, 0, 0)
m
0
tends to zero in any ¸p as jPR, we must only prove that
xv
j2
m l s
I"
­ G x, v, 1; , ,
j
m
j3 j j2
0
!
P PPA A
B
B
!­ G(x, v, 1; 0, 0, 0) E(m, s) f (m, l, s) dm dl ds
m
goes to zero in ¸p as jPR. We split the integral as follows:
xv
I "I1 #I2 #J2 #I3 #J3 #I4 #I5 ,
d,j
d,j
d,j
d,j
d,j
d,j
d,j
j
where
P P
j2d
I1 "
d,j
0
DmD)j3d
DlD)jd
A A
B
m l s
­ G x, v, 1; , ,
m
j3 j j2
B
!­ G(x, v, 1; 0, 0, 0) E(m, s) f (m, l, s) dm dl ds,
m
P P
P P A
j2
I2 "
d,j
j2d
j2
J2 "
d,j
j2d
R6
­ G(x, v, 1; 0, 0, 0) E(m, s) f (m, l, s) dm dl ds,
m
B
m l s
­ G x, v, 1; , ,
E(m, s) f (m, l, s) dm dl ds,
m
j3
j j2
R6
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
1011
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
P P
P P
A
P P A A
j2d
I3 "
d,j
DmD'j3d
DlD'jd
0
j2d
J3 "
d,j
B
B
m l s
­ G x, v, 1; , ,
E(m, s) f (m, l, s) dm dl ds,
m
j3
j j2
DmD'j3d
0
DlD'jd
j2d
I4 "
d,j
­ G(x, v, 1; 0, 0, 0) E(m, s) f (m, l, s) dm dl ds,
m
DmD)j3d
DlD'jd
0
m l s
­ G x, v, 1; , ,
m
j3 j j2
B
!­ G(x, v, 1; 0, 0, 0) E(m, s) f (m, l, s) dm dl ds,
m
P P A A
j2d
I5 "
d,j
DmD'j3d
DlD)jd
0
B
m l s
­ G x, v, 1; , ,
m
j3 j j2
B
!­ G(x, v, 1; 0, 0, 0) E(m, s) f (m, l, s) dm dl ds.
m
Given e'0, taking d small enough we get
DDI2 DD )C
d,j p
P P KA A
j2d
0
DmD)j3d
DlD)jd
B
PP
BKK
m l s
­ G x, v, 1; , ,
!­ G(x, v, 1; 0, 0, 0)
m
m
j3 j j2
]DE(m, s) f (m, l, s)D dm dl ds)Ce
=
0
R6
p
DE(m, s) f (m, l, s)D dm dl ds.
Now, for d fixed
P P
P A B P A
P
P A
B
P
P P
P A B P A
P A B P A
B
DDI2 DD )DD­ G(x, v, 1; 0, 0, 0)DD
d,j p
m
p
DDJ2 DD )C
d,j p
)C
1
j2
j2d
s ~1@2
1!
j2
G
(1!s)~1@2
R6
d
)C
1
j2
j2d
G
R6
(1!s)~1@2
j2d
)C
0
B
x v
m l s
, , 1;
, ,
E(m, s) f (m, l, s) dm dl ds
2 2
2j3 2j j2
j2
j12(1~1@p)
C
)
,
(1#j2s)3 (1#j2d)12(1~1@p) (1#j2d)2
DDI3 DD )CDD­ G(x, v, 1; 0, 0, 0)DD
d,j p
m
p
j2d
DE(m, s) f (m, l, s)D dm dl ds,
x v
, ; 1; m, l, s DE(2j3m, j2s) f (2j3m, 2jl, j2s)D j14 dm dl ds
2 2
d
DDJ3 DD )C
d,j p
R6
j2d
DE(m, s) f (m, l, s)D dm dl ds,
DmD'j3d
0 DlD'jd
x v
s ~1@2
m l s
1!
G , , 1;
, ,
E(m, s) f (m, l, s) dm dl ds
2 2
j2
2j3 2j j2
R6
0
s ~1@2
1!
j2
G
R6
B
x v
s
, , 1; m, l,
DE(2j3m, s) f (2j3m, 2jl, s)Dj12 dm dl ds
2 2
j2
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
1012
A. Carpio
P A B
P
P P
P P A A
j2d
)C
s ~(1@2)~6(1~1@p)
1!
j2
0
j2d
)C
DmD'j3d/2
DlD'jd/2
0
DmD)j3d
DlD'jd
0
DE(m, s) f (m, l, s)D dm dl ds,
DE(m, s) f (m, l, s)D dm dl ds,
j2d
DDI4 DD )C
d,j p
DmD'j3d/2
DlD'jd/2
B
B
m l s
­ G x, v, 1; , ,
!­ G(x, v, 1; 0, 0, 0)
m
m
j3 j j2
]DE(m, s) f (m, l, s)D dm dl ds
P P
j2d
)C(DD­ G(x, v, 1; 0, 0, 0)DD
m
p
P A
P P
P P
j2d
#
0
B
0
DDI5 DD )C
d,j p
DmD)j3d/2
DlD'jd/2
j2d
0
DmD)j3d
DlD'jd
s ~(1@2)~6(1~1@p)
1!
j2
j2d
)C
0
DmD'j3d/2
DlD'jd/2
DE(m, s) f (m, l, s)D dm dl ds
P
DmD)j3d/2
DlD'jd/2
DE(m, s) f (m, l, s)D dm dl ds
DE(m, s) f (m, l, s)D dm dl ds,
DE(m, s) f (m, l, s)D dm dl ds.
All these integrals tend to 0 as jPR, since := : R6 DE(m, s) f (m, l, s)D dm dl ds is finite.
0
Going back to the original variables we get
K
AP P
t1`6(1~1@p) w(t)!
=
0
R6
B
E( f ) f dx dv ds + G(x, v, t; 0, 0, 0)
m
K
P0
¸p
xv
as tPR.
In conclusion, we have proved the following result:
Theorem 6. ¸et us assume that
(i) f is the unique solution to (VPFP) taking f as initial data satisfying (I1);
0
(ii) the fundamental solution ! corresponding to E"E( f ) exists and satisfies
E
A
A
B
B
x v m l
0)! (x, v, t; m, l, q))C(DDEDD¸=) G , , t; , , q ,
xt
E
2 2 2 2
x v m l
D+ ! (x, v, t; m, l, q))C(DDEDD¸=) G , , t; , , q
xt
l E
2 2 2 2
for 0(t!q(R, x, v, m, l3R3;
(iii) f 3¸1W¸=(R6), f 3¸1 (¸= ).
0
0 v x
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
1013
Solutions of the Vlasov—Poisson—Fokker—Planck Equation
¹hen, the solution to (VPFP) taking f as initial data satisfies
0
=
E( f ) f dm dl ds + G(x, v, t) P0
t1`6(1~1@p) f (t)!MG(x, v, t)! m #
l
m
¸p
xv
0 R6
as tPR
K
A PP
B
K
where
P
M" f (m, l) dl dm,
0
P
m " l f (m, l) dl dm
l
0
and G(x, v, t)"G(x, v, t; 0, 0, 0).
Corollary 4. ¹heorem 5 applies to the solutions for (VPFP) constructed in
(1) Reference 16 for small initial data in the class
f 3¸1W¸=, 0)f )Ah(x)g(v), h, g3¸13¸=;
0
0
(2) Reference 2 for initial data in the class
f *0, f 3(¸1W¸=)(R6), f 3¸1 (¸= ), DvDmf 3¸1(R6) for some m'6
0
0
0
0 v x
provided that the data are small enough. By Reference 6 we can take m'15. By
4
Reference 3 we can take m"2 if o( f )3¸= .
x
0
Acknowledgements
The author would like to thank E. Zuazua for having suggested this subject. This research was supported
by DGICYT Project PB93-1203.
References
1. Batt, J., ‘Global symmetric solutions of the initial value problem of stellar dynamics’, J. Differential
Equations 25, 342—364 (1984).
2. Bouchut, F., ‘Existence and Uniqueness of a global smooth solution for the Vlasov—Poisson—
Fokker—Plank system in three dimensions’, J. Funct. Anal., 111, 239—258 (1993).
3. Bouchut, F., ‘Smoothing effect for the non-linear Vlasov—Poisson—Fokker—Plank system’, J. Differential Equations 122, 225—238 (1995).
4. Carpio, A., ‘Large-time behavior in incompressible Navier—Stokes equations’, SIAM J. Math. Anal., 27
(2), 449— 475 (1996).
5. Carrillo, J. A. and Soler, J., ‘On the initial value problem for the Vlasov—Poisson—Fokker—Plank system
with initial data in ¸p spaces’, Math. Meth. in the Appl. Sci., 18, 825—839 (1995).
6. Carrillo, J. A. and Soler, J., ‘On the Vlasov—Poisson—Fokker—Plank equations with measures in Morrey
spaces as initial data’, to appear.
7. Carrillo, J. A., Soler, J. and Vazquez, J. L., ‘Asymptotic behavior for the frictionless Vlasov—
Poisson—Fokker—Plank system’, C.R. Acad. Sci., 21, 1195—1200 (1995).
8. Carrillo, J. A., Soler, J. and Vazquez, J. L., ‘Asymptotic behavior and selfsimilarity for the three
dimensional Vlasov—Poisson—Fokker—Plank system’, to appear.
9. Degond, P., ‘Global existence of smooth solutions for the Vlasov—Fokker—Plank equation in 1 and
2 space dimensions’, Ann. Sci. Ecole Norm. Sup., 19, 519—542 (1986).
10. Duoandikoetxea, J. and Zuazua, E.,‘Moments, masses de Dirac et décomposition de fonctions’, C.R.
Acad Sci., 315, 693—698 (1992).
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)
1014
A. Carpio
11. Giga, Y., Miyakawa, T., ‘Navier—Stokes flow in R3 with measures as initial vorticity and Morrey
spaces’, Comm. in P.D.E., 14, 577—618 (1989).
12. Horst, E., ‘On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov
equation, I and II’, Math. Meth. in the Appl. Sci., 3, 229—248 (1981) and 4, 19—32 (1982).
13. Il’in, A. M., ‘On a class of ultraparabolic equations’, Soviet Math. Dokl., 5, 1673—1676 (1964).
14. Il’in, A. M. and Kasminsky, R. Z., ‘On the equations of Brownian motion’, ¹heory Probab. Appl., 9,
421— 444 (1964).
15. Rein, G. and Weckler, J., ‘Generic global classical solutions of the Vlasov—Fokker—Plank—Poisson
system in three dimensions’, J. Differential Equations 99, 59—77 (1992).
16. Triolo, L., ‘Global existence for the Vlasov—Poisson—Fokker—Plank equation in many dimensions for
small data’, Math. Meth. in the Appl. Sci., 10, 487— 497 (1988).
17. Victory, H. D., ‘On the existence of global weak solutions for Vlasov—Poisson—Fokker—Plank systems’,
J. Math. Anal. Appl., 160, 525—555 (1991).
18. Victory, H. D. and O’Dwyer, B. P., ‘On classical solutions of Vlasov—Poisson—Fokker—Plank systems’,
Ind. ºniv. Math. Math. J., 3, 1, 105—155 (1990).
19. Weber, M., ‘The fundamental solution of a degenerate partial differential equation of parabolic type’,
¹rans. Amer. Math. Soc., 71, 24—37 (1951).
( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.
Math. Meth. Appl. Sci., 21, 985—1014 (1998)