Commun.
Math.
Phys.
154, 245-263 (1993)
Communications in
Mathematical
Physics
©
Springer-Verlag
1993
Global
Weak
Solutions
of the
Vlasov-Maxwell
System
with
Boundary
Conditions*
Yan Guo
Department
of
Mathematics, Brown University, Providence,
RI
02912,
USA
Received September
30,
1991;
in
revised form December
26, 1991
Abstract.
Boundaries occur naturally in physical systems which
satisfy
the
Vlasov-
Maxwell system. Assume perfect conductor boundary conditions for Maxwell, and
either
specular reflection or partial absorption for
Vlasov.
Then weak solutions
with finite energy
exist
for all time.
§0.
Introduction
We study the initial and boundary value problem of both the non-relativistic and
relativistic
Vlasov-Maxwell
system. We shall prove the global existence of weak
solutions under various boundary conditions.
Let Ω be an open set in R
3
with C
1
^ boundary, for some μ > 0. Consider the
non-relativistic
Vlasov-Maxwell
system:
c
d
t
E - ccuήB = - j = - 4πΣe
β
J
f
β
dv
, (VM)
β R
3
ccurl£
= 0 ,
where 0<f<oo, x e Ω and v e R
3
, with the constraints
div
E = p =
AπY^ββ
j
f
fi
dv
,
β
R3
(0.1)
This research
is
supported
in
part
by NSF
Grant
DMS 90-23864
246 Y. Guo
The
initial conditions are
f/,(0, x, v) =
f
Oβ
{x,
υ)
for
1
^ β ^
N
9
£(0, x) =
E
0
(x)
9
3(0, x) = B
0
(x)
,
[div£
0
= Po
and
div£
0
=
0 .
The
boundary conditions are
'
£ x
n =
0 ,
f
p
(t,
x, v) =
a
p
(t
9
x, Ό)(Kf
p
(t, x, υ)) +
g
p
(t
9
x, υ\ 1
S β g
N,
(
*
)
for
XG3Ω
and n u
<
0, where n is the outward normal vector of 3Ω at x. Here the
reflection operator
is
defined as
Kf(t
9
x, v) =
/(r,
x
9
υ-2(υ
n)n)
9
(0.4)
where
v
—
2(υ n )n is the reflected vector of ϊ; respect to H.
Also
N is the number of
different types of particles with charges e
β
and masses m
β
, c is the speed of light. The
absorption coefficient a
β
(t, x, v) and the boundary source g
β
(t, x, t;) are two given
functions on n- v
< 0
satisfying
either one of the following conditions:
1. Purely specular reflection condition:
a
β
(t
9
x, v)
= 1,
g
β
(t
9
x
9
υ)
= 0. (0.5)
2. Partially absorbing condition:
0
S
a
β
(t, x, v)^a
o
< 1, <^(ί, x, t;)
^ 0 , (0.6)
where
a
0
is a
constant. The purely absorbing condition
is a
β
=
0
and gf^ = 0.
These are two typical kinds of the boundary conditions for transport equations.
The
assumed condition E
x
n =
0
comes naturally from physics when
Ω is
surrounded by
a
perfect conductor. The integrated energy for the non-relativistic
case
is
<^τ
= 4πΣ
J (l
+ \υ\
2
)m
p
f
β
dtdxdυ+
f
(E
2
+ B
2
)dtdx
. (0.7)
/? (0,Γ)xΩxl?
3
(OJ)χβ
Let Xτ(') be the characteristic function of [0, T]. Our main results are as follows.
Theorem
0.1
{Non-relativistic
case). Let dΩ
e
C
liμ
9
for
some
μ >
0. Let
f
oβ
^ 0
a.e.,
for 1
^
j8
^
iV, αnrf
feί £
0
flwd
^o
^
^
2
(Ω)
satisfy
div E
o
=
ρ
0
and di\ B
o
=
0 in
the
sense
of
distributions.
Assume
f
Oβ
(l
+ \υ\
2
)
e
L
1
. In the
purely
specular
case
(0.5),
assume
f
β
e
L°°
n
L
1
.
In the
partially
absorbing
case
(0.6),
assume
f
Oβ
e
L
p
,
χ
τ
g
β
e
L
p
,χ
τ
g
β
(l + |f|)
2
e
L
ί
,for
some
2
^ p ^
oo and all T < oo. TTien ίherβ
^xisί
α
w^α/c
solution
o/(VM) m
0 < t <
oo,
x e
Ω,
v e
i^
3
with
finite
energy
S
τ
,for all
T
< oo.
Moreover,
if f
Oβ
e
L
q
, χ
τ
g
β
e
L
q
,
for
all T < oo, ί/*en χ
Γ
/^
e
L
9
, /or all
T
< oo,
where
2 ^ q ^
oo.
Global
Weak Solutions of the Vlasov-Maxwell System 247
The
relativistic
Vlasov-Maxwell
system (RVM) is the same as (VM) except that
—
is replaced by v =
—=====.
(See [GS1].) The energy is the same as for (VM)
Imj
+ —
r
v
c
/ \v\
2
\ϊ
except that (1 + \v\
2
)m
β
is replaced by 2c
2
1 m\ Λ γ I . We have a parallel
V ^ /
theorem
(Theorem 5.1).
This paper is a
first
attempt to describe the plasma-wall interaction. An
important
potential application is to a tokamak. However, there are several
sources of particle
fluxes
to the
wall,
such as ions and electrons that
diffuse
across
the
confining field, runaway electrons, and neutral particles that are injected into
the
plasma from the
wall.
According to [St], "the physics of the transport processes
within the plasma core and boundary regions and the atomic physics of the
plasma-wall interaction are sufficiently complex and the experimental evidence is
sufficiently limited, that it is
very
difficult to confidently predict the magnitude and
energy distribution of the particle
fluxes
to the
wall."
Because of this uncertainty, it
is useful to remark that our proof works if we replace the second condition in (0.3)
by//? = Jffβ + gβ, and eliminate (0.4), (0.5), (0.6), where X is any linear operator:
L
p
({n-v >0})-+L
p
{{n v <0}), with
\\jf\\
< 1, assuming that 2 <Ξ p < oo.
Arsenev [A]
first
proved the global existence of weak solutions of the
Vlasov-
Poisson system. Using a velocity averaging argument, DiPerna and Lions [DL]
proved the global existence of the weak solutions of the Cauchy problem of the
Vlasov-Maxwell
system. Regularity of the global weak solutions with regular
initial data for the
Vlasov-Maxwell
systems (VM) and (RVM) were proved earlier
by Glassey, Strauss and Schaeffer in [GS1, GS2 and GSc], but they require some
restrictions on the data. In the Vlasov-Poisson case, regularity without extra
restrictions on the data have recently been proved by [Pf, H, Sc and PL].
Greengard
and Raviart [GR] proved the uniqueness and existence of weak
solutions for the one-dimensional stationary Vlasov-Poisson system with bound-
ary conditions. The case of linear transport equations have been studied by many
mathematicians.
In particular, Beals and Protopopescu [BP]
gave
a unified formu-
lation
in a general setting. Cooper and Strauss [CS] treated the general initial-
boundary value problem for the Maxwell system in time-dependent domains.
Even in the case of the full (VM) or (RVM) system without the boundary, as in
[DL],
the questions of uniqueness, regularity and conservation of energy are open,
unless the data is restricted as in [GS1, GS2 and GSc]. We have some positive and
negative results on these questions, which
will
appear in a later paper.
To
prove the existence of the weak solution, we
first
approximate the phase
space Ω x R
3
by a sequence of bounded domains. In each bounded domain, we
approximate a cut-off problem by a sequence of linear
Vlasov
equations and linear
Maxwell systems with suitable new initial and boundary conditions. Using the
results of [BP], we get a sequence of weak solutions (Sect. 2). We take the weak
limits of the solutions of the linear problems and obtain the energy estimate by the
compactness results of [DL] (Sect. 3). Then we get the weak solution of the partial
absorption problem as the limit of the solutions of the cut-off problems. We
approximate the purely specular problem by partial absorption problems (Sect. 4).
Finally we treat the relativistic case (Sect. 5).
248
Y. Guo
1.
Notation and
Weak
Formulation
Definition 1.1. Let Π = (0, oo) x Ω x R
3
, w/zere Ω is an open set in R
3
with
C
ι
'
μ
bioundary, μ > 0. Lei n be the outward normal vector of dΩ at x. Let
y
±
= {(ί,x,ι;)e(0,
oo)xdί2xR
3
|
+ «
ι>>0}
, (1.1)
y° = {(ί, x, υ) e (0, oo) x dΩ x R
3
1 n v = 0} . (1.2)
For
any Γ > 0, let Π
x
= (0, T)xΩ
ί
xV
ί
e Π. Let
.
(1.3)
p!
the LPnorm on Π
u
and let |
|
p;y
±(^)
be the L
p
norm
ony±
with respect to
the
measure \dy
β
l where
dγ
β
= (n —)dσ
x
dvdt, (1.4)
where dσ
x
is the standard surface measure of dΩ
u
and 1 ^ p ^ + oo.
Most
of the
estimates
in
this
paper
depend
on any fixed T, but the
solutions
are
defined
for 0 ^ t < oo .
Definition 1.2. The integrated energy in a region is
δ{f
β9
E, B, Ω
l9
V
u
T) = 4πX m
β
J (1 + \v\
2
)f
β
dtdxdv + J (E
2
+ B
2
)dtdx .
β
Πj_
(OJ)xΩi
(1.5)
We
also define the initial-boundary energy as
β
J (1 + \v\
2
)f
0β
dxdv
/?
ΩxR
3
Ω
+
\v\
2
)0βdy
p
, for a
fixed
Γ>0 .
β
y~
(1.6)
Definition
1.3.
77ιe
ίesί
function
spaces
are
TT
= {α(ί, x, ϋ)
G
C
c
°°([0,
oo)xR
3
xR
3
;R
1
)|
suppαczc {[0,
oo)xΩxR
3
}\{(0xM)uy
0
}}
, (1.7)
t/
#
= {(^φ)|^
G
C
c
α)
([0, oo)xΩ;R
3
),φeC
c
ω
([0, oo)xR
3
;R
3
)} . (1.8)
Definition 1.4.
(Test
functionals). Let Π
1
be as in Definition 1.1. Let f
β
e
LiocC^iX
/o^eLUΩxR
3
),
fβ
eL
ioΛyΐ), and g
β
sLϊ
oc
(yϊ\ with respect to dy
β
, for
l^β^N. Let E and B e
Lt
oc
((0
9
T) x Ω), and E
o
and B
o
e
Lf
oc
(Ω).
Let a
β
e
TT,
Global
Weak
Solutions
of
the
Vlasov-Maxwell
System
249
and
(ψ, φ) e Jί. Define
A
β
(fβJβ,E,
B, (x
β
, Ω
u
V
1
)= - J
foβθίβ{0,
x, v)dxdv - J {δ
t
a
β
+ W
x
a
β
'V
+ (E + vxB)
V
v
a
β
)f
β
dtdxdv+
J
<x
β
f£
dy
β
+
l*
β
{a
β
Kfϊ+g
β
}dy
β9
(1.9)
Γ
C(E
9
B
9
φ
9
j
9
Ω
l9
V
1
)= - J J E-δ
t
φdtdx- J
φ{Q
>
x)
E
o
dx
0
Ωi Ωi
T
Γ
-
J J
cmlψ-Bdtdx
+ J
Iψ-jdtdx,
(1.10)
Ofii
OΩi
where j = 4π§
Vί
Σ
β
ve
β
f
β
dv, and
T
,B,φ,Ω
u
V
t
)= - J J
B-d
t
φdtdx-
J
φ(0,x)
B
o
^
OΩi
Γ
+ J J
cuήφΈdtdx.
(1.11)
0
Ωi
Definition
1.5.
(PTeα/c
solutions).
Let /^ ^ 0 a.e, /^ e
L}
OC
{Π)
9
f
β
^ 0 a.e.,
/7
e
L
1
1
0C
(y
+
) and £,ΰe
Aoc((0»
°°)
x β
) ^^
flr
^ «
weak
solution
of
(VM)
witft
conditions
(0.1)
ί/irowgf/z
(0.3),
ι/Va^ £ ^, V(^, φ) 6 ^T, 1 ^ β ^ N,
,φ,Ω,R
3
) = 0, (1.12)
I
div E = p and div 5 = 0 m ί/z^ sense of distributions .
Since
γ° has zero surface measure (see
[GMP]),
it is
omitted.
Lemma
1.1. div B = 0 is implied by the other conditions in (1.12).
Proof.
For any ζ e C
c
°°([0,
oo)xί2;
R). Assume ζ(t
9
x) = 0 when t > T. Plug
φ
=
p
o
Vζdτ
- $lVζdt
into
£>(£,
B,
φ,Ω,R
3
)
= 0 and the
lemma
follows.
Lemma
1.2. Suppose that f
Όβ
, E
0
,g and a are smooth, that E
o
x n = 0, on dΩ,
thatf
β
,
//,
E and B are a weak solution o/(VM),
ίfcαί/^
e
C
X
(Π),
ίfcαί//
GC
1
^
E,
B e C^O, oo) x Ω), ί/zαί /^
/zαt;^
continuous extensions to y~ u y
+
u {t = 0},
ί/zαί
£ and 5 /zai e continuous extensions to [0, Γ] x Ω,for all T < oo. Γ/ϊβn fβ, fβ ,
E
and B is a classical solution o/(VM) with classical initial and boundary conditions
on
γ
+
u y" u {ί = 0}.
The
proof
is
standard.
Remark.
If n 5
0
= 0 in the
weak
sense
on dΩ,
then
n-B =
0follows
for all t.
Since
div
B
o
= 0, the
weak
form
of n £
0
= 0 is J
o
V ζ β
0
dx_^=
0, V
C
6 C
c
°°
(#
3
).
Choose
a
test
function
ζ
such
that
C(ί,
x) = 0
when
t > T.
Plug
φ = f
0
V ζ dz - j J V ζ dt
into
D
= 0. We get -
\H
Ω
Vζ-Bdtdx
= 0,
which
is the
weak
form
oΐn-B
= 0.