Diffusion limits 213
It is necessary to point out that many of the forthcoming results are deeply connected to the linearity of the problem, and they are not easily extendible to nonlinear situations. This will be
the object of our future research.
2. A maximum principle
In this section we prove a maximum principle for system 1. Local existence of solutions of such kind of hyperbolic systems is well known see, for example, [12] and the references within.
Therefore, a maximum principle implies that the solution is global and unique. Since the problem is linear, we will consider some different sub-problems, which are easier to study.
P
ROBLEM
5.1. We study the system
4
∂ u
1
∂ t
+ 1
ε ∂
u
1
∂ x
=
1 ε
2
v
1
− u
1
∂v
1
∂ t
− 1
ε ∂v
1
∂ x
=
1 ε
2
u
1
− v
1
, with the initial and boundary conditions:
u
1
x, 0 = u x ∈ L
∞
v
1
x, 0 = v x ∈ L
∞
u
1
− L , t = 0
v
1
+ L , t = 0.
P
ROBLEM
5.2. We study the system 5
1
ε ∂
u
2
∂ x
=
1 ε
2
v
2
− u
2
1 ε
∂v
2
∂ x
=
1 ε
2
v
2
− u
2
, with boundary conditions:
u
2
− L , t = ϕ
−
t ∈ W
1,∞
0, T v
2
+ L , t = ϕ
+
t ∈ W
1,∞
0, T . P
ROBLEM
5.3. We study the system 6
∂
u
3
∂ t
+ 1
ε ∂
u
3
∂ x
=
1 ε
2
v
3
− u
3
+ f
ε
x, t ∂v
3
∂ t
− 1
ε ∂v
3
∂ x
=
1 ε
2
u
3
− v
3
+ g
ε
x, t , where f
ε
x, t and g
ε
x, t are suitable functions that will be specified later, with the following initial and boundary conditions:
214 F. Salvarani
u
3
x, 0 = 0 v
3
x, 0 = 0 u
3
− L , t = 0
v
3
+ L , t = 0.
The functions u
ε
x, t = u
1
x, t + u
2
x, t + u
3
x, t , v
ε
x, t = v
1
x, t + v
2
x, t + v
3
x, t satisfy, by linearity, the differential system 1, with the correct initial-boundary conditions, pro-
vided that f
ε
x, t = − ∂
u
2
∂ t
and g
ε
x, t = − ∂v
2
∂ t
. In addition, if a maximum principle holds separately for Problem 5.1, 5.2, and 5.3, then the
original problem admits itself a maximum principle. In order to obtain bounds for Problem 5.1, we multiply the first equation of system 4 by
2 pu
2 p−1 1
and the second one by 2 pv
2 p−1 1
p ∈ N:
7
∂ u
2 p 1
∂ t
+ 1
ε ∂
u
2 p 1
∂ x
=
2 ε
2
pu
2 p−1 1
v
1
− u
1
∂v
2 p 1
∂ t
− 1
ε ∂v
2 p 1
∂ x
=
2 ε
2
pv
2 p−1 1
u
1
− v
1
. By integration over , adding the resulting equations and using the boundary conditions, we
have: d
dt Z
u
2 p 1
+ v
2 p 1
d x + 1
ε h
u
2 p 1
L , t + v
2 p 1
− L , t
i =
8 =
2 p ε
2
Z
u
1
− v
1
v
2 p−1 1
− u
2 p−1 1
d x ≤ 0. Thus we deduce, at least formally, that
d dt
Z
h u
2 p 1
+ v
2 p 1
i d x ≤ 0
for all t ≥ 0. Letting p go to +∞, we find that 9
max {ku
1
t k
∞
, kv
1
t k
∞
} ≤ max {ku k
∞
, kv k
∞
} .
Diffusion limits 215
This proves the following lemma. L
EMMA
5.1. Let u
1
x, 0 = u x, v
1
x, 0 = v x ∈ L
∞
and u
1
− L , t = 0,
v
1
+ L , t = 0, for all t 0. Then u
1
x, t , v
1
x, t ∈ L
∞
and
max {ku
1
t k
∞
, kv
1
t k
∞
} ≤ max {ku k
∞
, kv k
∞
} . In order to study the so called “stationary problem”, we subtract the two equations of sys-
tem 5, finding ∂
∂ x
u
2
− v
2
= 0;
this means that 10
u
2
x, t = v
2
x, t + αt almost everywhere in , where αt is a function that will be determined later.
Moreover, adding the two equations of system 5, we find that ∂
∂ x
u
2
+ v
2
= 2
ε v
2
− u
2
. By using 10 we have
11 ∂v
2
∂ x
x, t = − α
t ε
. If we integrate 11 on the interval L , x, we obtain
12 v
2
x, t = ϕ
+
t − α
t ε
x − L. Using 10 at x = −L, we have v
2
− L , t = ϕ
−
t − αt ; this result, joined to 12, leads to conclude that
α t = ε
ϕ
−
t − ϕ
+
t ε +
2L .
Similarly, by integrating on −L , x, we can prove that u
2
x, t = ϕ
−
t − α
t ε
x + L. Thus we have proved the following lemma.
L
EMMA
5.2. The solution of Problem 5.2 is given by the two functions u
2
x, t = ϕ
−
t − ϕ
−
t − ϕ
+
t ε +
2L x + L
216 F. Salvarani
v
2
x, t = ϕ
+
t − ϕ
−
t − ϕ
+
t ε +
2L x − L.
Thanks to the hypotheses on the boundary conditions ϕ
−
t and ϕ
+
t , u
2
x, t , v
2
x, t ∈ W
1,∞
0, T ; C
∞
¯
, uniformly in ε. Problem 5.3 needs a slightly more complicated proof, which will be given in several steps.
We first notice that f
ε
, g
ε
belong to L
∞
0, T ; C
∞
¯
by Lemma 5.2. Then, we multiply the first equation of system 6 for 2 pu
2 p−1 3
and the second one by 2 pv
2 p−1 3
; then we integrate on and add the two obtained equations:
13 d
dt Z
| u
3
|
2 p
+ |v
3
|
2 p
d x ≤ 2 p Z
| f
ε
|| u
3
|
2 p−1
+ | g
ε
||v
3
|
2 p−1
d x. Let M be
M = ess sup
x ∈, t ∈0,T
{| f
ε
|, | g
ε
|}. Then, inequality 13 becomes
d dt
Z
| u
3
|
2 p
+ |v
3
|
2 p
d x ≤ 2 pM Z
| u
3
|
2 p−1
+ |v
3
|
2 p−1
d x. Now, by the H¨older-inequality we have:
Z
| u
3
|
2 p−1
d x ≤ 2L
12 p
Z
| u
3
|
2 p
d x
2 p−1 2 p
. By the algebraic inequality
a
c
+ b
c
≤ 4a + b
c
a, b ≥ 0, 1
2 ≤
c ≤ 1, we obtain
d dt
Z
| u
3
|
2 p
+ |v
3
|
2 p
d x ≤ 8 pM2L
12 p
Z
| u
3
|
2 p
+ |v
3
|
2 p
d x
2 p−1 2 p
. Letting
yt = Z
[|u
3
x, t |
2 p
+ |v
3
x, t |
2 p
]d x, we must now solve the ordinary differential inequality:
d dt
yt ≤ 8 pM2L
12 p
yt
2 p−1 2 p
. Its solution is
yt
12 p
≤ y0
12 p
+ 4M2L
12 p
t, i. e., thanks to the initial conditions u
3
x, 0 = v
3
x, 0 = 0
Diffusion limits 217
14 Z
| u
3
|
2 p
+ |v
3
|
2 p
d x
12 p
≤ 4M2L
12 p
t. Finally, letting p → +∞, it is possible to show that, for any finite time, the solution of
Problem 5.3 is essentially bounded. In conclusion, considering our global problem, by linearity we have proved the following
theorem. T
HEOREM
5.1. Let u
ε
x, 0 = u x, v
ε
x, 0 = v x ∈ L
∞
and u
ε
− L , t =
ϕ
−
t , v
ε
+ L , t = ϕ
+
t ∈ W
1,∞
0, T , for all T 0. Then u
ε
x, t , v
ε
x, t ∈ L
∞
for
all t ∈ [0, T ], uniformly in ε.
3. The solution on the boundary
