Abstract
We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition:
u
t
+
div
Φ
(
u
)
=
f
on
Q
=
(
0
,
T
)
×
Ω
,
u
(
0
,
⋅
)
=
u
0
on
Ω and “
u
=
a
on some part of the boundary
(
0
,
T
)
×
∂
Ω
.” Existence and uniqueness of the entropy solution is established for any
Φ
∈
C
(
R
;
R
N
)
,
u
0
∈
L
∞
(
Ω
)
,
f
∈
L
∞
(
Q
)
,
a
∈
L
∞
(
(
0
,
T
)
×
∂
Ω
)
. In the
L
1
-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.