The entropy solutions to the partial differential equation
\[
(
∂
/
∂
t
)
u
(
t
,
x
,
y
)
+
(
∂
/
∂
x
)
f
(
u
(
t
,
x
,
y
)
)
+
(
∂
/
∂
y
)
g
(
u
(
t
,
x
,
y
)
)
=
0
,
(\partial /\partial t)u(t,x,y) + (\partial /\partial x)f(u(t,x,y)) + (\partial /\partial y)g(u(t,x,y)) = 0,
\]
with initial data constant in each quadrant of the
(
x
,
y
)
(x,y)
plane, have been constructed and are piecewise smooth under the condition
f
(
u
)
≠
0
,
g
(
u
)
≠
0
,
(
f
(
u
)
/
g
(
u
)
)
′
≠
0
f(u) \ne 0, g(u) \ne 0, (f(u)/g(u))\prime \ne 0
. This problem generalizes to several space dimensions the important Riemann problem for equations in one-space dimension. Although existence and uniqueness of solutions are well known, little is known about the qualitative behavior of solutions. It is this with which we are concerned here.