We consider the semigroup on
L
1
(
R
n
)
{L^1}\left ( {\mathbb {R}^n} \right )
defined by the nonlinear transport equation for the scalar
s
s
,
\[
∂
t
s
+
d
i
v
(
f
(
s
)
u
)
=
0
i
n
(
0
,
∞
)
×
R
n
{\partial _t}s + div\left ( f\left ( s \right )u \right ) = 0 \qquad in \left ( 0, \infty \right ) \times {\mathbb {R}^n}
\]
for given velocity field
u
u
. We show that this nonlinear semigroup is Hölder continuous for
t
>
0
t > 0
in the uniform operator topology, provided the graph of
f
f
has no linear segments. This continuity property—which expresses a regularizing effect of the nonlinearity in the transport equation—is robust with respect to the spatial behaviour of the time-independent velocity field
u
u
.