  This paper provides results on Wasserstein gradient flows between measures on
the real line. Utilizing the isometric embedding of the Wasserstein space
$\mathcal P_2(\mathbb R)$ into the Hilbert space $L_2((0,1))$, Wasserstein
gradient flows of functionals on $\mathcal P_2(\mathbb R)$ can be characterized
as subgradient flows of associated functionals on $L_2((0,1))$. For the maximum
mean discrepancy functional $\mathcal F_\nu := \mathcal D^2_K(\cdot, \nu)$ with
the non-smooth negative distance kernel $K(x,y) = -|x-y|$, we deduce a formula
for the associated functional. This functional appears to be convex, and we
show that $\mathcal F_\nu$ is convex along (generalized) geodesics. For the
Dirac measure $\nu = \delta_q$, $q \in \mathbb R$ as end point of the flow,
this enables us to determine the Wasserstein gradient flows analytically.
Various examples of Wasserstein gradient flows are given for illustration.
