  We analyze optimal transport problems with additional entropic cost evaluated
along curves in the Wasserstein space which join two probability measures
$m_0,m_1$. The effect of the additional entropy functional results into an
elliptic regularization for the (so-called) Kantorovich potentials of the dual
problem. Assuming the initial and terminal measures to be positive and smooth,
we prove that the optimal curve remains smooth for all time. We focus on the
case that the transport problem is set on a convex bounded domain in the
$d$-dimensional Euclidean space (with no-flux condition on the boundary), but
we also mention the case of Gaussian-like measures in the whole space. The
approach follows ideas introduced by P.-L. Lions in the theory of mean-field
games \cite{L-college}. The result provides with a smooth approximation of
minimizers in optimization problems with penalizing congestion terms, which
appear in mean-field control or mean-field planning problems. This allows us to
exploit new estimates for this kind of problems by using displacement convexity
properties in the Eulerian approach.
