  We establish the well-posedness and some quantitative stability of the
spatially homogeneous Landau equation for hard potentials, using some specific
Monge-Kantorovich cost, assuming only that the initial condition is a
probability measure with a finite moment of order $p$ for some $p>2$. As a
consequence, we extend previous regularity results and show that all
non-degenerate measure-valued solutions to the Landau equation, with a finite
initial energy, immediately admit analytic densities with finite entropy. Along
the way, we prove that the Landau equation instantaneously creates Gaussian
moments. We also show existence of weak solutions under the only assumption of
finite initial energy.
