We propose a gradient flow perspective to the spatially homogeneous Landau
equation for soft potentials. We construct a tailored metric on the space of
probability measures based on the entropy dissipation of the Landau equation.
Under this metric, the Landau equation can be characterized as the gradient
flow of the Boltzmann entropy. In particular, we characterize the dynamics of
the PDE through a functional inequality which is usually referred as the Energy
Dissipation Inequality. Furthermore, analogous to the optimal transportation
setting, we show that this interpretation can be used in a minimizing movement
scheme to construct solutions to a regularized Landau equation.
