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.

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