Understanding the Landau equation as a gradient flow

This thesis provides and investigates the rigorous gradient flow viewpoint of the spatially homogeneous Landau equation for soft potentials. This extends the similar perspective recently developed for the Boltzmann equation with Maxwellian potentials by Erbar. Taking advantage of the H-theorem for the Landau equation, we construct a metric from a dynamic optimal transportation viewpoint (Benamou-Brenier and Dolbeault-Nazaret-Savaré) for which the Landau equation can be viewed as the gradient flow of the Boltzmann entropy with respect to this metric. The gradient flow description is further reinforced by recovering the grazing collision limit from the Boltzmann equation to the Landau equation with simpler arguments and intuition in the spirit of Gamma-convergence (Sandier-Serfaty). Finally, a robust and efficient particle approximation is numerically analysed for a regularised version of the Landau equation which preserves the gradient flow structure. This thesis invites and advertises the use of gradient flow techniques to pursue some of the future research directions discussed here.