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We give an explicit bound for the Wasserstein distance with quadratic costbetween the solutions of Boltzmann's and Landau's equations in the case of softand Coulomb potentials. This gives an explicit rate of convergence for thegrazing collisions limit. Our result is local in time for very soft and Coulombpotentials and global in time for moderately soft potentials.

Asymptotic of grazing collisions for the spatially homogeneous Boltzmann equation for soft and Coulomb potentials