From Fisher information decay for the Kac model to the Landau-Coulomb hierarchy

We consider the Kac model for the space-homogeneous Landau equation with the Coulomb potential. We show that the Fisher information of the Liouville equation for the unmodified $N$-particle system is monotonically decreasing in time. The monotonicity ensures the compactness to derive a weak solution of the Landau hierarchy.