Dissipation estimates of the Fisher information for the Landau equation

We establish an a priori estimate for the dissipation of the Fisher information for the space-homogeneous Landau equation with very soft potentials. This work is motivated by the recent breakthrough by Guillen and Silvestre, which proves that the Fisher information is monotone decreasing. As a direct consequence, we show that the Fisher information becomes instantaneously bounded, even if it is not initially bounded. This leads to a proof of the global existence of smooth solutions for the space-homogeneous Landau equation with very soft potentials, given initial data $f_0\in L_{2-\gamma }^1\cap L\log L$. This result includes the case of the Coulomb potential.