On the existence of solutions to the multi-species Landau equation

We consider the spatially homogeneous Landau equation for multiple species with different masses. As in the single-species case, the singularity of the collision operator is determined by a parameter $\gamma \in [-3,1]$, where $\gamma = -3$ corresponds to Coulomb interactions. We prove that if $\gamma\geq -\sqrt{8}$ in the cross-interaction operators, then there exists a natural multi-species generalization of the Fisher information which is a Lyapunov functional for the multi-species Landau system. On the other hand, we give a counterexample showing that the Fisher information is in general no longer a Lyapunov functional below the threshold $(\gamma < - \sqrt{8})$ for the two-species system if one species has infinite mass. However, we are able to provide a new method to show global well-posedness, by constructing a different Lyapunov functional based on the spherical Fisher information.