On the existence of solutions to the multi-species Landau equation
Jonathan Junn\'e, Raphael Winter, Havva Yolda\c{s}

TL;DR
This paper investigates the existence of solutions to the multi-species Landau equation with different masses, establishing conditions under which a Lyapunov functional exists and introducing a new method for proving global well-posedness.
Contribution
It extends the analysis of the Landau equation to multiple species, identifying a threshold for the collision parameter and developing a novel Lyapunov functional for global well-posedness.
Findings
Existence of a multi-species Fisher information as a Lyapunov functional for certain ; values.
Counterexample showing Fisher information fails as Lyapunov below ; threshold when one species has infinite mass.
Introduction of a new Lyapunov functional based on spherical Fisher information for global well-posedness.
Abstract
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 , where corresponds to Coulomb interactions. We prove that if 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 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…
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Taxonomy
TopicsGas Dynamics and Kinetic Theory · Mathematical Biology Tumor Growth · Nonlinear Partial Differential Equations
