Spectral gap and exponential convergence to equilibrium for a multi-species Landau system

TL;DR

This paper establishes spectral gap and exponential convergence to equilibrium for a multi-species Landau system with soft potentials, providing new coercivity estimates and analyzing entropy decay.

Contribution

It introduces new constructive coercivity estimates and proves spectral gap existence for the linearized system, demonstrating exponential convergence to equilibrium.

Findings

Spectral gap established for the linearized Landau operator.

Exponential decay of solutions towards equilibrium proven.

Conservation of mass, momentum, and energy verified for the system.

Abstract

In this paper we prove new constructive coercivity estimates and convergence to equilibrium for a spatially non-homogeneous system of Landau equations with soft potentials. We show that the nonlinear collision operator conserves each species' mass, total momentum, total energy and that the Boltzmann entropy is nonincreasing along solutions of the system. The entropy decay vanishes if and only if the Boltzmann distributions of the single species are Maxwellians with the same momentum and energy. A linearization of the collision operator is computed, which has the same conservation properties as its nonlinear counterpart. We show that the linearized system dissipates a quadratic entropy, and prove existence of spectral gap and exponential decay of the solution towards the global equilibrium.