On optimal decay estimates for ODEs and PDEs with modal decomposition

TL;DR

This paper constructs optimal Lyapunov functionals for the Goldstein-Taylor model, providing sharp decay estimates for convergence to steady state, and characterizes optimal decay rates for 2D linear ODE systems.

Contribution

It introduces a method to construct optimal Lyapunov functionals for the Goldstein-Taylor model and characterizes decay rates for 2D ODE systems with positive stable matrices.

Findings

Optimal Lyapunov functional yields sharp exponential decay rates.

Complete characterization of decay rates for 2D ODE systems.

Partial results for higher-dimensional ODE systems.

Abstract

We consider the Goldstein-Taylor model, which is a 2-velocity BGK model, and construct the "optimal" Lyapunov functional to quantify the convergence to the unique normalized steady state. The Lyapunov functional is optimal in the sense that it yields decay estimates in $L^2$-norm with the sharp exponential decay rate and minimal multiplicative constant. The modal decomposition of the Goldstein-Taylor model leads to the study of a family of 2-dimensional ODE systems. Therefore we discuss the characterization of "optimal" Lyapunov functionals for linear ODE systems with positive stable diagonalizable matrices. We give a complete answer for optimal decay rates of 2-dimensional ODE systems, and a partial answer for higher dimensional ODE systems.