# Sharp Decay Estimates in Local Sensitivity Analysis for Evolution Equations with Uncertainties: from ODEs to Linear Kinetic Equations

## TL;DR

This paper develops sharp decay estimates for evolution equations with uncertainties by extending Lyapunov functional methods, providing uniform long-term behavior analysis for systems including ODEs, convection-diffusion, BGK, and Fokker-Planck equations.

## Contribution

It introduces explicit Lyapunov functionals for defective systems and applies them to derive uniform decay estimates in uncertain evolution equations.

## Key findings

- Derived sharp decay estimates of order (1+t^M)e^{-or systems with uncertainties.
- Extended Lyapunov functional method to defective ODE systems.
- Proved uniform decay behavior in the non-defective limit.

## Abstract

We review the Lyapunov functional method for linear ODEs and give an explicit construction of such functionals that yields sharp decay estimates, including an extension to defective ODE systems. As an application, we consider three evolution equations, namely the linear convection-diffusion equation, the two velocity BGK model and the Fokker-Planck equation. Adding an uncertainty parameter to the equations and analyzing its linear sensitivity leads to defective ODE systems. By applying the Lyapunov functional construction, we prove sharp long time behavior of order $(1+t^{M})e^{-\mu t}$, where $M$ is the defect and $\mu$ is the spectral gap of the system. The appearance of the uncertainty parameter in the three applications makes it important to have decay estimates that are uniform in the non-defective limit.