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

2019/04/01 by Anton Arnold, Shi Jin, Arnold, Anton +3
Economics, Econometrics and Finance · Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Stability and Controllability of Differential Equations #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1904.01190

openalex publication_date 2019/04/01 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

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

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