Sharp decay estimates in local sensitivity analysis for evolution equations with uncertainties: From ODEs to linear kinetic equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:268

Article

Arnold, Anton¹  Jin, Shi²  Woehrer, Tobias¹ 

[1] TU Vienna, Inst Anal & Sci Comp, Wiedner Hauptstr 8-10, A-1040 Vienna, Austria

[2] Shanghai Jiao Tong Univ, Sch Math Sci, Inst Nat Sci, MOE LSC, Shanghai 200240, Peoples R China

关键词: Long time behavior;    Defective ODEs;    Kinetic equations;    Lyapunov functionals;    Uncertainty quantification;    Sensitivity analysis;   

DOI  :  10.1016/j.jde.2019.08.047

来源: Elsevier

【 摘 要 】

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. (C) 2019 Published by Elsevier Inc.

【 授权许可】

Free