On fast and slow times in models with diffusion

TL;DR

This paper analyzes the interplay of wave and diffusion effects in models with the Kelvin-Voigt operator, deriving asymptotic behavior of Green functions to understand how solutions transition from wave-like to diffusion-dominated over different timescales.

Contribution

It introduces a rigorous analysis of the asymptotic behavior of solutions in models with diffusion perturbations, highlighting the interaction between fast wave and slow diffusion timescales.

Findings

Wave behavior is a good approximation for small times as psilon 0.

Diffusion effects dominate for large times, especially when t > 1/psilon.

Pure waves are quasi-undamped in the interval (psilon, 1/psilon).

Abstract

The linear Kelvin{Voigt operator L_\epsilon is a typical example of wave operator L_0 perturbed by higher-order viscous terms as \epsilonu_xxt. If P\epsilon is a prefixed boundary value problem for L_\epsilon, when \epsilon = 0, L_\epsilon turns into L_0 and P_\epsilon into a problem P_0 with the same initial{boundary conditions of P\epsilon. Boundary layers are missing and the related control terms depending on the fast time are negligible. In a small time interval, the wave behavior is a realistic approximation of u_\epsilon when \epsilon \rightarrow 0. On the contrary, when t is large, diffusion effects should prevail and the behavior of u_\epsilon for \epsilon \rightarrow 0 and t \rightarrow 1 should be analyzed. For this, a suitable functional correspondence between the Green functions G_\epsilon and G_0 of P_epsilon and P_0 is derived and its asymptotic behavior is rigorously examined. As a consequence, the interaction between diffusion effects and pure waves is evaluated by means of the slow time \epsilont; the main results show that in time intervals as (\epsilon; 1/epsilon) pure waves are quasi-undamped, while damped oscillations predominate as from the instant t > 1/\epsilon.