On the sharpness of Green's function estimates for a convection-diffusion problem

TL;DR

This paper investigates the sharpness of Green's function estimates for 3D convection-diffusion problems, providing both upper and lower bounds that explicitly depend on the singular perturbation parameter, thus confirming the bounds' tightness.

Contribution

The paper establishes the lower bounds for Green's function estimates, complementing previous upper bounds, and explicitly characterizes their dependence on the singular perturbation parameter.

Findings

Upper bounds for Green's function are sharp.

Lower bounds confirm the tightness of these estimates.

Explicit dependence on the perturbation parameter is demonstrated.

Abstract

Linear singularly perturbed convection-diffusion problems with characteristic layers are considered in three dimensions. We demonstrate the sharpness of our recently obtained upper bounds for the associated Green's function and its derivatives in the $L_1$ norm. For this, in this paper we establish the corresponding lower bounds. Both upper and lower bounds explicitly show any dependence on the singular perturbation parameter.