Green's function estimates for a 2d singularly perturbed convection-diffusion problem: extended analysis

TL;DR

This paper provides sharp $L_1$ estimates for the Green's function and its derivatives in a 2D singularly perturbed convection-diffusion problem, including mixed derivatives and Neumann boundary conditions, enhancing understanding of boundary layer behaviors.

Contribution

It extends previous work by estimating the mixed second-order derivatives of the Green's function and addresses Neumann boundary conditions, offering more comprehensive analytical tools.

Findings

Sharp $L_1$ estimates for Green's function derivatives

Explicit dependence on the small diffusion parameter

Addresses Neumann boundary conditions along characteristic boundaries

Abstract

This paper presents an extended version of the article [Franz, S., Kopteva, N.: J. Differential Equations, 252 (2012)]. The main improvement compared to the latter is in that here we additionally estimate the mixed second-order derivative of the Green's function. The case of Neumann conditions along the characteristic boundaries is also addressed. A singularly perturbed convection-diffusion problem is posed in the unit square with a horizontal convective direction. Its solutions exhibit parabolic and exponential boundary layers. Sharp estimates of the Green's function and its first- and second-order derivatives are derived in the $L_1$ norm. The dependence of these estimates on the small diffusion parameter is shown explicitly. The obtained estimates will be used in a forthcoming numerical analysis of the considered problem.