Asymptotic Solutions of 1-D Singularly Perturbed Convection-Diffusion Equations with a Turning Point: The Compatible Case

Abstract

In this article, we consider a convection-diffusion equation with a small diffusion coefficient . It is a version of a linearized Navier-Stokes equation. Due to the small parameter multiplied to the highest order of differential operators, the so-called turning point transition layers are displayed where flows in opposite directions collide. For example, a turning point can be observed where the Kuroshio and Kurile Currents meet, from opposite directions, in the North Pacific. Unlike boundary layers, turning point transition layers occur where the convective flows collide and more delicate analysis is necessitated. Especially, we consider a single turning point with multiple-orders in one-dimensional spaces and provide sharp estimations for the solution with compatible conditions. A main difficulty when solving the problem arises from the fact that the diffusion coefficient is very small in comparison with other terms and it causes a singularity in the solution. We use the asymptotic analysis that is different from typical methods in the singular perturbation problem considered here. The matching technique has been typically used, but this method brings about the difficulty in constructing a globally matched solution. Our method is relatively easy to analyze, and turning point transition layers are systematically and easily constructed.