Exponential decay phenomenon of the principal eigenvalue of an elliptic operator with a large drift term of gradient type

TL;DR

This paper investigates how the principal eigenvalue of an elliptic operator with a large gradient-based drift term decays exponentially under certain conditions, revealing asymptotic behavior as the drift magnitude increases.

Contribution

It establishes the exponential decay of the principal eigenvalue for elliptic operators with large gradient-driven advection under potential well conditions.

Findings

Principal eigenvalue decays exponentially with increasing drift magnitude.

Decay rate depends on the potential well condition.

Provides asymptotic analysis of eigenvalue behavior for large advection.

Abstract

We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the velocity potential well condition as the parameter tends to infinity.