Refined Limiting Profiles of the Principal Eigenvalue Problems with Large Advection

TL;DR

This paper investigates how large advection influences the principal eigenvalue and eigenfunction profiles in a bounded domain, providing refined asymptotic descriptions as the advection coefficient grows infinitely large.

Contribution

It offers a detailed analysis of the limiting behavior of the principal eigenpair under large advection, extending understanding of eigenvalue problems with strong advection effects.

Findings

Refined asymptotic profiles of eigenvalues and eigenfunctions as advection becomes large.

Demonstrates the dominant effect of advection on eigenpair behavior.

Potential applicability to broader principal eigenvalue problems.

Abstract

In this paper, we are concerned with the following eigenvalue problem with an advection term: \begin{equation}\label{0.1} \left\{ \begin{split} -\epsilon\Delta \phi-2\alpha\nabla m(x)\cdot\nabla \phi+V(x)\phi&=\lambda \phi\ \ \text{in}\ \ \Omega,\\ \phi&=0\ \ \hbox{on}\ \ \partial\Omega, ~~~\text{(0.1)} \end{split} \right. \end{equation} where $\Omega\subset\mathbb{R}^N~(N\geq1)$ satisfying $\partial\Omega\in C^{2}$ is a bounded domain and contains the origin as an interior point, the constants $\epsilon>0$ and $\alpha>0$ are the diffusive and advection coefficients, respectively, and $m(x)\in C^{2}(\bar{\Omega})$, $V (x)\in C^{\gamma}(\bar{\Omega})~(0<\gamma<1)$ are given functions. We analyze the refined limiting profiles of the principal eigenpair $(\lambda, \phi)$ for (0.1) as $\alpha\rightarrow\infty$, which display the visible effect of the large advection on $(\lambda, \phi)$. It expects that our argument is applicable to investigating the refined expansions of the general principal eigenvalue problems.