Asymptotic Behavior of the Principal Eigenvalue Problems with Large Divergence-Free Drifts

TL;DR

This paper studies the asymptotic behavior of the principal eigenvalue and eigenfunction in a PDE with large divergence-free drift, proving convergence and describing refined limiting profiles as the drift magnitude increases.

Contribution

It establishes the convergence of the principal eigenpair for large divergence-free drifts and analyzes the detailed limiting profiles, addressing an open question in the field.

Findings

Proved convergence of eigenpair as drift magnitude tends to infinity.

Described refined limiting profiles showing the drift's effects.

Addressed a specific open problem in eigenvalue asymptotics.

Abstract

In this paper, we consider the following principal eigenvalue problem with a large divergence-free drift: \begin{equation}\label{0.1} -\varepsilon\Delta \phi-2\alpha\nabla m(x)\cdot\nabla \phi+V(x)\phi=\lambda_\alpha \phi\ \,\ \text{in}\, \ H_0^1(\Omega),\tag{0.1} \end{equation} where the domain $\Omega\subset \mathbb{R}^N (N\ge 1)$ is bounded with smooth boundary $\partial\Omega$, the constants $\varepsilon>0$ and $\alpha>0$ are the diffusion and drift coefficients, respectively, and $m(x)\in C^{2}(\bar{\Omega})$, $V (x)\in C^{\gamma}(\bar{\Omega})~(0<\gamma<1)$ are given functions. For a class of divergence-free drifts where $m$ is a harmonic function in $\Omega$ and has no first integral in $H_{0}^{1}(\Omega)$, we prove the convergence of the principal eigenpair $(\lambda_\alpha, \phi)$ for (0.1) as $\alpha\rightarrow+\infty$, which addresses a special case of the open question proposed in [H. Berestycki, F. Hamel and N. Nadirashvili, CMP, 2005]. Moreover, we further investigate the refined limiting profiles of the principal eigenpair $(\lambda_\alpha, \phi)$ for (0.1) as $\alpha\rightarrow+\infty$, which display the visible effects of the large divergence-free drifts on the principal eigenpair $(\lambda_\alpha, \phi)$.