Non-convergence of the principal eigenvalue of elliptic operators for large advection

TL;DR

This paper demonstrates that the principal eigenvalue of certain elliptic operators with strong advection can diverge, challenging previous assumptions of convergence and revealing complex solution behaviors in oscillatory advection fields.

Contribution

It constructs the first example showing non-convergence of the principal eigenvalue for elliptic operators with strong advection, and explores implications for oscillatory advection-reaction-diffusion models.

Findings

Constructed an example of divergent principal eigenvalue as advection strength increases.

Showed eigenvalue changes sign infinitely often in oscillatory potential fields.

Revealed complex solution behaviors in advection-reaction-diffusion models with oscillatory velocity.

Abstract

This paper investigates the limit of the principal eigenvalue $\lambda(s)$ as $s\to+\infty$ for the following elliptic equation \begin{align*} -\Delta\varphi(x)-2s\mathbf{v}\cdot\nabla\varphi(x)+c(x)\varphi(x)=\lambda(s)\varphi(x), \quad x\in \Omega \end{align*} in a bounded domain $\Omega\subset \mathbb{R}^d (d\geq 1)$ with the Neumann boundary condition. Previous studies have shown that under certain conditions on $\mathbf{v}$, $\lambda(s)$ converges as $s\to\infty$ (including cases where $\lim\limits_{s \to\infty }\lambda(s)=\pm\infty$). This work constructs an example such that $\lambda(s)$ is divergent as $s\to+\infty$. This seems to be the first rigorous result demonstrating the non-convergence of the principal eigenvalue for second-order linear elliptic operators with some strong advection. As an application, we demonstrate that for the classical advection-reaction-diffusion model with advective velocity field $\mathbf{v}=\nabla m$, where $m$ is a potential function with infinite oscillations, the principal eigenvalue changes sign infinitely often along a subsequence of $s\to\infty$. This leads to solution behaviors that differ significantly from those observed when $m$ is non-oscillatory.