A Counterexample for the Principal Eigenvalue of An Elliptic Operator with Large Advection

TL;DR

This paper constructs a specific oscillating advection term in an elliptic eigenvalue problem, demonstrating that the principal eigenvalue can fail to converge as the advection strength increases, which was previously unproven.

Contribution

It provides the first example showing the non-convergence of the principal eigenvalue for an elliptic operator with large oscillating advection.

Findings

Principal eigenvalue does not necessarily converge as advection grows large.

Constructs an oscillating gradient advection term causing non-convergence.

First known example of non-convergence in this context.

Abstract

There are numerous studies focusing on the convergence of the principal eigenvalue $\lambda(s)$ as $s\to+\infty$ corresponding to the elliptic eigenvalue problem \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*} where $\Omega$ is a bounded domain and the advection term $\mathbf{v}$ under some certain restrictions. In this paper, we construct an infinitely oscillating gradient advection term $\mathbf{v}=\nabla m(x)\in C^1(\Omega)$ such that the principal eigenvalue $\lambda(s)$ does not converge as $s\to+\infty$. As far as we know, this is the first result that guarantee the non-convergence of the principal eigenvalue.