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

This paper investigates the limit of the principal eigenvalue $\lambda (s)$ as $s\to +\mathrm{\infty }$ for the following elliptic equation \begin{align*} -Δ\varphi(x)-2s$\mathbf{v}$\cdot\nabla\varphi(x)+c(x)\varphi(x)=λ(s)\varphi(x), \quad x\in Ω \end{align*} in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d(d\ge 1)$ with the Neumann boundary condition. Previous studies have shown that under certain conditions on $\mathbf{v}$, $\lambda (s)$ converges as $s\to \mathrm{\infty }$ (including cases where $\underset{s\to \mathrm{\infty }}{\lim }\lambda (s)=\pm \mathrm{\infty }$). This work constructs an example such that $\lambda (s)$ is divergent as $s\to +\mathrm{\infty }$. 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 $\mathbf{v}=\mathrm{\nabla }m$, where $m$ is a potential function with infinite oscillations, the principal eigenvalue changes sign infinitely often along a subsequence of $s\to \mathrm{\infty }$. This leads to solution behaviors that differ significantly from those observed when $m$ is non-oscillatory.