  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.
