Non-convergence of the principal eigenvalue of elliptic operators for large advection
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908437029322752 |
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| author | Bai, Xueli Wang, Zhi-An Xu, Xin Zhang, Kexin Zhou, Maolin |
| author_facet | Bai, Xueli Wang, Zhi-An Xu, Xin Zhang, Kexin Zhou, Maolin |
| contents | This paper investigates the limit of the principal eigenvalue $λ(s)$ as $s\to+\infty$ for the following elliptic equation
\begin{align*} -Δφ(x)-2s\mathbf{v}\cdot\nablaφ(x)+c(x)φ(x)=λ(s)φ(x), \quad x\in Ω
\end{align*} in a bounded domain $Ω\subset \mathbb{R}^d (d\geq 1)$ with the Neumann boundary condition. Previous studies have shown that under certain conditions on $\mathbf{v}$, $λ(s)$ converges as $s\to\infty$ (including cases where $\lim\limits_{s \to\infty }λ(s)=\pm\infty$). This work constructs an example such that $λ(s)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04418 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-convergence of the principal eigenvalue of elliptic operators for large advection Bai, Xueli Wang, Zhi-An Xu, Xin Zhang, Kexin Zhou, Maolin Analysis of PDEs 35P15, 35P20, 34C25 This paper investigates the limit of the principal eigenvalue $λ(s)$ as $s\to+\infty$ for the following elliptic equation \begin{align*} -Δφ(x)-2s\mathbf{v}\cdot\nablaφ(x)+c(x)φ(x)=λ(s)φ(x), \quad x\in Ω \end{align*} in a bounded domain $Ω\subset \mathbb{R}^d (d\geq 1)$ with the Neumann boundary condition. Previous studies have shown that under certain conditions on $\mathbf{v}$, $λ(s)$ converges as $s\to\infty$ (including cases where $\lim\limits_{s \to\infty }λ(s)=\pm\infty$). This work constructs an example such that $λ(s)$ 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. |
| title | Non-convergence of the principal eigenvalue of elliptic operators for large advection |
| topic | Analysis of PDEs 35P15, 35P20, 34C25 |
| url | https://arxiv.org/abs/2507.04418 |