Refined Limiting Profiles of the Principal Eigenvalue Problems with Large Advection
Yujin Guo, Yuan Lou, Hongfei Zhang
TL;DR
This work analyzes the principal eigenvalue problem with large advection in a bounded domain, establishing refined asymptotic profiles for the principal eigenpair as the advection parameter $\alpha$ grows. By combining Pohozaev-type identities with an $L^2$ constraint, the authors show that the eigenvalue scales like $\lambda(\alpha) \sim \alpha^{2/p}\hat{\lambda}$ with subsequent corrections determined by the local behavior of $V$ near the origin or by a prescribed homogeneous function $\hat{h}$. The corresponding eigenfunction concentrates near the origin of $m(x)$, with a precise scaled limit $\tilde{w}_{\alpha}(x)=\alpha^{-N/(2p)}u_{\alpha}(\alpha^{-1/p}x) \to \hat{u}(x)$ and higher-order corrections $\hat{\psi}_{i}$. The results cover both smooth $V$ near the origin and weaker regularity through a homogeneous-function framework, providing a systematic approach to refined expansions for general principal eigenvalue problems with large advection.
Abstract
In this paper, we are concerned with the following eigenvalue problem with an advection term: \begin{equation}\label{0.1} \left\{ \begin{split} -εΔφ-2α\nabla m(x)\cdot\nabla φ+V(x)φ&=λφ \ \text{in}\ \ Ω,\\ φ&=0\ \ \hbox{on}\ \ \partialΩ, ~~~\text{(0.1)} \end{split} \right. \end{equation} where $Ω\subset\mathbb{R}^N~(N\geq1)$ satisfying $\partialΩ\in C^{2}$ is a bounded domain and contains the origin as an interior point, the constants $ε>0$ and $α>0$ are the diffusive and advection coefficients, respectively, and $m(x)\in C^{2}(\barΩ)$, $V (x)\in C^γ(\barΩ)~(0<γ<1)$ are given functions. We analyze the refined limiting profiles of the principal eigenpair $(λ, φ)$ for (0.1) as $α\rightarrow\infty$, which display the visible effect of the large advection on $(λ, φ)$. It expects that our argument is applicable to investigating the refined expansions of the general principal eigenvalue problems.