Asymptotic Behavior of the Principal Eigenvalue Problems with Large Divergence-Free Drifts

TL;DR

This work analyzes the principal eigenvalue problem with a large divergence-free drift on a bounded smooth domain and reveals how the drift dominates the asymptotics when the drift is of quadratic form with zero sum of coefficients. Using blow-up and variational techniques, the authors show that $\alpha^{-1}\lambda_\alpha$ converges to $\mu=2\sum|a_i|$ and the rescaled eigenfunction converges to an explicit Gaussian profile $Q$, independent of the potential to leading order. They then derive refined limiting profiles: higher-order expansions of $\lambda_\alpha$ and $u_\alpha$ in terms of derivatives of the potential $V$ at the origin, with $w_\alpha=Q+\alpha^{-3/2}\varphi_1+\alpha^{-2}\varphi_2+\cdots$, where $\varphi_i$ solve linear problems. Further, the results are extended to lower regularity near the origin and to the case where $V$ behaves like a homogeneous function $h_0$, yielding explicit second-order corrections. Altogether, the paper provides a detailed description of how large divergence-free drifts shape the principal eigenpairs and their concentration behavior, with explicit limiting profiles and higher-order asymptotics.

Abstract

In this paper, we consider the following principal eigenvalue problem with a large divergence-free drift: \begin{equation}\label{0.1} -\varepsilonΔφ-2α\nabla m(x)\cdot\nabla φ+V(x)φ=λ_αφ \,\ \text{in}\, \ H_0^1(Ω),\tag{0.1} \end{equation} where the domain $Ω\subset \mathbb{R}^N (N\ge 1)$ is bounded with smooth boundary $\partialΩ$, the constants $\varepsilon>0$ and $α>0$ are the diffusion and drift coefficients, respectively, and $m(x)\in C^{2}(\barΩ)$, $V (x)\in C^γ(\barΩ)~(0<γ<1)$ are given functions. For a class of divergence-free drifts where $m$ is a harmonic function in $Ω$ and has no first integral in $H_{0}^{1}(Ω)$, we prove the convergence of the principal eigenpair $(λ_α, φ)$ for (0.1) as $α\rightarrow+\infty$, which addresses a special case of the open question proposed in [H. Berestycki, F. Hamel and N. Nadirashvili, CMP, 2005]. Moreover, we further investigate the refined limiting profiles of the principal eigenpair $(λ_α, φ)$ for (0.1) as $α\rightarrow+\infty$, which display the visible effects of the large divergence-free drifts on the principal eigenpair $(λ_α, φ)$.

Theorems & Definitions (13)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Definition 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• Corollary 2.5