Resolvent estimates for a two-dimensional non-self-adjoint operator
type: Item
instance of: scholarly article
title: Resolvent estimates for a two-dimensional non-self-adjoint operator
zbMATH Open document ID: 1276.47060
author: Wen Deng
published in: Communications on Pure and Applied Analysis
publication date: 2013-04-30T00:00:00Z
review text: The author considers a two-dimensional non-self-adjoint differential operator, originating from a stability problem for the two-dimensional Navier-Stokes equation, given by \[ {\mathcal L}_\alpha = -\Delta + |x|^2 + \alpha \sigma(|x|)\partial_\theta, \] where \(\sigma(r) = r^{-2}(1 - e^{-r^2})\), \(\partial_\theta = x_1\partial_2 - x_2\partial_1\), and \(\alpha\) is a positive parameter tending to \(+\infty\). A complete study is proposed of the resolvent of \({\mathcal L}_\alpha\) along the imaginary axis in the fast rotation limit \(\alpha \to +\infty\) and the optimal estimate \[ \sup_{\lambda \in {\mathbb R}}||({\mathcal L}_\alpha - i\lambda)^{-1}||_{{\mathcal L}(\tilde{L}^2({\mathbb R}^2))} \leq C\alpha^{-1/3} \] is derived. The proofs are based on a multiplier method, metrics on the phase space and localization techniques.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 6159441
zbMATH Keywords: multiplier method
zbMATH Keywords: metric on the phase space
zbMATH Keywords: localization technique
describes a project that uses: Eigtool
MaRDI profile type: Publication
full work available at URL: cpaa.2013.12.547
OpenAlex ID: W2312406004
DOI: 10.3934/CPAA.2013.12.547
description: scientific article; zbMATH DE number 6159441