Resolvent estimates for a two-dimensional non-self-adjoint operator

Wen Deng

doi:10.3934/cpaa.2013.12.547

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2013, Volume 12, Issue 1: 547-596. Doi: 10.3934/cpaa.2013.12.547

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Resolvent estimates for a two-dimensional non-self-adjoint operator

Wen Deng^1,

1.
Institut de Mathématiques de Jussieu, Université Pierre-et-Marie-Curie (Paris 6), 4 place Jussieu, 75005 Paris

Received: April 30, 2011

Revised: December 31, 2011

Published: December 2012

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Abstract

We consider a two-dimensional non-self-adjoint differential operator, originated from a stability problem in 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 $+\infty$. We give a complete study of the resolvent of ${\mathcal L}_\alpha$ along the imaginary axis in the fast rotation limit $\alpha\to+\infty$ and we prove $\sup_{\lambda\in \mathbb{R}}\|({\mathcal L}_\alpha-i\lambda)^{-1}\|_{{\mathcal L}(\tilde L^2(\mathbb{R}^2))}\leq C\alpha^{-1/3}$, which is an optimal estimate. Our proof is based on a multiplier method, metrics on the phase space and localization techniques.

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Mathematics Subject Classification: Primary: 47A10; Secondary: 35Q30.

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