Resolvent estimates for a two-dimensional non-self-adjoint operator
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.
scientific article; zbMATH DE number 6159441