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arXiv · 1804.08291

Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method

Emmanuel Grenier·Toan T. Nguyen·Frédéric Rousset·Avy Soffer
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Abstract

We study the large time behavior of solutions to two-dimensional Euler and Navier-Stokes equations linearized about shear flows of the mixing layer type in the unbounded channel $\mathbb{T} \times \mathbb{R}$. Under a simple spectral stability assumption on a self-adjoint operator, we prove a local form of the linear inviscid damping that is uniform with respect to small viscosity. We also prove a local form of the enhanced viscous dissipation that takes place at times of order $\nu^{-1/3}$, $\nu$ being the small viscosity. To prove these results, we use a Hamiltonian approach, following the conjugate operator method developed in the study of Schr\"odinger operators, combined with a hypocoercivity argument to handle the viscous case.

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Emmanuel Grenier, Toan T. Nguyen, Frédéric Rousset, Avy Soffer. 2018-04-23. Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method. https://arxiv.org/abs/1804.08291

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