# A Resolution of the Sommerfeld Paradox

@article{Li2011ARO, title={A Resolution of the Sommerfeld Paradox}, author={Y. Li and Zhiwu Lin}, journal={SIAM J. Math. Anal.}, year={2011}, volume={43}, pages={1923-1954} }

Sommerfeld paradox roughly says that mathematically Couette linear shear is linearly stable for all Reynolds number, but experimentally arbitrarily small perturbations can induce the transition from the linear shear to turbulence when the Reynolds number is large enough. The main idea of our resolution of this paradox is to show that there is a sequence of linearly unstable shears which approaches the linear shear in the kinetic energy norm but not in the enstrophy (vorticity) norm. These… Expand

#### 27 Citations

A resolution of the turbulence paradox: numerical implementation

- Mathematics, Physics
- 2009

Sommerfeld paradox (turbulence paradox) roughly says that mathematically the Couette linear shear flow is linearly stable for all values of the Reynolds number, but experimentally transition from the… Expand

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

- Mathematics, Physics
- 2013

We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on T×R. That is, given an initial perturbation of the Couette flow small in a suitable… Expand

Linear stability analysis for 2D shear flows near Couette in the isentropic Compressible Euler equations

- Physics, Mathematics
- 2020

In this paper, we investigate linear stability properties of the 2D isentropic compressible Euler equations linearized around a shear flow given by a monotone profile, close to the Couette flow, with… Expand

Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow

- Physics, Mathematics
- 2016

In this work we study the long time inviscid limit of the two dimensional Navier–Stokes equations near the periodic Couette flow. In particular, we confirm at the nonlinear level the qualitative… Expand

The ORR mechanism: Stability/instability of the Couette flow for the 2D Euler dynamic

- Physics
- 2019

We review our works on the nonlinear asymptotic stability and instability of the Couette flow for the 2D incompressible Euler dynamic. In the fits part of the work we prove that perturbations to the… Expand

Dynamics Near the Subcritical Transition of
the 3D Couette Flow I: Below Threshold Case

- Mathematics, Physics
- 2015

We study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. We prove that for sufficiently regular initial… Expand

Stability Criteria of 3

- 2009

The classical plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow share some universal 3D steady coherent structure in the form of “streak-roll-critical layer” [22] [21] [20]. As the… Expand

Asymptotic stability for the Couette flow in the 2D Euler equations

- Mathematics, Physics
- 2013

In this expository note we discuss our recent work [arXiv:1306.5028] on the nonlinear asymptotic stability of shear flows in the 2D Euler equations of ideal, incompressible flow. In that work it is… Expand

Stability Criteria of 3D Inviscid Shears

- Mathematics, Physics
- 2009

The classical plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow share some universal 3D steady coherent structure in the form of "streak-roll-critical layer". As the Reynolds number… Expand

Linear Inviscid Damping for Monotone Shear Flows

- Mathematics, Physics
- 2014

In this article, we prove linear stability, scattering and inviscid damping with optimal decay rates for the linearized 2D Euler equations around a large class of strictly monotone shear flows,… Expand

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