Enhanced dissipation in the Navier-Stokes equations near the Poiseuille flow

Abstract

We consider solutions to the 2d Navier-Stokes equations on $\mathbb{T}\times \mathbb{R}$ close to the Poiseuille flow, with small viscosity $\nu >0$. Our first result concerns a semigroup estimate for the linearized problem. Here we show that the $x$-dependent modes of linear solutions decay on a time-scale proportional to ${\nu }^{-1/2}|\log \nu |$. This effect is often referred to as \emph{enhanced dissipation} or \emph{metastability} since it gives a much faster decay than the regular dissipative time-scale ${\nu }^{-1}$ (this is also the time-scale on which the $x$-independent mode naturally decays). We achieve this using an adaptation of the method of hypocoercivity. Our second result concerns the full nonlinear equations. We show that when the perturbation from the Poiseuille flow is initially of size at most ${\nu }^{3/4+}$, then it remains so for all time. Moreover, the enhanced dissipation also persists in this scenario, so that the $x$-dependent modes of the solution are dissipated on a time scale of order ${\nu }^{-1/2}|\log \nu |$. This transition threshold is established by a bootstrap argument using the semigroup estimate and a careful analysis of the nonlinear term in order to deal with the unboundedness of the domain and the Poiseuille flow itself.

19 pages

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• Stability threshold of the 2D Couette flow in Sobolev spaces
• A dynamical approach to the study of instability near Couette flow
• Uniform linear inviscid damping and enhanced dissipation near monotonic shear flows in high Reynolds number regime (I): the whole space case
• The linearized Vlasov and Vlasov-Fokker-Planck equations in a uniform magnetic field
• Metastability for the dissipative quasi-geostrophic equation and the non-local enhancement
• Mixing and diffusion for rough shear flows
• Stirring Speeds Up Chemical Reaction
• Linear stability analysis for 2D shear flows near Couette in the isentropic Compressible Euler equations
• Enhanced dissipation, hypoellipticity for passive scalar equations with fractional dissipation
• Uniform structural stability and uniqueness of Poiseuille flows in a two dimensional periodic strip
• Separation of time-scales in drift-diffusion equations on ${\mathbb{R}}^2$
• Nonlinear enhanced dissipation in viscous Burgers type equations II
• Stable mixing estimates in the infinite Péclet number limit
• Global well-posedness for a generalized Keller-Segel system with degenerate dissipation and mixing
• Potential flows away from stagnation in infinite cylinders
• Nonlinear inviscid damping for zero mean perturbation of the 2D Euler Couette flow