Mixing and diffusion for rough shear flows

This article addresses mixing and diffusion properties of passive scalars advected by rough ($C^alpha$) shear flows. We show that in general, one cannot expect a rough shear flow to increase the rate of inviscid mixing to more than that of a smooth shear without critical points. On the other hand, diffusion may be enhanced at a much faster rate. This shows that in the setting of low regularity, the interplay between inviscid mixing properties and enhanced dissipation is more intricate, and in fact contradicts some of the natural heuristics that are valid in the smooth setting.

Stationary Structures near the Kolmogorov and Poiseuille Flows in the 2d Euler Equations

We study the behavior of solutions to the incompressible $2d$ Euler equations near two canonical shear flows with critical points, the Kolmogorov and Poiseuille flows, with consequences for the associated Navier-Stokes problems. We exhibit a large family of new, non-trivial stationary states of analytic regularity, that are arbitrarily close to the Kolmogorov flow on the square torus $mathbb{T}^2$. This situation contrasts strongly with the setting of some monotone shear flows, such as the Couette flow: in both cases the linearized problem exhibits an inviscid damping mechanism that leads to relaxation of perturbations of the base flows back to nearby shear flows. While this effect persists nonlinearly for suitably small and regular perturbations of some monotone shear flows, for the Kolmogorov flow our result shows that this is not possible. Our construction of these stationary states builds on a degeneracy in the global structure of the Kolmogorov flow on $mathbb{T}^2$. In this regard both the Kolmogorov flow on a rectangular torus and the Poiseuille flow in a channel are very different, and we show that the only stationary states near them must indeed be shears, even in relatively low regularity $H^3$ resp. $H^{5+}$. In addition, we show that this behavior is mirrored closely in the related Navier-Stokes settings: the linearized problems near the Poiseuille and Kolmogorov flows both exhibit an enhanced rate of dissipation. Previous work by us and others shows that this effect survives in the full, nonlinear problem near the Poiseuille flow and near the Kolmogorov flow on rectangular tori, provided that the perturbations lie below a certain threshold. However, we show here that the corresponding result cannot hold near the Kolmogorov flow on $mathbb{T}^2$.

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

We consider solutions to the 2d Navier-Stokes equations on $mathbb{T}timesmathbb{R}$ close to the Poiseuille flow, with small viscosity $ u>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 $ u^{-1/2}|log u|$. 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 $ u^{-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 $ u^{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 $ u^{-1/2}|log u|$. 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.

Singular limits of Voigt models in fluid dynamics

We investigate the long-term behavior, as a certain regularization parameter vanishes, of the three-dimensional Navier-Stokes-Voigt model of a viscoelastic incompressible fluid. We prove the existence of global and exponential attractors of optimal regularity. We then derive explicit upper bounds for the dimension of these attractors in terms of the three-dimensional Grashof number and the regularization parameter. Finally, we also prove convergence of the (strong) global attractor of the 3D Navier-Stokes-Voigt model to the (weak) global attractor of the 3D Navier-Stokes equation. Our analysis improves and extends recent results obtained by Kalantarov and Titi in [31].