Enhanced Dissipation, Hypoellipticity, and Anomalous Small Noise Inviscid Limits in Shear Flows

Abstract

ABSTRACT. We analyze the decay and instant regularization properties of the evolution semigroups generated by two-dimensional drift-diffusion equations in which the scalar is advected by a shear flow and dissipated by full or partial diffusion. We consider both the space-periodic T 2 setting and the case of a bounded channel T × [0, 1] with no-flux boundary conditions. In the infinite Péclet number limit (diffusivity ν → 0), our work quantifies the enhanced dissipation effect due to the shear. We also obtain hypoelliptic regularization, showing that solutions are instantly Gevrey regular even with only partial diffusion. The proofs rely on localized spectral gap inequalities and ideas from hypocoercivity with an augmented energy functional with weights replaced by pseudo-differential operators (of a rather simple form). As an application, we study small noise inviscid limits of invariant measures of stochastic perturbations of passive scalars, and show that the classical Freidlin scaling between noise and diffusion can be modified. In particular, although statistically stationary solutions blow up in H 1 in the limit ν → 0, we show that viscous invariant measures still converge to a unique inviscid measure.

Suppression of Plasma Echoes and Landau Damping in Sobolev Spaces by Weak Collisions in a Vlasov-Fokker-Planck Equation

“…It is caused by the interaction of the transport term and dissipation: conservative transport transfers information to high frequencies where the dissipation is increasingly dominant. The effect has been studied by many authors in the mathematics community [21,61,5,57,15] and in the fluid mechanics community [48,23,36]. That a similar effect is predicted to happen in plasmas due to the second-order smoothing of Coulomb collisions is classical in the physics community [37,52,43].…”

Stable mixing estimates in the infinite Péclet number limit

“…The proof of the enhanced diffusion estimate (1.10) is based on a modification of the so-called hypocoercivity method [38]. In the context of fluid mechanics, it was successfully used for the first time in [3] to study the Kolmogorov flow u(y) = sin y, and then for general shear flows in [6] on a periodic domain or a bounded periodic channel. Other examples include circular flows [15], the 2D Navier-Stokes equations [16,41] and the Boussinesq equations [37].…”

“…for every t ≥ 0. As noted in [6], this estimate is particularly relevant for the hypoelliptic equation (1.8), since it provides a quantification of the instantaneous regularization from L 2 to Gevrey- 3 2 even if no diffusion in x is present in the equation. For a precise statement, see Theorem 4.1 below.…”

“…It is therefore not surprising that a way to quantify this is through the decay of a negative Sobolev norm, as in (1.12). In the context of passive scalars, this point of view was introduced in [31], and it is deeply connected with the regularity of transport equations [12,18,26,45], the quantification and lower bounds on mixing rates [1,6,19,25,32,36,46], and the inviscid damping in the two-dimensional Euler equations linearized around shear flows [8,17,23,40,42,43,[47][48][49].…”

Selection of Quasi-stationary States in the Stochastically Forced Navier–Stokes Equation on the Torus

“…The support of the limiting invariant measure (i.e. as ν → 0), is in general still an open question, see [16], and it has been resolved in special cases in [3] and [4]. In our work, we study the long time behavior of the vorticity equation as ν → 0 and in particular the selection mechanism for the dominant quasi-stationary states.…”