Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid

Abstract

In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain $$\mathbb {T}\times \mathbb {R}$$ T × R . In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their $$L^2$$ L 2 norm grows as $$t^{1/2}$$ t 1 / 2 and this confirms previous observations in the physics literature. On the contrary, the solenoidal component of the velocity field experiences inviscid damping, namely it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order $$\nu ^{-1/6}$$ ν - 1 / 6 (with $$\nu ^{-1}$$ ν - 1 being proportional to the Reynolds number) on a time-scale $$\nu ^{-1/3}$$ ν - 1 / 3 , after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible flow, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.

Space quasi-periodic steady Euler flows close to the inviscid Couette flow

“…We start with n " 0. By (4.1), (4.4), (4.5), (3.1), Theorem 3.7, Lemma (3.6) and the mean value Theorem, we compute F p,η pψ m pyqq ´Fp,m pψ m pyqq " 1 2 χ η pϕq `Fp´1,m pψ m pyqq ´Fp,m pψ m pyq " 1 2 χ η pϕq `Fp´1,m pψ m py p,m ´δ´q q ´Fp,m pψ m py p,m `δ`q q " 1 2 χ η pϕq `ψ2 m py p,m ´δ´q ´ψ2 m py p,m `δ`q "…”

“…Let κ 0 P N. There exist ε 0 ą 0 small enough and a family of stationary solutions pψ ε px, yq " ψε px, yq| x"r ωx q εPr0,ε 0 s of the Euler equation (1.1) in the finite channel px, yq P R ˆr´1, 1s that are quasi-periodic in the horizontal direction x P R for some frequency vector r ω P R κ 0 , with x " r ωx P T κ 0 . Such family bifurcates from a shear equilibrium ψ m pyq and can be chosen to be arbitrarily close to the stream function of the Couette flow ψ cou pyq :" 1 2 y 2 in H s x H 7{2ý pT κ 0 ˆr´1, 1sq, with s ą 0 sufficiently large.…”

“…These results follow the work of Mouhot & Villani [41] on the nonlinear Landau damping for the Vlasov equation. Other extensions to the inviscid damping near the Couette flow include Ionescu & Jia [31] in the finite periodic channel, Yang & Lin [51], Bianchini, Coti Zelati & Dolce [13] for stratified fluids and Antonelli, Dolce & Marcati [1] in the compressible case. For other shear flows, we quote Zillinger [52] for monotonic shears and Coti Zelati, Elgindi & Widmayer [18] for non-monotone shears.…”

Taylor dispersion and phase mixing in the non-cutoff Boltzmann equation on the whole space

“…The idea of using mixed inner products to recover dissipation dates back at least to the PhD thesis of Kawashima [49] and has been successfully exploited in many different problems [7,31,42,44,59]. However, we also need to recover dissipation for terms involving Z-derivatives.…”

Asymptotic stability of two-dimensional Couette flow in a viscous fluid

“…See also [8,45,32] for similar depletion phenomena in various systems. Such damping caused by mixing is a common phenomenon in fluid dynamics, see [45,54,53,70] for the plasma fluid, see [6,66,46] for the stratified fluid, see [3,4,69] for the compressible fluid, and see [59,65] for the geophysical fluid.…”