The transition to instability for stable shear flows in inviscid fluids

Abstract

In this paper, we study the generation of eigenvalues of a stable monotonic shear flow under perturbations in C s with s < 2. More precisely, we study the Rayleigh operator LU m,γ = Um,γ ∂x − U ′′ m,γ ∂x∆ −1 associated with perturbed shear flow (Um,γ (y), 0) in a finite channel T2π × [−1, 1] where Um,γ (y) = U (y) + mγ 2 Γ(y/γ) with U (y) being a stable monotonic shear flow and {mγ 2 Γ(y/γ)} m≥0 being a family of perturbations parameterized by m. We prove that there exists m * such that for 0 ≤ m < m * , the Rayleigh operator has no eigenvalue or embedded eigenvalue, therefore linear inviscid damping holds. Otherwise, instability occurs when m ≥ m * . Moreover, at the nonlinear level, we show that asymptotic instability holds for m near m * and a growing mode exists for m > m * which also leads to instability.

Traveling Waves Near Couette Flow for the 2D Euler Equation

“…Recently, there has been a growing interest in the study of existence or not of invariant structures, and their stability for 2D Euler near other shear flows and for related equations, see [22,36,70,72,73,78,80,81]. 1.5.…”