Linear Inviscid Damping and Vorticity Depletion for Shear Flows

Abstract

In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the explicit decay estimates of the velocity, which is the same as one for monotone shear flows. We confirm a new dynamical phenomena found by Bouchet and Morita: the depletion of the vorticity at the stationary streamlines, which could be viewed as a new mechanism leading to the damping for the base flows with stationary streamlines.

Pseudospectral Bound and Transition Threshold for the 3D Kolmogorov Flow

“…see lemma 4.8 in [36]. As @ y c D u 0 .y c /@ c ; we can deduce that for k D 0; 1; 2; Finally, we present some estimates of E j .…”

“…jA.c/j Ä C˛sin y c ; j@ c A.c/j Ä C min ˛2;˛=sin y c ; @ 2 c A.c/ˇÄ C min.˛2= sin 2 y c ;˛= sin 3 y c /; @ c B.c/ D ; @ 2 c B.c/ D 0I see proposition 4.3 in [36] and [36], and for m D 0; 1; 2;…”

“…The construction of the wave operator is similar to [36]. Here we will directly use some lemmas and propositions from [36].…”

“…We first introduce some notations from [36]. Let˛> 1 and u.y/ D cos y, and let 1 .y; c/ be a function constructed in proposition 3.6 in [36], which satisfies the properties in lemma 3.7 and proposition 3.8, both in [36].…”

“…If D , U is independent of ; thus the transition thresholdˇÄ 7 4 . The enhanced dissipation rate of the linearized operator around the 2D Kolmogorov flow .sin y; 0/ was independently proved by Wei et al [36] and Ibrahim et al [19]. The transition threshold should be not optimal.…”