TY - JOUR

T1 - On the Stability of Shear Flows in Bounded Channels, I

T2 - Monotonic Shear Flows

AU - Ionescu, Alexandru D.

AU - Jia, Hao

N1 - Publisher Copyright:
© 2023, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.

PY - 2023

Y1 - 2023

N2 - We discuss some of our recent work on the linear and nonlinear stability of shear flows as solutions of the 2D Euler equations in the bounded channel T× [0 , 1] . More precisely, we consider shear flows u= (b(y) , 0) given by smooth functions b: [0 , 1] → R . We prove linear inviscid damping and linear stability provided that b is strictly increasing and a suitable spectral condition involving the function b is satisfied. Then we show that this can be extended to full nonlinear inviscid damping and asymptotic nonlinear stability, provided that b is linear outside a compact subset of the interval (0, 1) (to avoid boundary contributions which are not compatible with inviscid damping) and the vorticity is smooth in a Gevrey space. In the second article in this series we will discuss the case of non-monotonic shear flows b with non-degenerate critical points (like the classical Poiseuille flow b: [- 1 , 1] → R , b(y) = y2). The situation here is different, as nonlinear stability is a major open problem. We will prove a new result in the linear case, involving polynomial decay of the associated stream function.

AB - We discuss some of our recent work on the linear and nonlinear stability of shear flows as solutions of the 2D Euler equations in the bounded channel T× [0 , 1] . More precisely, we consider shear flows u= (b(y) , 0) given by smooth functions b: [0 , 1] → R . We prove linear inviscid damping and linear stability provided that b is strictly increasing and a suitable spectral condition involving the function b is satisfied. Then we show that this can be extended to full nonlinear inviscid damping and asymptotic nonlinear stability, provided that b is linear outside a compact subset of the interval (0, 1) (to avoid boundary contributions which are not compatible with inviscid damping) and the vorticity is smooth in a Gevrey space. In the second article in this series we will discuss the case of non-monotonic shear flows b with non-degenerate critical points (like the classical Poiseuille flow b: [- 1 , 1] → R , b(y) = y2). The situation here is different, as nonlinear stability is a major open problem. We will prove a new result in the linear case, involving polynomial decay of the associated stream function.

KW - Euler equations

KW - Linear inviscid damping

KW - Monotonic shear flows

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U2 - 10.1007/s10013-023-00656-w

DO - 10.1007/s10013-023-00656-w

M3 - Article

AN - SCOPUS:85173699501

SN - 2305-221X

JO - Vietnam Journal of Mathematics

JF - Vietnam Journal of Mathematics

ER -