On the Stability of Shear Flows in Bounded Channels, I: Monotonic Shear Flows

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Abstract

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.

Original languageEnglish (US)
JournalVietnam Journal of Mathematics
DOIs
StateAccepted/In press - 2023

All Science Journal Classification (ASJC) codes

  • General Mathematics

Keywords

  • Euler equations
  • Linear inviscid damping
  • Monotonic shear flows

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