## 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)=y^{2}). 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 language | English (US) |
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Pages (from-to) | 829-849 |

Number of pages | 21 |

Journal | Vietnam Journal of Mathematics |

Volume | 52 |

Issue number | 4 |

DOIs | |

State | Accepted/In press - 2023 |

### Bibliographical note

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

## Keywords

- 35B40
- 35P25
- 35Q31
- Euler equations
- Linear inviscid damping
- Monotonic shear flows