### Abstract

We review our works on the nonlinear asymptotic stability and instability of the Couette flow for the 2D incompressible Euler dynamic. In the fits part of the work we prove that perturbations to the Couette flow which are small in Gevrey spaces G^{s} of class 1/s with s > 1/2 converge strongly in L^{2} to a shear flow which is close to the Couette flow. Moreover in a well chosen coordinate system, the solution converges in the same Gevrey space to some limit profile. In a later work, we proved the existence of small perturbations in G^{s} with s < 1/2 such that the solution becomes large in Sobolev regularity and hence yields instability. In this note we discuss the most important physical and mathematical aspects of these two results and the key ideas of the proofs.

Original language | English (US) |
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Title of host publication | Invited Lectures |

Editors | Boyan Sirakov, Paulo Ney de Souza, Marcelo Viana |

Publisher | World Scientific Publishing Co. Pte Ltd |

Pages | 2155-2187 |

Number of pages | 33 |

ISBN (Electronic) | 9789813272927 |

State | Published - Jan 1 2018 |

Event | 2018 International Congress of Mathematicians, ICM 2018 - Rio de Janeiro, Brazil Duration: Aug 1 2018 → Aug 9 2018 |

### Publication series

Name | Proceedings of the International Congress of Mathematicians, ICM 2018 |
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Volume | 3 |

### Conference

Conference | 2018 International Congress of Mathematicians, ICM 2018 |
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Country | Brazil |

City | Rio de Janeiro |

Period | 8/1/18 → 8/9/18 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Invited Lectures*(pp. 2155-2187). (Proceedings of the International Congress of Mathematicians, ICM 2018; Vol. 3). World Scientific Publishing Co. Pte Ltd.