### Abstract

We consider the incompressible Euler equations on R^{d} or T^{d}, where d∈{2,3}. We prove that: (a) In Lagrangian coordinates the equations are locally well-posed in spaces with fixed real-analyticity radius (more generally, a fixed Gevrey-class radius). (b) In Lagrangian coordinates the equations are locally well-posed in highly anisotropic spaces, e.g. Gevrey-class regularity in the label a_{1} and Sobolev regularity in the labels a_{2},…,a_{d}. (c) In Eulerian coordinates both results (a) and (b) above are false.

Original language | English (US) |
---|---|

Pages (from-to) | 1569-1588 |

Number of pages | 20 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 33 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1 2016 |

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### Keywords

- Analyticity
- Euler equations
- Gevrey class
- Lagrangian and Eulerian coordinates

### ASJC Scopus subject areas

- Analysis
- Mathematical Physics

### Cite this

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*,

*33*(6), 1569-1588. https://doi.org/10.1016/j.anihpc.2015.07.002

**Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations.** / Constantin, Peter; Kukavica, Igor; Vicol, Vlad.

Research output: Contribution to journal › Article

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*, vol. 33, no. 6, pp. 1569-1588. https://doi.org/10.1016/j.anihpc.2015.07.002

}

TY - JOUR

T1 - Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations

AU - Constantin, Peter

AU - Kukavica, Igor

AU - Vicol, Vlad

PY - 2016/11/1

Y1 - 2016/11/1

N2 - We consider the incompressible Euler equations on Rd or Td, where d∈{2,3}. We prove that: (a) In Lagrangian coordinates the equations are locally well-posed in spaces with fixed real-analyticity radius (more generally, a fixed Gevrey-class radius). (b) In Lagrangian coordinates the equations are locally well-posed in highly anisotropic spaces, e.g. Gevrey-class regularity in the label a1 and Sobolev regularity in the labels a2,…,ad. (c) In Eulerian coordinates both results (a) and (b) above are false.

AB - We consider the incompressible Euler equations on Rd or Td, where d∈{2,3}. We prove that: (a) In Lagrangian coordinates the equations are locally well-posed in spaces with fixed real-analyticity radius (more generally, a fixed Gevrey-class radius). (b) In Lagrangian coordinates the equations are locally well-posed in highly anisotropic spaces, e.g. Gevrey-class regularity in the label a1 and Sobolev regularity in the labels a2,…,ad. (c) In Eulerian coordinates both results (a) and (b) above are false.

KW - Analyticity

KW - Euler equations

KW - Gevrey class

KW - Lagrangian and Eulerian coordinates

UR - http://www.scopus.com/inward/record.url?scp=84939833029&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939833029&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2015.07.002

DO - 10.1016/j.anihpc.2015.07.002

M3 - Article

AN - SCOPUS:84939833029

VL - 33

SP - 1569

EP - 1588

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 6

ER -