### Abstract

The 3D incompressible Euler equations with initial data characterized by uniformly large vorticity are investigated. We prove existence on long time intervals of regular solutions to the 3D incompressible Euler equations for a class of large initial data in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach is based on fast singular oscillating limits, nonlinear averaging and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, resonance conditions and a nonstandard small divisor problem, we obtain fully 3D limit resonant Euler equations. We establish the global regularity of the latter without any restriction on the size of 3D initial data and bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D Euler equations with weakly aligned uniformly large vorticity at t = 0.

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

Title of host publication | Progress in Nonlinear Differential Equations and Their Application |

Publisher | Springer US |

Pages | 161-185 |

Number of pages | 25 |

Volume | 61 |

DOIs | |

State | Published - Jan 1 2005 |

### Publication series

Name | Progress in Nonlinear Differential Equations and Their Application |
---|---|

Volume | 61 |

ISSN (Print) | 1421-1750 |

ISSN (Electronic) | 2374-0280 |

### Fingerprint

### Keywords

- Conservation laws
- Fast singular oscillating limits
- Three-dimensional Euler equations
- Vorticity

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Control and Optimization
- Mathematical Physics
- Computational Mechanics

### Cite this

*Progress in Nonlinear Differential Equations and Their Application*(Vol. 61, pp. 161-185). (Progress in Nonlinear Differential Equations and Their Application; Vol. 61). Springer US. https://doi.org/10.1007/3-7643-7317-2_13

**Regularity of euler equations for a class of three-dimensional initial data.** / Mahalov, Alex; Nicolaenko, B.; Bardos, C.; Golse, F.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Progress in Nonlinear Differential Equations and Their Application.*vol. 61, Progress in Nonlinear Differential Equations and Their Application, vol. 61, Springer US, pp. 161-185. https://doi.org/10.1007/3-7643-7317-2_13

}

TY - CHAP

T1 - Regularity of euler equations for a class of three-dimensional initial data

AU - Mahalov, Alex

AU - Nicolaenko, B.

AU - Bardos, C.

AU - Golse, F.

PY - 2005/1/1

Y1 - 2005/1/1

N2 - The 3D incompressible Euler equations with initial data characterized by uniformly large vorticity are investigated. We prove existence on long time intervals of regular solutions to the 3D incompressible Euler equations for a class of large initial data in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach is based on fast singular oscillating limits, nonlinear averaging and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, resonance conditions and a nonstandard small divisor problem, we obtain fully 3D limit resonant Euler equations. We establish the global regularity of the latter without any restriction on the size of 3D initial data and bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D Euler equations with weakly aligned uniformly large vorticity at t = 0.

AB - The 3D incompressible Euler equations with initial data characterized by uniformly large vorticity are investigated. We prove existence on long time intervals of regular solutions to the 3D incompressible Euler equations for a class of large initial data in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach is based on fast singular oscillating limits, nonlinear averaging and cancellation of oscillations in the nonlinear interactions for the vorticity field. With nonlinear averaging methods in the context of almost periodic functions, resonance conditions and a nonstandard small divisor problem, we obtain fully 3D limit resonant Euler equations. We establish the global regularity of the latter without any restriction on the size of 3D initial data and bootstrap this into the regularity on arbitrary large time intervals of the solutions of 3D Euler equations with weakly aligned uniformly large vorticity at t = 0.

KW - Conservation laws

KW - Fast singular oscillating limits

KW - Three-dimensional Euler equations

KW - Vorticity

UR - http://www.scopus.com/inward/record.url?scp=67649323941&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67649323941&partnerID=8YFLogxK

U2 - 10.1007/3-7643-7317-2_13

DO - 10.1007/3-7643-7317-2_13

M3 - Chapter

AN - SCOPUS:67649323941

VL - 61

T3 - Progress in Nonlinear Differential Equations and Their Application

SP - 161

EP - 185

BT - Progress in Nonlinear Differential Equations and Their Application

PB - Springer US

ER -