### Abstract

The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.

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

Pages (from-to) | 667-689 |

Number of pages | 23 |

Journal | Communications in Mathematical Physics |

Volume | 108 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1987 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*108*(4), 667-689. https://doi.org/10.1007/BF01214424

**Oscillations and concentrations in weak solutions of the incompressible fluid equations.** / DiPerna, Ronald J.; Majda, Andrew J.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 108, no. 4, pp. 667-689. https://doi.org/10.1007/BF01214424

}

TY - JOUR

T1 - Oscillations and concentrations in weak solutions of the incompressible fluid equations

AU - DiPerna, Ronald J.

AU - Majda, Andrew J.

PY - 1987/12

Y1 - 1987/12

N2 - The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.

AB - The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.

UR - http://www.scopus.com/inward/record.url?scp=0000863080&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000863080&partnerID=8YFLogxK

U2 - 10.1007/BF01214424

DO - 10.1007/BF01214424

M3 - Article

VL - 108

SP - 667

EP - 689

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 4

ER -