### Abstract

We consider the decay at high wavenumbers of the energy spectrum for weak solutions to the three-dimensional forced Navier-Stokes equation in the whole space. We observe that known regularity criteria imply that solutions are regular if the energy density decays at a sufficiently fast rate. This result applies also to a class of solutions with infinite global energy by localizing the Navier-Stokes equation. We consider certain modified Leray backward self-similar solutions, which belong to this class, and show that their energy spectrum decays at the critical rate for regularity. Therefore, this rate of decay is consistent with the appearance of an isolated self-similar singularity.

Original language | English (US) |
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Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Nonlinearity |

Volume | 18 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2005 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

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*Nonlinearity*, vol. 18, no. 1, pp. 1-19. https://doi.org/10.1088/0951-7715/18/1/001

**On the energy spectrum for weak solutions of the Navier-stokes equations.** / Mazzucato, Anna L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the energy spectrum for weak solutions of the Navier-stokes equations

AU - Mazzucato, Anna L.

PY - 2005/1/1

Y1 - 2005/1/1

N2 - We consider the decay at high wavenumbers of the energy spectrum for weak solutions to the three-dimensional forced Navier-Stokes equation in the whole space. We observe that known regularity criteria imply that solutions are regular if the energy density decays at a sufficiently fast rate. This result applies also to a class of solutions with infinite global energy by localizing the Navier-Stokes equation. We consider certain modified Leray backward self-similar solutions, which belong to this class, and show that their energy spectrum decays at the critical rate for regularity. Therefore, this rate of decay is consistent with the appearance of an isolated self-similar singularity.

AB - We consider the decay at high wavenumbers of the energy spectrum for weak solutions to the three-dimensional forced Navier-Stokes equation in the whole space. We observe that known regularity criteria imply that solutions are regular if the energy density decays at a sufficiently fast rate. This result applies also to a class of solutions with infinite global energy by localizing the Navier-Stokes equation. We consider certain modified Leray backward self-similar solutions, which belong to this class, and show that their energy spectrum decays at the critical rate for regularity. Therefore, this rate of decay is consistent with the appearance of an isolated self-similar singularity.

UR - http://www.scopus.com/inward/record.url?scp=12144260422&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=12144260422&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/18/1/001

DO - 10.1088/0951-7715/18/1/001

M3 - Article

VL - 18

SP - 1

EP - 19

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 1

ER -