### Abstract

For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space Β^{3}_{s} with S greater than 1/3. Β^{p}_{S} consists of functions that are Lip(s) (i.e., Hölder continuous with exponent s) measured in the L^{p} norm. Here this result is applied to a velocity field that is Lip(α_{0}) except on a set of co-dimension κ_{1} on which it is Lip(α_{1}), with uniformity that will be made precise below. We show that the Frisch-Parisi multifractal formalism is valid (at least in one direction) for such a function, and that there is energy conservation if minα(3α + κ(α)) > 1. Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and resistivity) for both energy and helicity . In addition, a necessary condition is derived for singularity development in ideal MHD generalizing the Beale-Kato-Majda condition for ideal hydrodynamics.

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

Pages (from-to) | 443-455 |

Number of pages | 13 |

Journal | Communications in Mathematical Physics |

Volume | 184 |

Issue number | 2 |

State | Published - 1997 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*184*(2), 443-455.

**Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD.** / Caflisch, Russel; Klapper, Isaac; Steele, Gregory.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 184, no. 2, pp. 443-455.

}

TY - JOUR

T1 - Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD

AU - Caflisch, Russel

AU - Klapper, Isaac

AU - Steele, Gregory

PY - 1997

Y1 - 1997

N2 - For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space Β3s with S greater than 1/3. ΒpS consists of functions that are Lip(s) (i.e., Hölder continuous with exponent s) measured in the Lp norm. Here this result is applied to a velocity field that is Lip(α0) except on a set of co-dimension κ1 on which it is Lip(α1), with uniformity that will be made precise below. We show that the Frisch-Parisi multifractal formalism is valid (at least in one direction) for such a function, and that there is energy conservation if minα(3α + κ(α)) > 1. Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and resistivity) for both energy and helicity . In addition, a necessary condition is derived for singularity development in ideal MHD generalizing the Beale-Kato-Majda condition for ideal hydrodynamics.

AB - For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space Β3s with S greater than 1/3. ΒpS consists of functions that are Lip(s) (i.e., Hölder continuous with exponent s) measured in the Lp norm. Here this result is applied to a velocity field that is Lip(α0) except on a set of co-dimension κ1 on which it is Lip(α1), with uniformity that will be made precise below. We show that the Frisch-Parisi multifractal formalism is valid (at least in one direction) for such a function, and that there is energy conservation if minα(3α + κ(α)) > 1. Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and resistivity) for both energy and helicity . In addition, a necessary condition is derived for singularity development in ideal MHD generalizing the Beale-Kato-Majda condition for ideal hydrodynamics.

UR - http://www.scopus.com/inward/record.url?scp=0031489424&partnerID=8YFLogxK

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M3 - Article

VL - 184

SP - 443

EP - 455

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -