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

Russel Caflisch, Isaac Klapper, Gregory Steele

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)443-455
Number of pages13
JournalCommunications in Mathematical Physics
Volume184
Issue number2
StatePublished - 1997

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energy conservation
Energy Dissipation
Hydrodynamics
dissipation
energy dissipation
hydrodynamics
Energy Conservation
Singularity
norms
Multifractal Formalism
Incompressible Euler Equations
conservation
velocity distribution
Helicity
Lp-norm
Besov Spaces
Resistivity
exponents
viscosity
formalism

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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

In: Communications in Mathematical Physics, Vol. 184, No. 2, 1997, p. 443-455.

Research output: Contribution to journalArticle

Caflisch, Russel ; Klapper, Isaac ; Steele, Gregory. / Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD. In: Communications in Mathematical Physics. 1997 ; Vol. 184, No. 2. pp. 443-455.
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