Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD |
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Authors: | Russel E Caflisch Isaac Klapper Gregory Steele |
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Institution: | (1) Mathematics Department, UCLA, Los Angeles, CA 90095-1555, USA., US |
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Abstract: | For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space
B
3
s
with s greater than 1/3. B
3
s
consists of functions that are Lip(s) (i.e., H?lder continuous with exponent s) measured in the L
p
norm. Here this result is applied to a velocity field that is Lip(α0) except on a set of co-dimension on which it is Lip($agr;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 . 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.
Received: 21 March 1995 / Accepted: 6 August 1996 |
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Keywords: | |
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