Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD
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For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B3s with s greater than 1/3. B3s 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 \(\) 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.
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