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

For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space Β³_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(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.