REMARKS ON SINGULARITIES, DIMENSION AND ENERGY-DISSIPATION FOR IDEAL HYDRODYNAMICS AND MHD

For weak solutions of the incompressible Euler equations, there is ene rgy conservation if the velocity is in the Besov space B-s(3) with s g reater than 1/3. B-s(P) consists of functions that are Lip(s) (i.e., H older continuous with exponent s) measured in the L-P norm. Here this result is applied to a velocity field that is Lip(alpha(0)) except on a set of co-dimension kappa(1) on which it is Lip(alpha(1)), with unif ormity 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 min(alpha)(3 alph a + kappa(alpha)) > 1. Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and re sistivity) for both energy and helicity. In addition, a necessary cond ition is derived for singularity development in ideal MHD generalizing the Beale-Kato-Majda condition for ideal hydrodynamics.