Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD

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., 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 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.