For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space Β3s with S greater than 1/3. ΒpS 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 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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics