Communications in Mathematical Physics

, Volume 184, Issue 2, pp 443-455

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

  • Russel E. CaflischAffiliated withMathematics Department, UCLA, Los Angeles, CA 90095-1555, USA.
  • , Isaac KlapperAffiliated withMathematics Department, UCLA, Los Angeles, CA 90095-1555, USA.
  • , Gregory SteeleAffiliated withMathematics Department, UCLA, Los Angeles, CA 90095-1555, USA.

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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.