Skip to main content
Log in

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

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Author information

Authors and Affiliations

Authors

Additional information

Received: 21 March 1995 / Accepted: 6 August 1996

Rights and permissions

Reprints and permissions

About this article

Cite this article

Caflisch, R., Klapper, I. & Steele, G. Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD . Comm Math Phys 184, 443–455 (1997). https://doi.org/10.1007/s002200050067

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s002200050067

Keywords

Navigation