TY - JOUR
T1 - Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD
AU - Caflisch, Russel E.
AU - Klapper, Isaac
AU - Steele, Gregory
N1 - Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 1997
Y1 - 1997
N2 - 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.
AB - 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.
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U2 - 10.1007/s002200050067
DO - 10.1007/s002200050067
M3 - Article
AN - SCOPUS:0031489424
SN - 0010-3616
VL - 184
SP - 443
EP - 455
JO - Communications In Mathematical Physics
JF - Communications In Mathematical Physics
IS - 2
ER -