Dissipation wavenumber and regularity for electron magnetohydrodynamics

We consider the electron magnetohydrodynamics (MHD) with static background ion flow. A special situation of B(x,y,t) = ∇× (a~e z) + b~e z with scalar-valued functions a(x,y,t) and b(x,y,t) was studied numerically in the physics paper [7]. The authors concluded from numerical simulations that there is no evidence of dissipation cutoff for the electron MHD. In this paper we show the existence of determining wavenumber for the electron MHD, and establish a regularity condition only on the low modes of the solution. Our results suggest that the conclusion of the physics paper on the dissipation cutoff for the electron MHD is debatable.