A generaliżation of the Arnol'd inequality in MHD

P. M. Akhmet'ev^{1, 4} - E. A. Kudryavtseva² - A. Yu. Smirnov^{3, 1}

¹ IZMIRAN, Troitsk 142190, Moscow, Russia
² Moscow State University, Moscow 119991, Russia
³ National University of Science and Technology "MISiS", Moscow 119049, Russia \and National Research University Higher School of Economics, Moscow Institute of Electronics and Mathematics, Moscow, Russia

Abstract
The Arnol'd inequality estimates the magnetic energy U⁽²⁾(B) = ∫_Ω (B,B) dR³ from below by means of the magnetic helicity χ(B)=∫_Ω(A,B) dR³. A new modification of the Arnol'd inequality is proved using the magnetic U⁽⁴⁾-energy and the magnetic quadratic helicity χ⁽²⁾. Figs 2, Refs 15.