Generalized MHD equations

Abstract

Solutions of the d-dimensional generalized MHD (GMHD) equations ∂ tu+u· ∇u=− ∇P+b· ∇b−ν(− Δ) αu, ∂ tb+u· ∇b=b· ∇u−η(− Δ) βb are studied in this paper. We pay special attention to the impact of the parameters ν, η, α and β on the regularity of solutions. Our investigation is divided into three major cases: (1) ν>0 and η>0, (2) ν=0 and η>0, and (3) ν=0 and η=0. When ν>0 and η>0, the GMHD equations with any α>0 and β>0 possess a global weak solution corresponding to any L 2 initial data. Furthermore, weak solutions associated with α⩾ 1 2 + d 4 and β⩾ 1 2 + d 4 are actually global classical solutions when their initial data are sufficiently smooth. As a special consequence, smooth solutions of the 3D GMHD equations with α⩾ 5 4 and β⩾ 5 4 do not develop finite-time singularities. The study of the GMHD equations with ν=0 and η>0 is motivated by their potential applications in magnetic reconnection. A local existence result of classical solutions and several global regularity conditions are established for this case. These conditions are imposed on either the vorticity ω=∇× u or the current density j=∇× b (but not both) and are weaker than some of current existing ones. When ν=0 and η=0, the GMHD equations reduce to the ideal MHD equations. It is shown here that the ideal MHD equations admit a unique local solution when the prescribed initial data is in a Hölder space C r with r>1.