On conditional regularity criteria for the 3D generalized MHD system via partial components

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Abstract

This paper focuses on the regularity criteria for the 3D generalized magnetohydrodynamic (MHD) equations, particularly those involving the partial components of the velocity u₃ and the magnetic field (b₁,b₂,b₃). For any l,i,j,k ∈{1, 2, 3}, we establish regularity criteria via ∂_l u₃ and (∂_i b₁, ∂_j b₂, ∂_k b₃), u₃ and (∂_i b₁, ∂_j b₂, ∂_k b₃), or ∂_l u₃ and (b₁,b₂,b₃). Furthermore, a comprehensive regularity condition is obtained via (ξu₃, ηb₁, ζb₂, σb₃) and (ξ^′∂_l u₃, η^′∂_i b₁, ζ^′∂_j b₂, σ^′∂_k b₃) with ξ + ξ^′ = η + η^′ = ζ + ζ^′ = σ + σ^′ = 1 and ξ, η, ζ, σ, ξ^′, η^′, ζ^′, σ^′ ∈{0, 1}.

Keywords

generalized MHD equations / regularity criterion / partial components / weak solutions

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