On global regularity of 2D generalized magnetohydrodynamic equations

Global regularity of generalized magnetic Benard problem

“…Because the result in was the case ν , η > 0, κ = 0 and α = β = 1, g 1 ≡ g 2 ≡1, our result indicates that the powers of α may be shifted to γ and therefore represents a complete generalization in this sense. The proof of Theorem was inspired by the recent developments on the two‐dimensional generalized MHD system with β = 1, and in particular in which the authors showed that the two‐dimensional generalized MHD system with ν , η > 0, β = 1, α > 0, g 1 ≡ g 2 ≡1 admits the global regularity result. Therefore, Theorem also represents an extension of the result in to the magnetic Bénard problem, interestingly without requiring −Δ θ but only Λ 2 γ , γ = 1− α , α > 0. The proof of Theorem was inspired by the recent work on the two‐dimensional generalized MHD system with ν = 0, and in particular in which the authors showed that the two‐dimensional generalized MHD system with ν = 0, η > 0, β > 1, g 2 ≡1 allows the global regularity result to hold.…”

“…The proof of Theorem was inspired by the recent developments on the two‐dimensional generalized MHD system with β = 1, and in particular in which the authors showed that the two‐dimensional generalized MHD system with ν , η > 0, β = 1, α > 0, g 1 ≡ g 2 ≡1 admits the global regularity result. Therefore, Theorem also represents an extension of the result in to the magnetic Bénard problem, interestingly without requiring −Δ θ but only Λ 2 γ , γ = 1− α , α > 0. The proof of Theorem was inspired by the recent work on the two‐dimensional generalized MHD system with ν = 0, and in particular in which the authors showed that the two‐dimensional generalized MHD system with ν = 0, η > 0, β > 1, g 2 ≡1 allows the global regularity result to hold. Thus, Theorem represents an extension of the results in to the magnetic B nard problem in the expense of requiring −Δ θ in the temperature field equation, while it also represents an extension of the work in on the Boussinesq system. The proof of Theorem was inspired by the recent work in on the MHD system; we also refer to other important work in .…”

Bilinear estimate on tent‐type spaces with application to the well‐posedness of fluid equations

“…They proved global regularity of solutions when the hyper‐viscosity is sufficiently strong. For more information on the MHD equations, we refer the readers to, Wu Wu , Tran‐Yu‐Zhai , and the reference therein.…”

On the regularity of the axially symmetric solutions to the three‐dimensional MHD equations

“…Fang Wang and Keyan Wang proved if the initial data satisfy , where ϵ is a sufficiently small positive number, then the 3D MHD equations with mixed partial dissipation and magnetic diffusion admit global smooth solutions. Chuong V. Tran and Xinwei Yu proved that if the dissipation terms are − ν (−Δ) α u and − κ (−Δ) β b , smooth solutions are global in three cases: ; ; α ≥2, β = 0. In the case of the cylindrical coordinate, Zhen Lei recently proved the global regularity of axially symmetric solution to the ideal MHD in three dimension for a family of nontrivial magnetic fields ( u θ = B r = B z =0).…”