A remark on the existence of a class of stretched 212D Magnetohydrodynamics flow. (December 2018) - Kwak, Minkyu; Lkhagvasuren, Bataa
Abstract: We consider the existence of a class of stretched solutions of 2 1 2 D Magnetohydrodynamics equations in R 3, which are sometimes called the columnar or two and half dimensional flows. The third components of the state variables have the form u 3 = x 3 γ 1 ( x 1, x 2 ) + φ 1 ( x 1, x 2 ) and B 3 = x 3 γ 2 ( x 1, x 2 ) + φ 2 ( x 1, x 2 ) and the first two components of the state variables depend on x 1 and x 2 only. We prove the local existence of such a flow in Sobolev spaces and give the regularity criteria in terms of 2 D vorticity ω or 2 D magnetic density j (but not both). Next, we identify the global existence of a class of axisymmetric flow without swirl for both velocity and magnetic field as a corollary of the main theorem. Finally, we present some exact global solutions as well as singular solutions with a special structure for viscous case and some exact global solutions for full inviscid case.