A remark on the existence of a class of stretched 21/2D Magnetohydrodynamics flow

作者：Kwak Minkyu; Lkhagvasuren Bataa^*

来源：Nonlinear Analysis: Real World Applications , 2018, 44: 365-384.

DOI：10.1016/j.nonrwa.2018.05.010

• 21/2D flow
• Incompressible MHD equations
• Sobolev spaces
• Regularity criteria
• Global solutions
• Blow up solutions

We consider the existence of a class of stretched solutions of 21/2D Magnetohydro-dynamics 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)gamma(1)(x(1), x(2)) + phi(1)(x(1), x(2)) and B-3 = x(3)gamma(2)(x(1), x(2)) + phi(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 2D vorticity omega or 2D 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.

• 出版日期2018-12