Regular solution and global attractor for a 2D slab of plasma with resistivity depending on a flux function (Q1863594)

The existence of a regular solution and the existence of a global attractor for the magnetohydrodynamics (MHD) equations for two-dimensional slab of plasma is shown. The velocity \(V\) and the magnetic field \(B\) can be expressed in terms of flux functions \(\phi\) and \(\psi\), respectively \(V= e_z\wedge \nabla\phi\) and \(B= e_z\wedge \nabla\psi\), where \(e_z\) denotes the invariant space direction. Then the vector equations of MHD are reduced to two nonlinear equations for the flux functions \(\phi\) and \(\psi\). First, using a fixed point method, the existence of a local regular solution is proved and later, getting further estimates, the existence of a global solution is proved. To study long time behaviour of the confined plasma, the existence of equilibrium state \((\psi_{eq}, \phi_{eq})\) is assumed and some perturbation of this state is considered.