Saint Venant’s estimates in anisotropic magnetohydrodynamics

We study the steady motion of an incompressible viscous fluid in a semi-infinite cylinder supposing the fluid to be electrically conducting and embedded in an external magnetic field which is constant and longitudinal. We state the rate of decay of the solution as the norm of x tends to infinity: under suitable hypotheses at infinity, the velocity and magnetic fields tend exponentially fast in the energy norm to the magnetohydrodynamic Poieseuille flow. We assume that on the lateral surface of the cylinder the velocity vanishes and that the tangential components of the magnetic field is the external magnetic field. Hall and ion-slip effects are taken into account.