An initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations

Abstract

We investigate an initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations. The Dirichlet boundary conditions are imposed only on the velocity, while no boundary condition is imposed on the height of the fluid or the magnetic field. We derive a series of a priori estimates for the approximate solution sequences to show that they are Cauchy in a suitable Sobolev space. The local well-posedness in time of strong solutions for the initial-boundary value problem is established by the strong convergence of the approximate solution sequences.