Wave Motion in a Conducting Fluid with a Boundary Layer. I. Hilbert Space Formulation

The wave motion of MI-ID systems can be quite complicated. In order to study the motion of waves in a perfectly conducting fluid under the influence of an external magnetic field with a boundary layer, we make the simplifying assumption that the pressure is constant (to first order). This is classical "cold plasma" approximation from the physical literature. This is still an interesting system and is not strongly propagative. Alfven waves are still present. The system is further simplified by assuming that the external field is either orthogonal or parallel to the boundary layer. While it may seem presumptuous to claim anything new about this problem the method introduced here is cumulative: it may be extended to more complex problems of the same type. In Part I the appropriate energy preserving boundary conditions are studied. For the parallel field case there are just two possible boundary conditions which preserve energy. For the orthogonal case, there are two one parameter families of energy preserving boundary conditions. One of the boundary conditions for the parallel case is selected and from this boundary condition, it is shown that the relevant operator is selfadjoint and data which propagates is characterized. A crucial density result is then proved.