Short Wave-Long Wave Interactions in Magnetohydrodynamics

Daniel Rodriguez Marroquin

**Keywords:**Compressible MHD equations | Nonlinear SchrÃ¶dinger Equation | Strong Solutions | Vanishing Viscosity | Weak Solutions

In this thesis we study several mathematical aspects of a system of equations modelling the interaction between short waves, described by a nonlinear Schrödinger equation, and long waves, described by the equations of magnetohydrodynamics for a compressible, heat conductive fluid. The system in question models an aurora-type phenomenon, where a short wave propagates along the streamlines of a magnetohydrodynamic medium. We address several problems in both the one dimensional and in the multidimensional versions of the model. Namely, existence and uniqueness of strong solutions, as well as the vanishing viscosity problem, in the 1-dimensional case; and existence of weak solutions with large data in the 2-dimensional case.