Bifurcation de solutions stationnaires pour un problème de magnétohydrodynamique (mhd), en état adiabatique

AbstractThe study of bifurcations of stationary solutions, from a given static equilibrium, for a system described by magnetohydrodynamic (mhd) equations with variable density can't be expressed in a simple way by a functional equation, as for an incompressible fluid problem. But we can obtain such a functional equation by using Nash–Moser technics, well adapted to problems of “loss of derivatives”. Then we begin by studying existence of stationary solutions for a simplified problem, which is used later to study the bifurcation. A similar work, but with perfect gaz equation (pressure=density) has been done in (Saramito, 1994). In this paper, with adiabatic law (pressure=a power α of the density), the difficulty comes from the nonlinearity in α. We need first to obtain estimates of nα depending on n in Hölder spaces (lemma). Then we can conclude on bifurcation for the mhd problem