Nonlocal elliptic-parabolic equation arising in the transient regime of a magnetically confined plasma in a Stellarator

We prove the existence and the regularity of solutions of an elliptic-parab olic equation involving the notions of relative rearrangement and monotone rearrangement. These equations were obtained from 3D MHD systems, taking, ( in particular) into account the Ohm and Faraday's laws and the averaging ar guments of Hender and Carreras for obtaining a 2D evolution model. Due to t he presence of a strong nonlinearity, we introduce a new notion of solution derived from a new property of the relative rearrangement. This allows us to improve the results obtained in the stationnary case by cancelling the c ondition of smallness of the parameter lambda in the pressure. (C) 1999 Aca demie des sciences/Editions scientifiques et medicales Elsevier SAS.