## Section: New Results

### Mathematical theory of reduced MHD models

Participant : Hervé Guillard.

One of the fundamental model used for fusion plasma simulations is the
magnetohydrodynamic (MHD) model. However, in practice, many theoretical and numerical works in this field
use specific
approximations of this model known as *reduced* MHD models. These models assume that in the presence
of a strong magnetic field, the main dynamic reduces to incompressible motion in the plane perpendicular
to the dominating magnetic field and to the propagation of Alfvén waves in the magnetic field direction.
In the framework of the slab approximation
for large aspect ratio tokamaks ($R/a>>1$ where $R$ and $a$ are respectively the major and minor radius of the
machine) we have studied last year the validity of this assumption using techniques coming from the asymptotic theory
of hyperbolic equations with a large parameter.
In particular, we have proved that
the solutions of the full MHD system converge in a weak sense to the solutions of an appropriate
reduced model even in the presence of ill-prepared initial data. This work continues with a tentative to relax the
large aspect ratio assumption that is not verified in modern machines.