## Section: New Results

### Mathematical theory of reduced MHD models

Participant : Hervé Guillard.

In the modelling of strongly magnetized plasma, one of the fundamental model used 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 plasma 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 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.