  The mathematical description of laboratory fusion plasmas produced in
Tokamaks is still challenging. Complete models for electrons and ions, as
Vlasov-Maxwell systems, are computationally too expensive because they take
into account all details and scales of magneto-hydrodynamics. In particular,
for most of the relevant studies, the mass electron is negligible and the
velocity of material waves is much smaller than the speed of light. Therefore
it is useful to understand simplified models. Here we propose and study one of
those which keeps both the complexity of the Vlasov equation for ions and the
Hall effect in Maxwell's equation. Based on energy dissipation, a fundamental
physical property, we show that the model is nonlinear stable and consequently
prove existence.
