Advances in secular magnetohydrodynamics of stellar interiors dedicated to asteroseismic spatial missions

With the first light of COROT, the preparation of KEPLER and the future helioseismology spatial projects such as GOLF-NG, a coherent picture of the evolution of rotating stars from their birth to their death is needed. We describe here the modelling of the macroscopic transport of angular momentum and matter in stellar interiors that we have undertaken to reach this goal. First, we recall in detail the dynamical processes that are driving these mechanisms in rotating stars and the theoretical advances we have achieved. Then, we present our new results of numerical simulations which allow us to follow in 2D the secular hydrodynamics of rotating stars, assuming that anisotropic turbulence enforces a shellular rotation law. Finally, we show how this work is leading to a dynamical vision of the Hertzsprung-Russel diagram with the support of asteroseismology and helioseismology, seismic observables giving constraints on the modelling of the internal transport and mixing processes. In conclusion, we present the different processes that should be studied in the next future to improve our description of stellar radiation zones.