There is compelling evidence showing that extragalactic jets are a crucial ingredient in the evolution of host galaxies and their environments. Extragalactic jets are well collimated and relativistic, both in terms of thermodynamics and kinematics at sub-parsec and parsec scales. They generate strong shocks in the ambient medium, associated with observed hotspots in FRII radio galaxies, and carve cavities that are filled with the shocked jet flow, dragging a large fraction of the interstellar gas along, in the form of slow, massive outflows within the host galaxies. In this paper, I discuss relevant processes associated to jet evolution in the frame of FRI-FRII dichotomy. In particular, I focus on the role of 1) the interaction between galactic atmospheres and the jet head on global FRII jet kinematics, and 2) mass load by stellar winds or small-scale instabilities on jet deceleration in FRI jets. The results presented are based on 3D relativistic hydrodynamical (RHD) and/or 2D axisymmetric, time-independent relativistic magnetohydrodynamical (RMHD) simulations.
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