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Context. We study the role of radial migration of stars on the chemical evolution of the Milky Way disk. Aims. We are interested in the impact of that process on the local properties of the disk (age-metallicity relation and its dispersion, metallicity distribution, evolution of abundance ratios) and on the morphological properties of the resulting thick and thin disks. Methods. We use a model with several new or up-dated ingredients: atomic and molecular gas phases, star formation that depends on molecular gas, yields from a recent homogeneous grid and observationally inferred SNIa rates. We describe radial migration with parametrised time-and radius-dependent diffusion coefficients, based on the analysis of an N-body+SPH simulation. We also consider parametrised radial gas flows, induced by the action of the Galactic bar. Results. Our model reproduces current values of most of the main global observables of the MW disk and bulge, and also the observed "stacked" evolution of MW-type galaxies. The azimuthally averaged radial velocity of gas inflow is constrained to less than a few tenths of km s -1 . Radial migration is constrained by the observed dispersion in the age-metallicity relation. Assuming that the thick disk is the oldest (>9 Gyr) part of the disk, we find that the adopted radial migration scheme can quantitatively reproduce the main local properties of the thin and thick disk: metallicity distributions, "two-branch" behaviour in the O/Fe vs. Fe/H relation and the local surface densities of stars. The thick disk extends up to 11 kpc and has a scale length of 1.8 kpc, which is considerably shorter than the thin disk, because of the inside-out formation scheme. We also show how, in this framework, current and forthcoming spectroscopic observations can constrain the nucleosynthesis yields of massive stars for the metallicity range of 0.1 Z to 2-3 Z .