Dust grains are classically thought to form in the winds of Asymptotic Giant Branch ( AGB ) stars .
However , nowadays there is increasing evidence for dust formation in Supernovae ( SNe ) .
In order to establish the relative importance of these two classes of stellar sources of dust it is important to know what is the fraction of freshly formed dust in SN ejecta that is able to survive the passage of the reverse shock and be injected in the interstellar medium .

With this aim , we have developed a new code , GRASH_Rev , that allows to follow the dynamics of dust grains in the shocked SN ejecta and to compute the time evolution of the mass , composition and size distribution of the grains .
We consider four well studied SNe in the Milky Way and Large Magellanic Cloud : SN 1987a , Cas A , the Crab Nebula , and N49 .
These sources have been observed with both Spitzer and Herschel and the multiwavelength data allow to better assess the mass of warm and cold dust associated with the ejecta .

For each SN , we first identify the best explosion model , using the mass and metallicity of the progenitor star , the mass of ^ { 56 } Ni , the explosion energy and the circumstellar medium density inferred from the data .
We then run a dust formation model to compute the properties of freshly formed dust ( Marassi et al .
2015 ) .
Starting from these input models , GRASH_Rev self-consistely follow the dynamics of the grains considering the effects of the forward and reverse shock and allows to predict the time evolution of the dust mass , composition and size distribution in the shocked and unshocked regions of the ejecta .

For all the simulated models , we find good agreement with observations .
Our study suggests that SN 1987A is too young for the reverse shock to have affected the dust mass .
Hence the observed dust mass of 0.7 - 0.9 M _ { \odot } in this source can be safely considered as indicative of the mass of freshly formed dust in SN ejecta .
Conversely , in the other three SNe , the reverse shock has already destroyed between 10 and 40 % of the initial dust mass .
However , the largest dust mass destruction is predicted to occur between 10 ^ { 3 } and 10 ^ { 5 } yr after the explosions .
Since the oldest SN in the sample has an estimated age of 4800 yr , current observations can only provide an upper limit to the mass of SN dust that will enrich the interstellar medium , the so-called effective dust yields .
We find that only between 1 and 8 % of the currently observed mass will survive , resulting in an average SN effective dust yield of ( 1.55 \pm 1.48 ) \times 10 ^ { -2 } M _ { \odot } .
This is in good agreement with the values adopted in chemical evolution models which consider the effect of the SN reverse shock .

We discuss the astrophysical implications of our results for dust enrichment in local galaxies and at high redshift .