  In the present study, considering the physical conditions that are relevant
interactions between supernova remnants (SNRs) and dense molecular clouds for
triggering star formation we have built a diagram of SNR radius versus cloud
density in which the constraints above delineate a shaded zone where star
formation is allowed. We have also performed fully 3-D radiatively cooling
numerical simulations of the impact between SNRs and clouds under different
initial conditions in order to follow the initial steps of these interactions.
We determine the conditions that may lead either to cloud collapse and star
formation or to complete cloud destruction and find that the numerical results
are consistent with those of the SNR-cloud density diagram. Finally, we have
applied the results above to the $\beta-$ Pictoris stellar association which is
composed of low mass Post-T Tauri stars with an age of 11 Myr. It has been
recently suggested that its formation could have been triggered by the shock
wave produced by a SN explosion localized at a distance of about 62 pc that may
have occurred either in the Lower Centaurus Crux (LCC) or in the Upper
Centaurus Lupus (UCL) which are both nearby older subgroups of that association
(Ortega and co-workers).Using the results of the analysis above we have shown
that the suggested origin for the young association at the proposed distance is
plausible only for a very restricted range of initial conditions for the parent
molecular cloud, i.e., a cloud with a radius of the order of 10 pc, a density
of the order of 10$-$20 cm$^{-3}$, and a temperature of the order of 10$-$100 K
}.
