Unité Mixte de Recherche du C.N.R.S. N°7534

Abstract : We establish the well-posedness of the Cauchy problem for the Smoluchowski coagulation equation in the homogeneous space $\dot L^1_1$ for a class of homogeneous coagulation rates of degree $\lambda \in [0,2)$. For any initial datum $f_{in} \in \dot L^1_1$ we build a weak solution which conserves the mass when $\lambda \le 1$ and loses mass in finite time (gelation phenomenon) when $\lambda > 1$. We then extend the existence result to a measure framework allowing dust source term. In that case we prove that the income dust instantaneously aggregates and the solution does not contain dust phase. On the other hand, we investigate the qualitative properties of self-similar solutions to the Smoluchowski's coagulation equation when $\lambda < 1$. We prove regularity results and sharp uniform small and large size behavior for the self-similar profiles. \

 Dust and self-similarity for the Smoluchowski coagulation