The singular kernel coagulation equation with multifragmentation

In this article we prove the existence of solutions to the singular coagulation equation with multifrag-mentation. We use weighted L1-spaces to deal with the singularities and to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak L1 compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also given.