Let \(k\) be a field of characteristic \(p\). For \(r\in\mathbb N\), let \(\Sigma_r\) be the symmetric group on \(r\) letters. Young modules for \(\Sigma_r\) are indexed by partitions of \(r\), and the classical Schur algebra can be exhibited as the endomorphism algebra of a direct sum of Young modules. The paper focusses on the decomposition of tensor products of Young modules as direct sums of Young modules, since it is known that understanding such multiplicities in the decomposition is equivalent to understanding the decomposition numbers of the Schur algebras. The author makes extensive use of the Brauer construction applied to \(p\)-permutation modules, as originally developed by Broué. Initially he provides some general results on tensor products of trivial source modules for arbitrary finite groups \(G\). Namely if \(U\) and \(V\) are trivial source \(kG\)-modules with vertices \(P\) and \(Q\), respectively, then there is a family \(X\) of subgroups of \(G\) depending only on \(P\) and \(Q\), such that for every member \(H\in X\), there exists a direct summand of \(U\otimes V\) with vertex \(H\); moreover, such summands are in one-to-one correspondence with indecomposable projective direct summands of a tensor product of certain Brauer quotients with respect to \(H\). Then consideration is given to the special case of Young modules for symmetric groups, including the derivation of results concerning Young vertices and some reduction theorems for multiplicities of Young modules in the tensor product of Young modules. As an application, some lower bounds on certain Cartan numbers for Schur algebras are obtained.