Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness

We introduce for the system of pressureless gases a new notion of solution, which consists in interpreting the system as two nonlinearly coupled linea r equations. We prove In this setting existence of solutions for the Cauchy problem as well as uniqueness under optimal conditions on initial data. Th e proofs rely on the detailed study of the relations between pressureless g ases, the dynamics of sticky particles and nonlinear scalar conservation la ws with monotone Initial data. We prove for the latter problem that monoton icity implies uniqueness, and a generalization of Oleinik's entropy conditi on.