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 consist in interpreting the system as two nonlinearly coupled linear equations. We prove In this setting existence of solutions for the Cauchy Problem, as well as uniqueness under optimal conditions on initlaffata. The proofs rely on the detailed study of the relations between pressureless gases, tie dynamics of sticky particles and nonlinear scalar conservation laws with monotone initial data. We prove for the latter problem that monotonicit implies uniqueness. and a generalization of Oleinik's entropy condition