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We introduce a notion of duality solution for a single or a system oftransport equations in spaces of probability measures reminiscent of theviscosity solution notion for nonlinear parabolic equations. Our notion ofsolution by duality is, under suitable assumptions, equivalent to gradient flowsolutions in case the single/system of equations has this structure. Incontrast, we can deal with a quite general system of nonlinear nonlocal,diffusive or not, system of PDEs without any variational structure.

Interpreting systems of continuity equations in spaces of probability measures through PDE duality