Equivalence between duality and gradient flow solutions for one-dimensional aggregation equations

Existence and uniqueness of global in time measure solution for aone dimensional nonlinear aggregation equation is considered.Such a system can be written as a conservation law with avelocity field computed through a self-consistent interaction potential.Blow up of regular solutions is now well established for suchsystem. In Carrillo et al. (Duke Math J (2011)) [18],a theory of existence and uniqueness based on the geometricapproach of gradient flows on Wasserstein space has been developed.We propose in this work to establish the link between this approachand duality solutions. This latter concept of solutionsallows in particular to define a flow associated to the velocity field.Then an existence and uniqueness theory for duality solutions isdeveloped in the spirit of James and Vauchelet (NoDEA (2013)) [26].However, since duality solutions are only known in one dimension, werestrict our study to the one dimensional case.