Connections between Optimal Transport, Combinatorial Optimization and Hydrodynamics

Yann Brenier (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

There are well-established connections between combinatorial optimization, optimal transport theory and Hydrodynamics, through the linear assignment problem in combinatorics, the Monge-Kantorovich problem in optimal transport theory and the model of inviscid, potential, pressure-less fluids in Hydrodynamics. Here, we consider the more challenging quadratic assignment problem (which is NP, while the linear assignment problem is just P) and find, in some particular case, a correspondence with the problem of finding stationary solutions of Euler's equations for incompressible fluids. For that purpose, we introduce and analyze a suitable "gradient flow" equation. Combining some ideas of P.-L. Lions (for the Euler equations) and Ambrosio-Gigli-Savaré (for the heat equation), we provide for the initial value problem a concept of generalized ''dissipative'' solutions which always exist globally in time and are unique whenever theyare smooth.

Mots clés: Optimal transport; PDEs of Fluid Mechanics; Calculus of Variations; Combinatorial Optimization

Domaines: Equations aux dérivées partielles [math.AP]

Yann Brenier. Connections between Optimal Transport, Combinatorial Optimization and Hydrodynamics. 2014. ⟨hal-01070575⟩