Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

Abstract

Abstract. We prove the equivalence between the notion of Wasserstein gradient flow for a onedimensional nonlocal transport PDE with attractive/repulsive Newtonian potential on one side, and the notion of entropy solution of a Burgers-type scalar conservation law on the other. The solution of the former is obtained by spatially differentiating the solution of the latter. The proof uses an intermediate step, namely the L 2 gradient flow of the pseudo-inverse distribution function of the gradient flow solution. We use this equivalence to provide a rigorous particle-system approximation to the Wasserstein gradient flow, avoiding the regularization effect due to the singularity in the repulsive kernel. The abstract particle method relies on the so-called wave-front-tracking algorithm for scalar conservation laws. Finally, we provide a characterization of the sub-differential of the functional involved in the Wasserstein gradient flow.

Scalar conservation laws seen as gradient flows: known results and new perspectives

“…A clear connection of a scalar conservation law with Wasserstein gradient flows in one space dimension was established years later in [8]. We shall briefly review this result in the next section.…”

“…In both the attractive and the repulsive case, the identification between the L 2 gradient flow and the Wasserstein gradient flow are rigorously recovered in [8].…”

“…The natural question arises on whether F is an entropy solution for (25) provided µ is a Wasserstein gradient flow for (22) and viceversa. This task has been addressed in [8].…”

“…The gradient flow approach of [45] was also used in chemotaxis modelling, see [6], and in nonlocal interaction equations with non smooth kernels, see [8,20]. Already in their review paper [28], Carrillo and Toscani started investigating the role of the Lagrangian formulation for a continuity equation of the form (1), with the aim of proving simple 'quasi-contraction inequalities' of the form d 2 (ρ 1 (t), ρ 2 (t)) ≤ d 2 (ρ 1 (0), ρ 2 (0))e Ct , C ∈ R.…”

“…The latter shows a link between the Burgers equation and a class of nonlocal interaction equations. It was developed in [8].…”