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

Abstract

Abstract. We review some results in the literature which attempted (only partly successfully) at linking the theory of scalar conservation laws with the Wasserstein gradient flow theory. In particular, we consider the problem of writing a scalar conservation law within the Wasserstein gradient flow theory. As a related problem, we also review results on contraction properties of scalar conservation laws in the p-Wasserstein distances. Moreover, we provide a particle-based approach to view a scalar conservation law as a gradient flow in a nonlinear-mobility sense. Finally, we propose a semi-implicit particle method based on the standard 2-Wasserstein distance.

Solutions to aggregation–diffusion equations with nonlinear mobility constructed via a deterministic particle approximation

“…10 Thanks to (22), (23) and (24) and the fact that the support of ρ N is uniformly bounded in time for every N , we then obtain…”

“…This term complicates the problem because it prevents the construction of the approximating sequence via discretization of the JKO-scheme, as it is done instead in most of the papers quoted above. The reason is that, in this case, the Wasserstein distance is not explicit but it can only be expressed in the Benamou-Brenier formulation, see [12], even if the equation preserves a gradient flow structure, see [22]. However, in presence of the nonlinear mobility, it is easy to reformulate the diffusion equation in Laplacian form by calling…”