Shock waves for the Burgers equation and curvatures of diffeomorphism groups

Abstract

We establish a simple relation between certain curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This relates the ideal Euler hydrodynamics (via Arnold’s approach) to shock formation in the multidimensional Burgers equation and the Kantorovich-Wasserstein geometry of the space of densities.