2007/02/07 by Boris Khesin, Khesin, Boris, Gerard Misiołek +2 · 1 citation
Mathematics · Physics and Astronomy · #35Q35 #58B25 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #math-ph #math.DG #math.MP #msc:35Q35 #msc:58B25
paper · pdf · doi:10.48550/arxiv.math/0702196
11 pages, 2 figures
arxiv created 2007/02/07 · openalex publication_date 2007/02/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish a simple relation between 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 establishes an intrinsic connection between ideal Euler hydrodynamics (via Arnold's approach), shock formation in the multidimensional Burgers equation and the Wasserstein geometry of the space of densities.