Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations

2012/08/03 by Marco Di Francesco, Di Francesco, Marco, Daniel Matthes +1
Mathematics · #35A02 #35A15 #35K65 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #math.AP #msc:35A02 #msc:35A15 #msc:35K65

paper · pdf · doi:10.48550/arxiv.1208.0789

arxiv created 2012/08/03 · openalex publication_date 2012/08/03 · arxiv updated 2012/08/06 · openalex created_date 2022/09/26 · openalex updated_date 2026/07/28

Abstract

We consider a nonlinear degenerate convection-diffusion equation with inhomogeneous convection and prove that its entropy solutions in the sense of Kružkov are obtained as the - a posteriori unique - limit points of the JKO variational approximation scheme for an associated gradient flow in the L²-Wasserstein space. The equation lacks the necessary convexity properties which would allow to deduce well-posedness of the initial value problem by the abstract theory of metric gradient flows. Instead, we prove the entropy inequality directly by variational methods and conclude uniqueness by doubling of the variables.

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