Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux I

2015/12/15 by Giovanni Alberti, Stefano Bianchini, Alberti, G. +3 · 1 citation
Mathematics · #35L60 #37C10 #58J45 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1512.04863

openalex publication_date 2015/12/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We discuss different notions of continuous solutions to the balance law u_t + (f(u ))_x =g with g bounded, f∈ C², extending previous works relative to the flux f(u)=u². We establish the equivalence among distributional solutions and a suitable notion of Lagrangian solutions for general smooth fluxes. We eventually find that continuous solutions are Kruzkov iso-entropy solutions, which yields uniqueness for the Cauchy problem. We also establish the ODE reduction on any characteristics under the sharp assumption that the set of inflection points of the flux f is negligible. The correspondence of the source terms in the two settings is matter of a companion work, where we also provide counterexamples when the negligibility on inflection points fails.

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