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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:261

Article

Alberti, G.¹  Bianchini, S.²  Caravenna, L.³ 

[1] Univ Pisa, Dipartimento Matemat, Largo Pontecorvo 5, I-56127 Pisa, Italy

[2] SISSA ISAS, Via Bonomea 265, I-34136 Trieste, Italy

[3] Univ Padua, Dipartimento Matemat Tullio Levi Civita, Via Trieste 63, I-35121 Padua, Italy

关键词: Balance law;    Lagrangian description;    Eulerian formulation;   

DOI  :  10.1016/j.jde.2016.06.026

来源: Elsevier

【 摘 要 】

We discuss different notions of continuous solutions to the balance law partial derivative(t)u + partial derivative(x) (f(u)) = g g bounded f is an element of C-2 extending previous works relative to the flux f (u) = u(2). 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 reduce the ODE 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 a matter of the companion work [2], where we include counterexamples when the negligibility on inflection points fails. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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