We discuss different notions of continuous solutions to the balance law∂tu+∂x(f(u))=gg 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 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.

Eulerian, Lagrangian and Broad continuous solutions to a balance law with non-convex flux I / Alberti, G.; Bianchini, S.; Caravenna, L.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 261:8(2016), pp. 4298-4337. [10.1016/j.jde.2016.06.026]

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

Bianchini S.;
2016-01-01

Abstract

We discuss different notions of continuous solutions to the balance law∂tu+∂x(f(u))=gg 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 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.
2016
261
8
4298
4337
https://www.sciencedirect.com/science/article/pii/S0022039616301620?via=ihub
https://arxiv.org/abs/1512.04863
Alberti, G.; Bianchini, S.; Caravenna, L.
File in questo prodotto:
File Dimensione Formato  
a+b+caravenna-ContinuousSolutions.pdf

Open Access dal 21/01/2018

Tipologia: Documento in Post-print
Licenza: Creative commons
Dimensione 1.16 MB
Formato Adobe PDF
1.16 MB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/110114
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 8
  • ???jsp.display-item.citation.isi??? 9
social impact