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

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• 作者：G. Alberti^a ; ^{galberti1@dm.unipi.it" class="auth_mail" title="E-mail the corresponding author} ; S. Bianchini^b ; ^{bianchin@sissa.it" class="auth_mail" title="E-mail the corresponding author} ; L. Caravenna^c ; ^{laura.caravenna@unipd.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35L60 ; 37C10 ; 58J45
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：261
• 期：8
• 页码：4298-4337
• 全文大小：1272 K

文摘

We discuss different notions of continuous solutions to the balance law extending previous works relative to the flux f(u)=u²$f(u)=u^2$. 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.