Eulerian, Lagrangian and broad continuous solutions to a balance law with non convex flux II
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| Format: | Preprint |
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2024
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| author | Alberti, Giovanni Bianchini, Stefano Caravenna, Laura |
| author_facet | Alberti, Giovanni Bianchini, Stefano Caravenna, Laura |
| contents | We consider a *continuous* solution $u$ of the balance law \[ \partial_{\mathit t} u + \partial_{\mathit x} (f(u)) = g\] in one space dimension, where the flux function $f$ is of class $C^2$ and the source term $g$ is bounded. This equation admits an Eulerian intepretation (namely the distributional one) and a Lagrangian intepretation (which can be further specified). Since $u$ is only continuous, these interpretations do not necessessarily agree; moreover each interpretation naturally entails a different equivalence class for the source term $g$. In this paper we complete the comparison between these notions of solutions started in the companion paper [Alberti-Bianchini-Caravenna I], and analize in detail the relations between the corresponding notions of source term. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_03544 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eulerian, Lagrangian and broad continuous solutions to a balance law with non convex flux II Alberti, Giovanni Bianchini, Stefano Caravenna, Laura Analysis of PDEs (2020):35L60 (primary), 37C10, 58C20, 76N10 (secondary) We consider a *continuous* solution $u$ of the balance law \[ \partial_{\mathit t} u + \partial_{\mathit x} (f(u)) = g\] in one space dimension, where the flux function $f$ is of class $C^2$ and the source term $g$ is bounded. This equation admits an Eulerian intepretation (namely the distributional one) and a Lagrangian intepretation (which can be further specified). Since $u$ is only continuous, these interpretations do not necessessarily agree; moreover each interpretation naturally entails a different equivalence class for the source term $g$. In this paper we complete the comparison between these notions of solutions started in the companion paper [Alberti-Bianchini-Caravenna I], and analize in detail the relations between the corresponding notions of source term. |
| title | Eulerian, Lagrangian and broad continuous solutions to a balance law with non convex flux II |
| topic | Analysis of PDEs (2020):35L60 (primary), 37C10, 58C20, 76N10 (secondary) |
| url | https://arxiv.org/abs/2401.03544 |