Eulerian, Lagrangian and broad continuous solutions to a balance law with non convex flux II

TL;DR

This work analyzes a continuous solution $u$ of the balance law $\partial_t u + \partial_x(f(u)) = g$ with $f\in C^2$ and bounded $g$, comparing Eulerian, Broad, and Lagrangian formulations and how each yields a different interpretation of the source term. It proves compatibility results under the negligibility of inflection points, constructs Cantor-type Lagrangian parameterizations, and exhibits positive-measure sets where $u$ is nondifferentiable along characteristics, demonstrating limits to the equivalence of formulations. The paper provides counterexamples showing that Broad, Eulerian, and Lagrangian sources can diverge when inflection points are non-negligible, and it clarifies when continuous sources lead to coherent, shared source terms across formulations. These findings clarify the structure of solution notions for non-convex flux balance laws and highlight the delicate role of inflection points and parameterizations in source identification and well-posedness. Overall, the results delineate the boundaries of equivalence among Eulerian, Broad, and Lagrangian approaches and offer guidance on when a common source can be identified.

Abstract

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.

Figures (5)

• Figure 1: We picture relations among the sources that we determine for a fixed continuous solution of the balance law \ref{['EE']} under the non-degeneracy Assumption \ref{['ass:h']}. When a Lagrangian source is continuous, it is also an Eulerian source, although, surprisingly, it might not be the Lagrangian source for a different Lagrangian parameterization, see Remark \ref{['Rem:cubico']} and Theorem \ref{['T:continousSourceGen']}. If a broad source is continuous, it is a right source with all the formulations.
• Figure 2: The initial region $Q_{0}$ and one of its strips $S_{1}$. Proportions are distorted. Dashed lines suggest the qualitative behavior of characteristic curves.
• Figure 3: From left to right, figures illustrate the iterative horizontal subdivision of the height $a_{0}$---left figure---first in two extremal horizontal strips of height $a_{1}$ (blue ones) and a central strip of height $b_{1}$ (central one), then---second figure---the subdivision of each horizontal strip of height $a_{1}$ into two horizontal strips of height $a_{2}$ (blue ones) and a central strip of height $b_{2}$, and so on al later iterations. $K$ lies within blue regions. The regions $L_{i}$ are so thin, even after two iterations, that they are not visible in such a picture.
• Figure 4: Flux function $f$ considered in § \ref{['Ss:nondifferentiabilityset']}. Close to the origin, $f$ is strictly convex, but not uniformly convex. This flux function is $C^{\infty}(\mathbb{R})$, with all derivatives vanishing at the origin, but of course it is not analytic
• Figure 5: The initial region $Q_{0}$ and one of its strips $S_{1}$. Proportions are distorted. Dotted lines suggest the qualitative behavior of characteristic curves. Dashed lines suggest the blocks $S_{i}$, $Q_{i}$, $L_{i}$, $R_{i}$ in the construction.

Theorems & Definitions (21)

• Remark 1.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4: file1ABC
• Theorem 2.6
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Corollary 3.5