Calculus of Variations and Geometric Measure Theory

F. Ancona - L. Talamini

Backward-forward characterization of attainable set for conservation laws with spatially discontinuous flux

created by talamini on 24 Jan 2025

[BibTeX]

Preprint

Inserted: 24 jan 2025
Last Updated: 24 jan 2025

Year: 2024

Abstract:

Consider a scalar conservation law with a spatially discontinuous flux at a single point $x=0$, and assume that the flux is uniformly convex when $x\neq 0$. Given an interface connection $(A,B)$, we define a backward solution operator consistent with the concept of $AB$-entropy solution. We then analyze the family $\mathcal A^{[AB]}(T)$ of profiles that can be attained at time $T>0$ by $AB$-entropy solutions with ${\mathbf L^\infty}$-initial data. We provide a characterization of $\mathcal A^{[AB]}(T)$ as fixed points of the backward-forward solution operator. As an intermediate step we establish for the first time a full characterization of $\mathcal A^{[AB]}(T)$ in terms of unilateral constraints and Oleinik-type estimates, valid for all connections. Building on such a characterization we derive uniform $BV$ bounds on the flux of $AB$-entropy solutions, which in turn yield the $\mathbf L^1_{\mathrm{loc}}$-Lipschitz continuity in time of these solutions.


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