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On the structure of entropy dissipation and regularity for quasi-entropy solutions to 1d scalar conservation laws and to isentropic Euler system with $γ=3$

Fabio Ancona, Elio Marconi, Luca Talamini

TL;DR

This work analyzes quasi-entropy solutions to 1D scalar conservation laws and the isentropic Euler system with $\gamma=3$ by establishing a kinetic formulation and a Lagrangian representation via Smirnov's decomposition of currents. It proves that entropy-dissipation measures concentrate on a 1-rectifiable set in 1D under a general non-degeneracy condition on the flux and extends the approach to the Euler system, where Riemann invariants become quasi-entropy Burgers solutions. A key contribution is the rectifiability result for the dissipation measures, along with a Besov regularity improvement for Burgers when the kinetic measure has a fixed sign, and the application to the Euler system showing that density and momentum inherit a precise Besov-type regularity. These results offer a unified, geometric-measure-theoretic framework for regularity and jump structure in conservation laws and related systems, with implications for well-posedness and stability analyses in low-regularity regimes.

Abstract

In this paper, we first investigate quasi-entropy solutions to scalar conservation laws in several space dimensions. In this setting, we introduce a suitable Lagrangian representation for such solutions. Next, we prove that, in one space dimension and for fluxes $f$ satisfying a general non-degeneracy condition, the entropy dissipation measures of quasi-entropy solutions are concentrated on a 1-rectifiable set. The same result is obtained for the isentropic Euler system with $γ= 3$, for which we also slightly improve the available fractional regularity by exploiting the sign of the kinetic measures.

On the structure of entropy dissipation and regularity for quasi-entropy solutions to 1d scalar conservation laws and to isentropic Euler system with $γ=3$

TL;DR

This work analyzes quasi-entropy solutions to 1D scalar conservation laws and the isentropic Euler system with by establishing a kinetic formulation and a Lagrangian representation via Smirnov's decomposition of currents. It proves that entropy-dissipation measures concentrate on a 1-rectifiable set in 1D under a general non-degeneracy condition on the flux and extends the approach to the Euler system, where Riemann invariants become quasi-entropy Burgers solutions. A key contribution is the rectifiability result for the dissipation measures, along with a Besov regularity improvement for Burgers when the kinetic measure has a fixed sign, and the application to the Euler system showing that density and momentum inherit a precise Besov-type regularity. These results offer a unified, geometric-measure-theoretic framework for regularity and jump structure in conservation laws and related systems, with implications for well-posedness and stability analyses in low-regularity regimes.

Abstract

In this paper, we first investigate quasi-entropy solutions to scalar conservation laws in several space dimensions. In this setting, we introduce a suitable Lagrangian representation for such solutions. Next, we prove that, in one space dimension and for fluxes satisfying a general non-degeneracy condition, the entropy dissipation measures of quasi-entropy solutions are concentrated on a 1-rectifiable set. The same result is obtained for the isentropic Euler system with , for which we also slightly improve the available fractional regularity by exploiting the sign of the kinetic measures.
Paper Structure (17 sections, 24 theorems, 257 equations)

This paper contains 17 sections, 24 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Results and plan of the paper
  3. Kinetic Formulation of quasi-entropy solutions
  4. Lagrangian Representation
  5. Definition of the Lagrangian Representation
  6. Existence of (good) Lagrangian Representations.
  7. Preliminaries About the Theory of Currents
  8. Smirnov's Theorem for Normal Currents
  9. Epigraph and Simultaneous Lagrangian Representations
  10. Lagrangian representation for the epigraph.
  11. Simultaneous Lagrangian Representations
  12. Good Selection of Curves
  13. Structure of the Kinetic Measures
  14. Rectifiability of J
  15. The One-Dimensional Case
  16. ...and 2 more sections

Key Result

Proposition 2.3

A function $u \in \mathbf L^\infty(\Omega; \mathbb R)$ is a quasi-entropy solution in the sense of Definition defi:fes if and only if, there exist locally finite measures $\mu_0, \mu_1 \in \mathscr{M}(\Omega \times \mathbb R)$ with $\mathrm{supp}\, \mu_i \subset \Omega \times [\inf u, \sup u]$, such where

Theorems & Definitions (61)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Definition 3.3
  • Remark 3.4
  • ...and 51 more