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Entropy Production in General Balance Laws

Rinaldo Colombo, Vincent Perrollaz

TL;DR

This work develops a comprehensive entropy-production framework for general scalar balance laws with space-time dependent flux and source, defining a linear operator $M_u(E)$ that acts on test functions and encodes entropy production for entropy and distributional solutions as well as $L^ obreakspace^ ext{\infty}$ data. It introduces a robust extension to regulated entropies via the Kurzweil–Stieltjes integral and a Fourier-analytic representation, enabling complex-valued entropies and an explicit relationship between $E''$ and the family of measures $\mu_k$. The authors then extend the construction to time-space entropies $E(t,x,u)$ with corresponding fluxes $F$ obeying $\partial_u F = \partial_u E \; \partial_u f$, defining $\mathcal{M}^{tx}_u(E)$ and proving linearity, flux-independence, and nonnegativity under convexity, thereby linking kinetic, measure-valued, and entropy-process formulations. A wealth of technical results establish continuity, stability, and representation formulas, including a faithful encoding of any $L^ obreakspace^ ext{\infty}$ function and a set of Fourier-based tools with potential for uniqueness and stability analyses in nonhomogeneous settings. Overall, the framework provides quantitative representations and flexible extensions to nonhomogeneous and complex-valued entropies, with broad implications for scalar conservation laws and related formulations.

Abstract

Given a general scalar balance law, i.e., in several space dimensions and with flux and source both space and time dependent, we focus on the functional properties of the entropy production. We apply this operator to entropy solutions, to distributional solutions or to merely L $\infty$ functions. Proving its analytical properties naturally leads to the projective tensor product of C 1 spaces and to further natural extensions to space and time dependent or complex valued ''entropies''. Besides various qualitative properties, this extended framework allows to obtain new quantitative formulae -also by means of Fourier transforms -that provide different representations of the entropy production. Remarkably, this operator also furnishes a faithful representation of any L $\infty$ function.

Entropy Production in General Balance Laws

TL;DR

This work develops a comprehensive entropy-production framework for general scalar balance laws with space-time dependent flux and source, defining a linear operator that acts on test functions and encodes entropy production for entropy and distributional solutions as well as data. It introduces a robust extension to regulated entropies via the Kurzweil–Stieltjes integral and a Fourier-analytic representation, enabling complex-valued entropies and an explicit relationship between and the family of measures . The authors then extend the construction to time-space entropies with corresponding fluxes obeying , defining and proving linearity, flux-independence, and nonnegativity under convexity, thereby linking kinetic, measure-valued, and entropy-process formulations. A wealth of technical results establish continuity, stability, and representation formulas, including a faithful encoding of any function and a set of Fourier-based tools with potential for uniqueness and stability analyses in nonhomogeneous settings. Overall, the framework provides quantitative representations and flexible extensions to nonhomogeneous and complex-valued entropies, with broad implications for scalar conservation laws and related formulations.

Abstract

Given a general scalar balance law, i.e., in several space dimensions and with flux and source both space and time dependent, we focus on the functional properties of the entropy production. We apply this operator to entropy solutions, to distributional solutions or to merely L functions. Proving its analytical properties naturally leads to the projective tensor product of C 1 spaces and to further natural extensions to space and time dependent or complex valued ''entropies''. Besides various qualitative properties, this extended framework allows to obtain new quantitative formulae -also by means of Fourier transforms -that provide different representations of the entropy production. Remarkably, this operator also furnishes a faithful representation of any L function.

Paper Structure

This paper contains 22 sections, 30 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. The Standard Entropy Operator
  3. Elementary Setting -- E in C1
  4. Extension to E' Regulated
  5. Fourier Transform
  6. Time and Space Dependent Entropy Operator
  7. Technical Details
  8. Proofs Related to Section \ref{['sec:time-space-dependent']}
  9. \ref{['item:9']}:
  10. \ref{['item:34']}:
  11. \ref{['item:1']}:
  12. Proofs Related to § \ref{['subsec:first-setting']}
  13. Proofs Related to § \ref{['subsec:case-class-entr']}
  14. Step 1. Equality \ref{['eq:3']} holds when $E_c (k) = {\left|k-c\right|}$, for a $c \in {\mathbb{R}}$.
  15. Step 2: Extension to a general $E$.
  16. ...and 7 more sections

Key Result

Proposition 2.3

Let $f \in \mathscr{F}^n$, $g \in \mathbf{C}^{0} (\overline{I} \times {\mathbb{R}}^n\times {\mathbb{R}};{\mathbb{R}})$, $u_o \in {\mathbf{L}^\infty} ({\mathbb{R}}; {\mathbb{R}})$ and $u \in {\mathbf{L}^\infty} (I \times {\mathbb{R}}^n; {\mathbb{R}})$. Then,

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 37 more