A Minimality Property for Entropic Solutions to Scalar Conservation Laws in 1 + 1 Dimensions

TL;DR

This paper investigates a minimality property of entropic solutions to scalar conservation laws, demonstrating that these solutions optimize the rate of entropy increase among all possible evolutions in 1+1 dimensions.

Contribution

It introduces a novel minimality property of entropic solutions, linking thermodynamic principles to the mathematical structure of scalar conservation laws with convex fluxes.

Findings

Entropic solutions maximize the rate of entropy increase.

The minimality property is established rigorously for convex flux functions.

The results connect thermodynamic laws with mathematical properties of hyperbolic PDEs.

Abstract

The Second Law of Thermodynamics asserts that the physical entropy of an adiabatic system is an increasing function in time. In this paper we will study a more stringent version of this law, according to which the entropy should not only increase in time, but the rate of increase is optimal in absolute value among all possible evolutions. We will establish this property in the framework of non-linear scalar hyperbolic conservation law with strictly convex fluxes.