Minimal Entropy Conditions for Scalar Conservation Laws with General Convex Fluxes

TL;DR

This paper establishes minimal entropy conditions for scalar conservation laws with convex fluxes, showing that a single convex entropy pair suffices to identify entropy solutions among weak solutions.

Contribution

It proves that one convex entropy-entropy flux pair uniquely characterizes entropy solutions for scalar conservation laws, extending to solutions in $L^p_{loc}$ and using Hamilton-Jacobi and compensated compactness techniques.

Findings

A single convex entropy pair suffices for solution uniqueness.

Extension of results to $L^p_{loc}$ solutions based on asymptotic flux behavior.

Use of Hamilton-Jacobi equivalence and commutator estimates in proofs.

Abstract

We are concerned with the minimal entropy conditions for one-dimensional scalar conservation laws with general convex flux functions. For such scalar conservation laws, we prove that a single entropy-entropy flux pair $(\eta(u),q(u))$ with $\eta(u)$ of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in $L^\infty_{\rm loc}$ that satisfy the inequality: $\eta(u)_t+q(u)_x\leq \mu$ in the distributional sense for some non-negative Radon measure $\mu$. Furthermore, we extend this result to the class of weak solutions in $L^p_{\rm loc}$, based on the asymptotic behavior of the flux function $f(u)$ and the entropy function $\eta(u)$ at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness.