Minimal Entropy Conditions for Scalar Conservation Laws with General Convex Fluxes

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 $L_{\mathrm{loc}}^{\infty }$ that satisfy the inequality: $\eta (u{)}_t+q(u{)}_x\le \mu$ in the distributional sense for some non-negative Radon measure $\mu$. Furthermore, we extend this result to the class of weak solutions in $L_{\mathrm{loc}}^p$, 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.

31 pages, to appear in Quarterly of Applied Mathematics, AMS, 2023

Cited by in corpus (1)

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