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arXiv · 2212.11430

Minimal Entropy Conditions for Scalar Conservation Laws with General Convex Fluxes

Gaowei Cao·Gui-Qiang G. Chen
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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 $(η(u),q(u))$ with $η(u)$ 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: $η(u)_t+q(u)_x\leq μ$ in the distributional sense for some non-negative Radon measure $μ$. 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 $η(u)$ 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.

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BibTeXRIS

Gaowei Cao, Gui-Qiang G. Chen. 2023-04-26. Minimal Entropy Conditions for Scalar Conservation Laws with General Convex Fluxes. https://arxiv.org/abs/2212.11430

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