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

Minimal Entropy Conditions for Scalar Conservation Laws with General Convex Fluxes