Minimal Entropy Conditions for Scalar Conservation Laws with General Convex Fluxes

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 $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}$ 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 $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p$, 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.