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 $L^\infty _{\mathrm { loc}}$ 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 $L^p_{\mathrm {loc}}$, 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.