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 ( η ( u ) , q ( u ) ) (\eta (u),q(u)) with η ( u ) \eta (u) of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in L l o c ∞ L^\infty _{\mathrm { loc}} that satisfy the inequality: η ( u ) t + q ( u ) x ≤ μ \eta (u)_t+q(u)_x\leq \mu in the distributional sense for some non-negative Radon measure μ \mu . Furthermore, we extend this result to the class of weak solutions in L l o c p L^p_{\mathrm {loc}} , based on the asymptotic behavior of the flux function f ( u ) f(u) and the entropy function η ( u ) \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.