Initial Layers and Uniqueness of Weak Entropy Solutions to Hyperbolic Conservation Laws

We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weak-star in L∞ as t→ 0+ and satisfy the entropy inequality in the sense of distributions for t > 0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss its applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation. Gui-Qiang Chen: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA Email: [email protected] Michel Rascle: Laboratoire J.A. Dieudonné, Université de Nice, Parc Valrose, F06108, Nice Cedex 02, France Email: [email protected]. Proposed Running Title: Initial Layers and Uniqueness 1991 Mathematics Subject Classification. Primary:35L65,35B30,35L45; Secondary:35B35,76E30.