Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Abstract

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959–1000] obtained the strong convergence of uniform L loc p bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein–Weierstrass theorem, and obtaining the strong convergence of uniform L ∞ or L loc p bounded viscosity solutions for scalar conservation law without convexity.