L1-Theory of Scalar Conservation Law with Continuous Flux Function

Abstract

Uniqueness of a generalized entropy solution (g.e.s.) to the Cauchy problem for N-dimensional scalar conservation laws ut+divxφ(u)=g, u(0, ·)=f with continuous flux function φ is still an open problem. For data (f, g) vanishing at infinity, we show that there exist a maximal and a minimal g.e.s. to the Cauchy problem and to the associated stationary problem u+divxφ(u)=f. In the case of L1 data, using the nonlinear semigroup theory, we prove that there is uniqueness for all data of a g.e.s. to the Cauchy problem if and only if there is uniqueness for all data of a g.e.s. to the related stationary problem. Applying this result and an induction argument on the dimension N, we extend uniqueness results of Bénilan, Kruzhkov (1996, Nonlinear Differential Equations Appl.3, 395–419) for flux having some monotonicity properties.