A new nonlinear functional for general scalar conservation laws (Q1019668)

The authors consider a new nonlinear functional for general scalar conservation laws. The approach based on the construction of some nonlinear functionals was proved to be robust in the study of the well-posedness theories of hyperbolic conservation laws, especially in one space dimensional case. In particular, a generalized entropy functional was constructed by \textit{T.-P. Liu} and \textit{T. Yang} [Commun. Pure Appl. Math. 52, No.~11, 1427--1442 (1999; Zbl 0941.35051)] for the \(L^1\) stability of weak solutions. However, this generalized functional is so far only defined for scalar equations with convex flux function. In this paper, the authors introduce a new nonlinear functional which is motivated by the new Glimm functional introduced in [\textit{J.-L. Hua, Z.-H. Jiang} and \textit{T. Yang}, A new Glimm functional and convergence rate of Glimm scheme for general systems of hyperbolic conservation laws, to appear in Arch. Rat. Mech. Anal. (2010: doi:10.1007/s00205-009-0266-1)] for general scalar conservation laws. This functional improves the one given in [\textit{H.-X. Liu} and \textit{T. Yang}, J. Differ. Equations 235, No.~2, 658--667 (2007; Zbl 1168.35392)] and it can be viewed as a better attempt for the generalized entropy functional for general equations.