The L 1 Theory for Scalar Conservation Laws

Abstract

The theory of the scalar balance law, in several spatial dimensions, has reached a state of virtual completeness. In the framework of classical solutions, the elementary, yet effective, method of characteristics yields a sharper version of Theorem 5.1.1, determining explicitly the life span of solutions with Lipschitz continuous initial data and thereby demonstrating that in general this life span is finite. Thus one must deal with weak solutions, even when the initial data are very smooth. In regard to weak solutions, the special feature that sets the scalar balance law apart from systems of more than one equation is the size of its family of entropies. It will be shown that the abundance of entropies induces an effective characterization of admissible weak solutions as well as very strong L 1-stability and L ∞-monotonicity properties. Armed with such powerful a priori estimates, one can construct admissible weak solutions in a number of ways. As a sample, construction by the method of vanishing viscosity, the theory of L 1-contraction semigroups, the layering method, a relaxation method and an approach motivated by the kinetic theory will be presented here. The method of vanishing viscosity will also be employed for solving the initial-boundary value problem. When the initial data are functions of locally bounded variation then so are the solutions. Remarkably, however, even solutions that are merely in L ∞ exhibit the same geometric structure as BV functions, with jump discontinuities assembling on “manifolds” of codimension one.