Comparison of Some Solution Concepts for Linear First-order Hyperbolic Differential Equations with Non-smooth Coefficients

We discuss solution concepts for linear hyperbolic equations with coefficients of regularity below Lipschitz continuity. Thereby our focus is on theories which are based either on a generalization of the method of characteristics or on refined techniques concerning energy estimates. We provide a series of examples both as simple illustrations of the notions and conditions involved but also to show logical independence among the concepts. 0. Introduction According to Hurd and Sattinger in [23] the issue of a systematic investigation of hyperbolic partial differential equations with discontinuous coefficients as a research topic has been raised by Gelfand in 1959. Here, we attempt a comparative study of some of the theories on that subject which have been put forward since. More precisely, we focus on techniques and concepts that build either on the geometric picture of propagation along characteristics or on the functional analytic aspects of energy estimates. In order to produce a set-up which makes the various methods comparable at all, we had to stay with the special situation of a scalar partial differential equation with real coefficients. As a consequence, for example, we do not give full justice to theories whose strengths lie in the application to systems rather than to a single equation. A further limitation in our choices comes from the restriction to concepts, hypotheses and mathematical structures which (we were able to) directly relate to distribution theoretic or measure theoretic notions. To illustrate the basic problem in a simplified lower dimensional situation for a linear conservation law, we consider the following formal differential equation for a density function (or distribution, or generalized function) u depending on time t and spatial position x ∂tu(t, x) + ∂x(a(t, x)u(t, x)) = 0. 2000 Mathematics Subject Classification: 35D05; 35D10, 46F10, 46F30. Supported by FWF grant Y237-N13.