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RESEARCH PRODUCT

Historischer Überblick zur mathematischen Theorie von Unstetigkeitswellen seit Riemann und Christoffel

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We give a brief historical account of the development of the mathematical theory of propagation of discontinuities in gases, fluids or elastic materials. The theory was initiated by Riemann who investigated the propagation of shocks in one-dimensional isentropic gas flow. Riemann’s results were used by Christoffel to treat, more generally, the propagation of (first order) discontinuity surfaces in three-dimensional flows of perfect fluids. Subsequently Christoffel applied his general theory to first order waves in certain elastic materials. Independently of Riemann and Christoffel significant contributions were made by Hugoniot. The theory was completed in Hadamard’s celebrated monograph [31] where among many other things acceleration waves in hyperelastic bodies were correctly treated. Later Prandtl, A. Busemann et al. attacked the problem of discontinuous flow from the more practical point of view of the engineer and obtained many important results. In the final section of our report we briefly survey some recent global weak existence theorems for Riemann and general Cauchy initial value problems of general strictly hyperbolic conservation laws.