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Hyperbolic systems of conservation laws in one space dimension. (English) Zbl 1032.35129
Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. I: Plenary lectures and ceremonies. Beijing: Higher Education Press. 159-178 (2002).
This paper is a survey on some basic ideas and recent developments in the theory of strictly hyperbolic systems of conservation laws in one space dimension. As it is well known, even with smooth initial data the solutions of these systems can develop shocks in finite time. This fact motivates the consideration of weak solutions that can be defined globally in time. On the other hand, when discontinuities are presented the weak solutions of the Cauchy problem may not be unique. The author discusses the problems of uniqueness and stability of entropy weak solutions to the Cauchy problem and the convergence of vanishing viscosity approximations in the BV setting, i.e. constructing the solutions within a space of functions with bounded variation.
For the entire collection see [Zbl 1011.00026].

35L65 Hyperbolic conservation laws
35L60 First-order nonlinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
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