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Conservation laws with time dependent discontinuous coefficients. (English) Zbl 1078.35071
The authors consider scalar conservation laws where the flux function depends discontinuously on both the spatial and temporal locations. The existence and well-posedness of an entropy solution to the Cauchy problem are proved. The existence is established by showing that a sequence of front tracking approximations is compact in \(L^1\), and that the limits are entropy solutions. Then, using the definition of an entropy solution taken from [K. H. Karlsen, N. H. Risebro and J. D. Towers, Skr. K. Nor. Vidensk. Selsk. 3, 1–49 (2003; Zbl 1036.35104)], the solution operator which is \(L^1\) contractive is proved. These results generalize the corresponding results from [S. N. Kruzhkov, Math. Sb. 10, 217–243 (1970; Zbl 0215.16203)] and also partially those from Karlsen, Risebro, and Tower.

MSC:
35L65 Hyperbolic conservation laws
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
35R05 PDEs with low regular coefficients and/or low regular data
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