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Well-Posedness of Hyperbolic Systems of Balance Laws of Temple Type on Networks
von Boby GunarsoConsider a Cauchy problem for hyperbolic systems of balance laws on a network. The systems of balance laws, given on each edge, are connected by suitable coupling conditions at the vertices. In this dissertation, we study the well-posedness of this problem, under appropriate assumptions on the convective part, the source part and the coupling conditions.
For the homogeneous system of conservation laws, we assume that it is of Temple type, in which shock and rarefaction curves coincide. In particular, no convexity of the flux and smallness assumption on the initial data are required. The existence of solutions is obtained by using the wave-front tracking method which yields a sequence of piecewise constant approximate solutions to the Cauchy problem on the network. At the vertex, the number of waves and the total variation of the solution might increase. Therefore, we construct appropriate functionals to obtain suitable estimates in case of interactions between waves and with the vertex. This allows us, by a compactness argument, to obtain a subsequence of the approximate solutions which converges to an admissible solution of the homogeneous coupling problem globally in time. If convexity is assumed, we show, by the technique of pseudopolygonals, that the entire sequence converges and that the solution is Lipschitz continuous with respect to the initial data.
To obtain approximate solutions for the system of balance laws, we combine the approximate solutions of the convective and the source part via an operator splitting method. Using the well-posedness results of the convective and the source part, the existence of an admissible solution for the Cauchy problem of balance laws on the network is obtained, globally in time. If genuine nonlinearity is assumed, we further show that the solution of the balance laws depends continuously on the initial data.
For the homogeneous system of conservation laws, we assume that it is of Temple type, in which shock and rarefaction curves coincide. In particular, no convexity of the flux and smallness assumption on the initial data are required. The existence of solutions is obtained by using the wave-front tracking method which yields a sequence of piecewise constant approximate solutions to the Cauchy problem on the network. At the vertex, the number of waves and the total variation of the solution might increase. Therefore, we construct appropriate functionals to obtain suitable estimates in case of interactions between waves and with the vertex. This allows us, by a compactness argument, to obtain a subsequence of the approximate solutions which converges to an admissible solution of the homogeneous coupling problem globally in time. If convexity is assumed, we show, by the technique of pseudopolygonals, that the entire sequence converges and that the solution is Lipschitz continuous with respect to the initial data.
To obtain approximate solutions for the system of balance laws, we combine the approximate solutions of the convective and the source part via an operator splitting method. Using the well-posedness results of the convective and the source part, the existence of an admissible solution for the Cauchy problem of balance laws on the network is obtained, globally in time. If genuine nonlinearity is assumed, we further show that the solution of the balance laws depends continuously on the initial data.