Abstract
We are concerned with the Riemann problem ofp-system, i.e.,v t −u x =0,u t +p(v) x =0. We know that ifp is a monotone function, the solution can be constructed by shock waves and rarefaction waves. Therefore we consider the functionp which is not monotone. In this case, this system is a mixed type, and very little is known about this type of the system. Furthermore, we know that we can not construct the solution only by the above waves because of the ellipticity of the system. In order to overcome this difficulty, we consider the phase boundary, which is a shock wave with shock speed 0 and changes phases. Using this and the above waves, we describe the solution explicitly, for arbitrary initial step data.
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Yanagi, S. The Riemann problem for a class of conservation laws of van der waals fluid. Japan J. Indust. Appl. Math. 9, 239 (1992). https://doi.org/10.1007/BF03167567
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DOI: https://doi.org/10.1007/BF03167567