Abstract
The exact solutions of the Riemann problem for a symmetric system of Keyfitz–Kranzer type are obtained in fully explicit forms. Furthermore, the global solutions of the double Riemann problems are also constructed explicitly when the initial data are taken to be three piecewise constant states. During the process of constructing the global solutions, all occurring wave interactions have been dealt with in detail by using the method of characteristics. In addition, it is shown that the Riemann solutions are stable with respect to the specific small perturbations of the Riemann initial data.
Highlights
1 Introduction In this paper, we are concerned with the following hyperbolic system of conservation laws in the form
Which is named a symmetric system of Keyfitz–Kranzer type [1,2,3,4,5]
It is worthwhile to notice that the shock curve has the same expression formula as the rarefaction one in the quarter (u, v) phase plane, so that system (1.1) belongs to the so-called Temple class [6, 7]
Summary
We are concerned with the following hyperbolic system of conservation laws in the form. We restrict ourselves only to considering the situation φ (r) > 0 and φ (r) > 0 in the quarter (u, v) phase plane, where u ≥ 0 and v ≥ 0 are required. We have λ1 < λ2 under our assumption φ (r) > 0 when r > 0, which implies that system (1.1) is strictly hyperbolic except for the origin in the quarter (u, v) phase plane. It is worthwhile to notice that the shock curve has the same expression formula as the rarefaction one in the quarter (u, v) phase plane, so that system (1.1) belongs to the so-called Temple class [6, 7]
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