We consider a system of two equations derived from isentropic gas dynamics with no classical Riemann solutions. We show existence of unbounded self-similar solutions (singular shocks) of the Dafermos regularization of the system. Our approach is based on the blowing-up approach of geometric singular perturbation theory. 1. Introduction. This paper brings together two ideas that have been studied before: the study of large-data solutions to systems of conservation laws, for which wellposedness when the Cauchy data are small in the sense of total variation has been established (see for example [1]), and the study of singular shocks, a type of weak solution of very low regularity, originally described by Keyfitz and Kranzer [11, 12, 13] and later analysed more systematically by Sever [20]. To these constructions, we bring some new ingredients. Change of type. The system we study is not hyperbolic for all states, but exhibits change of type, with a degeneracy corresponding to vacuum states, and nonhyperbolic states corresponding to negative densities. For some Riemann data, we find the solution includes vacuum states, on the degenerate curve. A physical model. Whereas the system examined in [11] was derived from a onedimensional model for isothermal, isentropic gas dynamics, a model that is a limit whose correspondence to a genuine physical situation is ambiguous, here we deal with a more standard model, the equations of isentropic gas dynamics for an ideal fluid with γ, the
We present an example in which the Glimm estimate for a strictly hyperbolic system of two conservation laws is violated. 1. Introduction. Consider the Cauchy problem for a strictly hyperbolic system of conservation laws in one space dimension:
We consider a system of two equations that can be used to describe nonlinear chromatography and produce a coherent explanation and description of the unbounded solutions (singular shocks) that appear in Mazzotti's model [28]. We use the methods of Geometric Singular Perturbation Theory, to show existence of a viscous solution to Dafermos-DiPerna regularization.2000 Mathematics Subject Classification. Primary 35L65, 35L67; Secondary 34E15, 34C37.
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