Abstract

This paper deals with the so-called \(p\)-system describing the dynamics of isothermal and compressible fluids. The constitutive equation is assumed to have the typical convexity/concavity properties of Van der Waals equation. The authors search for discontinuous solutions constrained by the associated mathematical entropy inequality. First, following a strategy proposed by Abeyarante and Knowles and by Hayes and LeFloch, here the whole family of nonclassical Riemann solutions for this model is described. Second, the authors supplement the set of equations with a kinetic relation for the propagation of nonclassical undercompressive shocks, and they arrive at a uniquely defined solution of the Riemann problem. The authors also prove that the solutions depend continuously upon their data. The main novelty of the present paper is the precence of two inflection points in the constitutive equation. The Riemann solver constructed here is relevant for fluids in which viscosity and capillarity effects are kept in ballance.