The phenomena of concentration and cavitation and the formation of $\delta$-shocks and vacuum states in solutions to the Euler equations for isentropic fluids are identified and analyzed as the pressure vanishes. It is shown that, as the pressure vanishes, any two-shock Riemann solution to the Euler equations for isentropic fluids tends to a $\delta$-shock solution to the Euler equations for pressureless fluids, and the intermediate density between the two shocks tends to a weighted $\delta$-measure that forms the $\delta$-shock. By contrast, any two-rarefaction-wave Riemann solution of the Euler equations for isentropic fluids is shown to tend to a two-contact-discontinuity solution to the Euler equations for pressureless fluids, whose intermediate state between the two contact discontinuities is a vacuum state, even when the initial data stays away from the vacuum. Some numerical results exhibiting the formation process of $\delta$-shocks are also presented.

We develop a well-posedness theory for solutions in L^1 to the Cauchy problem of general degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; and its advantage is that the kinetic equations in the kinetic formulation are well defined even when the macroscopic fluxes are not locally integrable, so that L^1 is a natural space on which the kinetic solutions are posed. Based on this notion, we develop a new, simpler, more effective approach to prove the contraction property of kinetic solutions in L^1 , especially including entropy solutions. It includes a new ingredient, a chain rule type condition, which makes it different from the isotropic case. Résumé: Nous développons une théorie d’existence et unicité pour les solutions L^1 seulement du problème de Cauchy pour un problème de Cauchy hyperbolique-parabolique avec diffusion non-isotrope générale. Des notions de formulations entropique et cinétique sont introduites qui incorporent un nouvel ingrédient, une condition de type dérivation composée, qui montre la différence fondamentale avec le cas d’une diffusion isotrope. L’avantage de la notion de solution cinétique est de travailler directement dans l’espace naturel L^1 .