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We are concerned with the isothermal limit of entropy solutions in L∞, containing the vacuum states, of the Euler equations for isentropic gas dynamics. We prove that the entropy solutions in L∞ of the isentropic Euler equations converge strongly to the corresponding entropy solutions of the isothermal Euler equations, when the adiabatic exponent γ→1. This is achieved by combining careful entropy analysis and refined kinetic formulation with compensated compactness argument to obtain the required uniform estimates for the limit. The entropy analysis involves careful estimates for the relation between the corresponding entropy pairs for the isentropic and isothermal Euler equations when the adiabatic exponent γ→1. The kinetic formulation for the entropy solutions of the isentropic Euler equations with the uniformly bounded initial data is refined, so that the total variation of the dissipation measures in the formulation is locally uniformly bounded with respect to γ>1. The explicit asymptotic analysis of the Riemann solutions containing the vacuum states is also presented. [ABSTRACT FROM AUTHOR]

We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional nonisentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint a necessary and sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum—there exists a compression in the initial data. For the nonisentropic Euler equations, we identify a sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum—there exists a strong compression in the initial data. Furthermore, we identify two new phenomena—decompression and de-rarefaction—for the nonisentropic Euler flows, different from the isentropic flows, via constructing two respective solutions. For the decompression phenomenon, we construct a first global continuous nonisentropic solution, even though the initial data contain a weak compression, by solving a backward Goursat problem, so that the solution is smooth, except on several characteristic curves across which the solution has a weak discontinuity (i.e., only Lipschitz continuity). For the de-rarefaction phenomenon, we construct a continuous nonisentropic solution whose initial data contain isentropic rarefactions (i.e., without compression) and a locally stationary varying entropy profile, for which the solution still forms a shock wave in a finite time. [ABSTRACT FROM AUTHOR]

We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are governed by the steady isothermal Euler equations for potential flow with axisymmetry so that the equations contain a singular geometric source term. We first formulate the shock stability problem as an initial boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and structural/asymptotic stability of global entropy solutions of bounded variation (BV) of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via self-similar solutions as building blocks in order to incorporate with the geometric source term. Then we introduce the Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions, as well as the estimates of the center changes of the self-similar solutions. To make sure of the decrease of the Glimm-type functional, we choose appropriate weights by careful asymptotic analysis of the reflection coefficients in the interac-tion estimates, when the Mach number of the incoming flow is sufficiently large. Finally, we establish the existence of global entropy solutions involving a strong leading conical shock-front, besides weak waves, under the conditions that the Mach number of the incoming flow is sufficiently large and the weighted total variation of the slopes of the generating curve of the Lipschitz perturbed cone is sufficiently small. Furthermore, the entropy solution is shown to approach asymptotically the self-similar solution that is determined by the incoming flow and the asymptotic tangent of the cone boundary at infinity. [ABSTRACT FROM AUTHOR]

We are concerned with the suitability of the main models of compressible fluid dynamics for the Lighthill problem for shock diffraction by a convex corned wedge, by studying the regularity of solutions of the problem, which can be formulated as a free boundary problem. In this paper, we prove that there is no regular solution that is subsonic up to the wedge corner for potential flow. This indicates that, if the solution is subsonic at the wedge corner, at least a characteristic discontinuity (vortex sheet or entropy wave) is expected to be generated, which is consistent with the experimental and computational results. Therefore, the potential flow equation is not suitable for the Lighthill problem so that the compressible Euler system must be considered. In order to achieve the nonexistence result, a weak maximum principle for the solution is established, and several other mathematical techniques are developed. The methods and techniques developed here are also useful to the other problems with similar difficulties. [ABSTRACT FROM AUTHOR]

The evolution of discontinuity and formation of triple-shock pattern in solutions to a two-dimensional hyperbolic system of conservation laws are studied. When the initial discontinuity is a convex curve, it is discovered that the structure of the global solution changes dramatically around a critical time: After the critical time, a triple-shock pattern forms, while, before the critical time, only two shocks are developed. The envelope surface of intersections and the evolution of discontinuity are analyzed by developing new ideas and approaches. The global structure of the entropy solution is presented. [ABSTRACT FROM AUTHOR]

The phenomena of concentration and cavitation and the formation of &shocks and vacuum states in solutions to the Euler equations for isentropic fluids arm 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 δ-shock solution to the Euler equations for pressureless fluids, and the intermediate density between the two shocks tends to a weighted 6measure that forms the δ-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, ever, when the initial data stays away from the vacuum. Some numerical results exhibiting the formation process of δ-shocks are also presented. [ABSTRACT FROM AUTHOR]

Presents a study that considered the Cauchy problem for the given nonlinear hyperbolic-parabolic equation. Main issues relating to the Cauchy problem; Definition of entropy solutions; Remarks on the stability.