We consider the complete Euler system describing the time evolution of an inviscid nonisothermal gas. We show that the rarefaction wave solutions of the 1D Riemann problem are stable, in particular unique, in the class of all bounded weak solutions to the associated multiD problem. This may be seen as a counterpart of the non-uniqueness results of physically admissible solutions emanating from ...

We analize the structure of the general solution of the Riemann problem for a striclty hyperbolic system of conservation laws whose characteristic elds are neither genuinely non-linear nor linearly degenerate in the sense of Lax.

where p(t, z), ~(2, t) are the mean density and velocity of the flow, p(z, t) is the pressure. Assume the gas under consideration is polytropic. Then the temperature, density, and pressure are related with RT = p/p where R is a positive constant proportional to the molecular weight of gas and assumed to be unit for simplicity. So p = Tp. Roughly speaking, the pressure p vanishes as T goes to ze...

We consider the semidiscrete upwind scheme (0.1) u(t, x)t + 1 ( f ( u(t, x) ) − f ( u(t, x− ) )) = 0. We prove that if the initial data ū of (0.1) has small total variation, then the solution u (t) has uniformly bounded BV norm, independent of t, . Moreover by studying the equation for a perturbation of (0.1) we prove the Lipschitz continuous dependence of u (t) on the initial data. Using a tec...

In a previous paper (Transp. Porous Media, 55(1): 47–70), algorithms are given for computing the analytical solution to the three-phase Riemann problem. Application of those algorithms requires that the wave configuration is known. The purpose of this note is to provide a procedure to determine the wave structure for any initial and injected saturation states.

We consider the Riemann problem for non-genuinely nonlinear conservation laws where the flux function admits two inflection points. This is a simplification of van der Waals fluid pressure, which can be seen as a function of the specific volume for a specific entropy at which the system lacks the non-genuine nonlinearity. Corresponding to each inflection point, A nonclassical Riemann solver can...

An extension of the Colombo phase transition model is proposed. The congestion phase is described by a two-dimensional zone defined around an equilibrium flux known as the classical fundamental diagram. General criteria to build such a set-valued fundamental diagram are enumerated, and instantiated on several equilibrium fluxes with different concavity properties. The solution of the Riemann pr...

Abstract. In this paper, we investigate a traffic model similar to the Lighthill–Whitham– Richards model, consisting of a hyperbolic conservation law with a discontinuous, non-convex, piecewise-linear flux. Using Dias and Figueira’s mollification framework, analytical solutions to the corresponding Riemann problem are derived. For certain initial data, these Riemann problems can generate zero w...

In this paper, we establish some Hermite-Hadamard type inequalities for function whose n-th derivatives are logarithmically convex by using Riemann-Liouville integral operator.