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We consider a hyperbolic system of three conservation laws in one space variable. The system is a model for fluid flow allowing phase transitions; in this case the state variables are the specific volume, the velocity, and the mass density fraction of the vapor in the fluid. For a class of initial data having large total variation we prove the global existence of solutions to the Cauchy problem.

The system of conservation lawsWe prove two global existence theorems for (1) -(2) for two distinct types of boundary conditions, with data of small total variation.

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This paper deals with a coupled system consisting of a scalar conservation law and an eikonal equation, called the Hughes model. Introduced in [24], this model attempts to describe the motion of pedestrians in a densely crowded region, in which they are seen as a 'thinking' (continuum) fluid. The main mathematical difficulty is the discontinuous gradient of the solution to the eikonal equation appearing in the flux of the conservation law. On a one dimensional interval with zero Dirichlet conditions (the two edges of the interval are interpreted as 'targets'), the model can be decoupled in a way to consider two classical conservation laws on two sub-domains separated by a turning point at which the pedestrians change their direction. We shall consider solutions with a possible jump discontinuity around the turning point. For simplicity, we shall assume they are locally constant on both sides of the discontinuity. We provide a detailed description of the local-in-time behavior of the solution in terms of a 'global' qualitative property of the pedestrian density (that we call 'relative evacuation rate'), which can be interpreted as the attitude of the pedestrians to direct towards the left or the right target. We complement our result with explicitly computable examples.

We consider the Cauchy problem for the (strictly hyperbolic, genuinely nonlinear) system of conservation laws with relaxation Assume there exists an equilibrium curve A(u), such that r(u,A(u)) = 0. Under some assumptions on σ and r, we prove the existence of global (in time) solutions of bounded variation, uε, υε, for ε > 0 fixed.As ε → 0, we prove the convergence of a subsequence of uε, υε to some u, υ that satisfy the equilibrium equations

We consider a simple nonlinear hyperbolic system modeling the flow of an inviscid fluid. The model includes as state variable the mass density fraction of the vapor in the fluid and then phase transitions can be taken into consideration; moreover, phase interfaces are contact discontinuities for the system. We focus on the special case of initial data consisting of two different phases separated by an interface. We find explicit bounds on the (possibly large) initial data in order that weak entropic solutions exist for all times. The proof exploits a carefully tailored version of the front tracking scheme.