Global existence of solutions for a multi-phase flow: a bubble in a liquid tube and related cases

In this paper we study the problem of the global existence (in time) of weak, entropic solutions to a system of three hyperbolic conservation laws, in one space dimension, for large initial data. The system models the dynamics of phase transitions in an isothermal fluid; in Lagrangian coordinates, the phase interfaces are represented as stationary contact discontinuities. We focus on the persistence of solutions consisting in three bulk phases separated by two interfaces. Under some stability conditions on the phase configuration and by a suitable front tracking algorithm we show that, if the BV-norm of the initial data is less than an explicit (large) threshold, then the Cauchy problem has global solutions

Global existence of solutions for a multi-phase flow: a drop in a gas-tube

In this paper we study the flow of an inviscid fluid composed by three different phases. The model is a simple hyperbolic system of three conservation laws, in Lagrangian coordinates, where the phase interfaces are stationary. Our main result concerns the global existence of weak entropic solutions to the initial-value problem for large initial data

Global weak solutions for a model of two-phase flow with a single interface

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

Instability of travelling wave profiles for the Lax-Friedrichs scheme

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Uniqueness of Classical and Nonclassical Solutions for Nonlinear Hyperbolic Systems

AbstractIn this paper we establish a general uniqueness theorem for nonlinear hyperbolic systems of partial differential equations in one-space dimension. First of all we introduce a new notion of admissible solutions based on prescribed sets of admissible discontinuitiesΦ and admissible speedsψ. Our definition unifies in a single framework the various notions of entropy solutions known for hyperbolic systems of conservation laws, as well as for systems in nonconservative form. For instance, it covers the nonclassical (undercompressive) shock waves generated by a vanishing diffusion-dispersion regularization and characterized via a kinetic relation. It also covers Dal Maso, LeFloch, and Murat's definition of weak solutions of nonconservative systems. Under certain natural assumptions on the prescribed sets Φ and ψ and assuming the existence of a L1-continuous semi-group of admissible solutions, we prove that, for each Cauchy datum at t=0, there exists at most one admissible solution to the Cauchy problem depending L1-continuously upon the initial data. In particular, our result shows the uniqueness of the L1-continuous semi-group of admissible solutions. In short, this paper proves that supplementing a hyperbolic system with the “dynamics” of elementary discontinuities characterizes at most one L1-continuous and admissible solution

Stability of L∞L^\infty solutions for hyperbolic systems with coinciding shocks and rarefactions

We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction curves and the existence of a set of Riemann coordinates $w$, we prove that there exists a semigroup of solutions $u(t) = \mathcal{S}_t u_0$, defined on initial data $u_0 \in L^\infty$. The semigroup $\mathcal{S}$ is continuous w.r.t. time and the initial data $u_0$ in the $L^1_{\text{loc}}$ topology. Moreover $\mathcal{S}$ is unique and its trajectories are obtained as limits of wave front tracking approximations.Comment: 19 pages, 13 figure