On a model of multiphase flow

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.Comment: 32 pages. Revised and corrected versio

Unconditional flocking for weak solutions to self-organized systems of Euler-type with all-to-all interaction kernel

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish that the global entropy weak solutions, constructed in [2] to the Cauchy problem for any $BV$ initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein, admit unconditional time-asymptotic flocking without any further assumptions on the initial data. In addition, we show that the convergence to a flocking profile occurs exponentially fast.Comment: 31 pages, 2 figure

Front tracking approximations for slow erosion

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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

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

An Integro-Differential Conservation Law arising in a Model of Granular Flow

We study a scalar integro-differential conservation law. The equation was first derived in [2] as the slow erosion limit of granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for which one can not adapt the standard theory of conservation laws. We construct approximate solutions with a fractional step method, by recomputing the integral term at each time step. A-priori L^\infty bounds and BV estimates yield convergence and global existence of BV solutions. Furthermore, we present a well-posedness analysis, showing that the solutions are stable in L^1 with respect to the initial data

Stringent error estimates for one-dimensional, space-dependent 2×2 relaxation systems

International audienceSharp and local $L^1$ {\it a-posteriori} error estimates are established for so--called "well-balanced" $BV$ (hence possibly discontinuous) numerical approximations of $2 \times 2$ space-dependent Jin-Xin relaxation systems under sub-characteristic condition. According to the strength of the relaxation process, one can distinguish between two complementary regimes: 1/ a weak relaxation, where local $L^1$ errors are shown to be of first order in \DX and uniform in time, 2/ a strong one, where numerical solutions are kept close to entropy solutions of the reduced scalar conservation law, and for which Kuznetsov's theory indicates a behavior of the $L^1$ error in t\cdot \sqrt{\DX}. The uniformly first-order accuracy in weak relaxation regime is obtained by carefully studying interaction patterns and building up a seemingly original variant of Bressan-Liu-Yang's functional, able to handle $BV$ solutions of arbitrary size for these particular inhomogeneous systems. The complementary estimate in strong relaxation regime is proven by means of a suitable extension of methods based on entropy dissipation for space-dependent problem