Global existence of solutions by path decomposition for a model of multiphase flow

Abstract

We consider a strictly hyperbolic system of three\ud conservation laws, in one space dimension. The system is a simple\ud model for a fluid flow undergoing liquid-vapor phase transitions.\ud We prove, by a front-tracking algorithm, that weak solutions exist\ud for all times under a condition on the (large) variation\ud of the initial data. An original issue is the control of\ud interactions by means of decompositions of shock waves into paths

The path decomposition technique for systems of hyperbolic conservation laws

“…Links to similar ideas are also given in a sort of short survey. Some recent results obtained by the authors [7,8] are then reported in Sections 3 and 4. In particular, in Section 3 we show with full details this technique when applied to a simple system of three equations arising in phase transition modeling; moreover, a slightly stronger result than that given in [7] is proved.…”

“…The decay property of Nishida solutions obtained in the case γ = 1 still holds for the solutions of system 6 and is interpreted as the pathwise Nishida lemma. This method has been used in subsequent papers, see Asakura-Corli [7,8]. We note that also Temple and Young [25] introduced a (different) notion of path; a short account of their work is found in Section 2.…”

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

“…System (1.1) is strictly hyperbolic in Ω with eigenvalues e 1 = − √ −p v , e 2 = 0, e 3 = √ −p v ; the first and the third characteristic fields are genuinely nonlinear, while the second one is linearly degenerate. The first result on the existence of global solutions to system (1.1), provided with suitably large initial data, is given in [4]; a different proof is given in [7]. In particular, in the case where λ is constant, the classical result by Nishida [16] is recovered.…”

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

“…We also refer to [14] for the extension of Nishida's result to the initial-value problem in Special Relativity. As far as (1.1)-(1.3) is concerned, a positive answer was first given in [3] and then in [5]. In particular, in the former paper an explicit threshold of the BV-norm of the initial data was provided in order to have the global existence of solutions.…”