Fumioki Asakura

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

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

Wave-front tracking for the equations of isentropic gas dynamics

We study the 2 x 2 system of conservation laws of the form v t - v x = v t + p(v) x = 0, p = k 2 v - γ (γ > 1), which are the model equations of isentropic gas dynamics. Weak global in time solutions are obtained by Nishida-Smoller (CPAM 1973) provided (γ - 1) times the total variation of the initial data is sufficiently small. The aim of this paper is to give an alternative proof by using the Dafermos-Bressan-Risebro wave-front tracking scheme. We obtain new estimates of the total amount of interactions, which also imply the asymptotic decay of the solution. The main idea is to define appropriate amplitude to the path that is a continuation of shock fronts.

Large time stability of propagating phase boundaries

We study the Cauchy problem for a 2 × 2-system of conservation laws: v t − u x = 0, u t − σ(v) t = 0 which describe the phase transition. Two constant states satisfying the Maxwell equal-area principle constitute an admissible stationary solution; a small perturbation of these Maxwell states will be our initial data. We shall show that: there exists a global in time propa gating phase boundary which is admissible in the sense that it satisfies the Abeyaratne-Knowles kinetic condition; the states outside the phase boundary tend to the Maxwell states as time goes to infinity.

Stability of Maxwell states in thermo-elasticity

We study the Cauchy problem for model equations of 1-D thermoelasticity that admit a solid-solid phase transition. The Maxwell states are defined to be two constant states such that the entropy is equal in the both states. A small perturbation of these Maxwell states will be our initial data. In the isothermal model, we shall show that: there exists a global in time admissible phase boundary satisfying the Abeyaratne-Knowles kinetic condition; as time goes to infinity, the strain and the velocity outside the phase boundary tend to the Maxwell states and the entropy tends to a certain limit function. In the polytropic case (the internal energy is proportional to the temperature), we shall show that there exist unique Maxwell states for given temperature and there exist at least two admissible solutions to the Riemann problem with certain initial data in a neighborhood of the Maxwell states.

The reflection of an ionized shock wave

In a previous paper, we studied the thermodynamic and kinetic theory for an ionized gas, in one space dimension; in this paper, we provide an application of those results to the reflection of a shock wave in an electromagnetic shock tube. Under some reasonable limitations, which fully agree with experimental data, we prove that both the incident and the reflected shock waves satisfy the Lax entropy conditions; this result holds even outside genuinely nonlinear regions, which are present in the model. We show that the temperature increases in a significant way behind the incident shock front but the degree of ionization does not undergo a similar growth. On the contrary, the degree of ionization increases substantially behind the reflected shock front. We explain these phenomena by means of the concavity of the Hugoniot loci. Therefore, our results not only fit perfectly but explain what was remarked in experiments.

VISCOUS SHOCK PROFILES FOR 2 × 2 SYSTEMS OF HYPERBOLIC CONSERVATION LAWS WITH AN UMBILIC POINT

This article analyzes the existence of viscous shock profiles joining two states satisfying the Rankine–Hugoniot condition that comes from hyperbolic 2 × 2 systems of conservation laws having quadratic flux functions with an isolated umbilic point: the point where the characteristic speeds coincide and the Jacobian matrix of the flux functions is diagonalizable. The systems studied in this note are particularly in Schaeffer and Shearers cases I and II which are relevant to the three-phase Buckley–Leverett model for oil reservoir flow. It is shown that any compressive and overcompressive shocks have a viscous shock profile provided that there are no undercompressive shock with viscous profile having the same propagation speed. The idea of the proof is a generalization of the first theorem of Morse to noncompact level sets. It is also shown that there exists a shock satisfying the Liu–Oleĭnik condition but having no viscous shock profile. In this case, there is an undercompressive shock with viscous shock profile.

Admissibility of Shock Waves and Uniqueness of the Riemann Problem

We study the admissibility of shock waves and the uniqueness of the Riemann problem for a general 2 × 2 hyperbolic system of conservation laws in one space dimension: U t + F(U) x =0 with initial data having large amplitude. We assume that that the characteristic fields are strictly separated; that is: there exist two disjoint (open) conical neighborhoods such that the first characteristic field is confined to one of these neighborhoods and the second characteristic field to the other. We show that, together with some technical assumptions, the viscous profile exists for states U−, U+ not necessarily close and there exists at most one solution to the Riemann problem. These results will generalize admissibility and uniqueness theorems of Liu and Smoller, by giving descriptions free from particular choice of rectangular coordinates.

Riemann Problem for Conservation Laws with an Umbilic Point

We study the Riemann problems for 2 × 2 conservation laws with a hyperbolic singularity. The flux are a pair of quadratic functions where the char acteristic speeds are equals and the Jacobian matrix is diagonal at the hyperbolic singularity i.e. umbilic point. Discontinuous solutions will be considered. They are characterized by 2 points on the Hugoniot curves which consist of 1-Hugoniot curve, 2-Hugoniot curve and a detached curve. The parts of compressible and overcompressible waves on the wave curves will be determined.