Mitsuru Yamazaki

Entropy solutions of the Euler equations for isothermal relativistic fluids

We investigate the initial-value problem for the relativistic Euler equations of isothermal perfect fluids, and generalise an existence result due to LeFloch and Shelukhin for the non-relativistic setting. We establish the existence of globally defined, bounded measurable, entropy solutions with arbitrary large amplitude. An earlier result by Smoller and Temple covered solutions with bounded variation that avoid the vacuum state. Our new framework provides solutions in a larger function space and allows for the mass density to vanish and the velocity field to approach the light speed. The relativistic Euler equations become strongly degenerate in both regimes, as the conservative or the flux variables vanish or blow up. Our proof is based on the method of compensated compactness and takes advantage of a scaling invariance property of the Euler equations.

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.

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.

Existence globale à données d'entropie localement finie pour les modèles discrets de l'équation de Boltzmann avec termes linéaires et quadratiques

Abstract We prove the global existence in time of solution for the discrete Boltzmann equation with linear and quadratic terms in 1 space dimension, provided that the velocities are distinct and that initial data are nonnegative and of locally finite entropy.

Generalized Broadwell models for the discrete Boltzmann equation with linear and quadratic terms

A generalization of the Broadwell models for the discrete Boltzmann equation with linear and quadratic terms is investigated. We prove that there exists a time-global solution to this model in one space-dimension for locally bounded initial data, using a maximum principle of solutions. The boundedness of solutions is established by analyzing the system of ordinary equations related to the linear term.