Guy Boillat

On the evolution law of weak discontinuities for hyperbolic quasi-linear systems

Abstract We show that the law of propagation of weak discontinuities obtained by A. Jeffrey is in agreement with the Bernoullis law found by other authors.

Simple Waves in N‐Dimensional Propagation

A field is assumed to satisfy a quasilinear system of partial differential equations. Solutions are sought which only depend on a single function of the time and space coordinates.

Relativistic gas: Moment equations and maximum wave velocity

For a rarefied relativistic gas we consider the N-moment equations associated with the relativistic Boltzmann–Chernikov equation and we require the compatibility with the entropy principle thus obtaining a closed symmetric hyperbolic system. This interesting form permits one to deduce a lower and an upper bound for the maximum velocity of a wave propagating in a monoatomic or a degenerate gas of fermions or bosons and to prove that when this number N increases this velocity tends to the speed of light.

Covariant disturbances and exceptional waves

The disturbances of a field are determined for each kind of wave by the hyperbolic field equations. In a space‐time formulation one may consider tensors which also depend on the 4‐normal to the wavefront. It is shown that the disturbance of such a tensor has a definite covariant form only when the wave is exceptional. For the sake of illustration a perfect fluid is considered. It appears that the sound waves are exceptional not only when the fluid is incompressible but also when a special equation of state is used. This relation between pressure and density corresponds in the nonrelativistic limit to a usual approximation for subsonic flow.

On the symmetric conservative form of Landau's superfluid equations

SummaryIt is shown how to obtain Landaus equations for a superfluid in a symmetric conservative form. On account of the constraints involved, this case is more complicated than those of the classical theory developed by Godunov and by other authors.SuntoSi dimostra che è possibile porre in forma simmetrica conservativa il sistema di equazioni di Landau del superfluido. A causa dei vincoli a cui é soggetto il moto del sistema la teoria che si usa risulta piú complessa di quella classica sviluppata da Godunov e da altri autori.

Ray Velocity and Exceptional Waves: A Covariant Formulation

In a nonlinear field an accelerated wave sooner or later turns into a shock. When this is not the case, the wave is exceptional (e.g., Alfven waves of magnetohydrodynamics). Then the normal speed of the wave remains undisturbed. This criterion is given a convenient covariant form in terms of the ray velocity. As an example a thermodynamical relativistic fluid is studied.

On the Born-Infeld electron: Spin effects

We start from a natural generalization of the Born–Infeld Lagrangian which involves two constants h,k and two parameters s,θ0 and show, by investigating static solutions of a first order approximation with respect to a small parameter (e=h/k) that s=1/2 and that h is directly proportional to Planck’s constant. It seems reasonable to interpret s=1/2 as the spin of the electron and the angle θ0 as its orientation. Thus we obtain solutions that appear to reflect the influence of the states of spin on the electromagnetic field.

Energy momentum, wave velocities and characteristic shocks in Euler’s variational equations with application to the Born–Infeld theory

We consider the Euler’s variational equations deriving from a general Lagrangian L(∂αqr,qs). Under the assumption of convexity of energy, we write down some inequalities for the energy-momentum tensor including Hawking–Ellis energy conditions. We show that there exists the same number of positive and negative wave velocities and no velocity can change sign. Finally, we study the structure of the characteristic shocks with particular attention to the generalized Born–Infeld Lagrangian describing the electron with spin.

On a special symmetric form of the hydrodynamic superfluid equations

SummaryA particular case of Landaus superfluid equations is investigated by assuming as viewpoint the theory of the symmetric hyperbolic systems with constraints. Some results on the symmetrization and hyperbolicity are obtained; then they are compared with those of the general case, studied in a previous paper [5].SuntoSi studia un caso particolare delle equazioni di Landau del superfluido dal punto di vista della teoria dei sistemi iperbolici quasi lineari con vincoli. Si ottengono alcuni risultati sulla simmetrizzazione e sulla iperbolicità del sistema di equazioni; tali risultati sono poi confrontati con quelli già ottenuti, nel caso generale, in un lavoro precedente [5].