Résumé
We consider the Euler equations of a perfect fluid having only two independent space-like variables, which account for the stationary 2-dimensional or axisymmetrical 3-dimensional cases as well as the 2-dimensional Riemann problem. We show that the pressure and the angle between an axis and the velocity field satisfy a first-order system which turns out to be elliptic in the subsonic zone. In particular, the pressure satisfies a maximum principle which has not been stated before, to the best of my knowledge. Using this and the Bernouilli law, we give various a priori estimates of the pressure, the density, the enthalpy, and the velocity in the problem of the reflection of a shock wave by a wedge. We also bound the size of the subsonic region and the force that the fluid applies to the boundary.
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Serre, D. Écoulements de fluides parfaits en deux variables indépendantes de type espace. Réflexion d'un choc plan par un diédre compressif. Arch. Rational Mech. Anal. 132, 15–36 (1995). https://doi.org/10.1007/BF00390347
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DOI: https://doi.org/10.1007/BF00390347