Compressible 1d Euler equations with large data: A case study

Consider 1D flow of a compressible, ideal, and polytropic gas on a bounded interval in Lagrangian variables. We study the Cauchy problem when the initial data consist of four constant states that yield two contact waves bounding an interval of lower density, together with an admissible shock between them. To render the solution tractable for direct calculations, we also impose absorbing boundary conditions, at fixed locations (in Lagrangian coordinates) to the left and to the right of the two contacts. By estimating the wave strengths in shock-contact interactions, we show that the resulting flow is defined for all times. In particular, the pressure, density, particle velocities, and shock speeds are all uniformly bounded in time. We also record a scaling invariance of the system and comment on its relevance to large data solutions of the Euler system.