Existence of measure-valued solutions to a complete Euler system for a perfect gas (Mathematical Analysis of Viscous Incompressible Fluid)

Abstract

The concept of renormalized dissipative measures-valued (rDMV) solutions to a complete Euler system for a perfect gas was introduced in [8] and further discussed in [9]. Moreover it was shown there that rDMV solutions satisfy the weak (measure-valued)-strong uniqueness principle that makes them a useful tool. In this paper we prove the existerrce of rDMV solutions. Namely, we formulate the complete Euler system in conservative variables usual for numerical analysis and recall the concept of rDMV solutions based on the total energy balance and renormalization of entropy inequality for the physical entropy presented in [8]. We then give two different ways how to generate rDMV solutions. First via vanishing viscosity limit using Navier-Stokes equations coupled with entropy transport and second via the vanishing dissipation limit of the two-velocity model proposed by H. Brenner. Finally, we recall the weak-strong uniqueness principle for rDMV solutions proved in [8] and [9].