Measure-valued solutions to the complete Euler system revisited

TL;DR

This paper introduces a new class of measure-valued solutions for the Euler system of fluid dynamics, capturing limits of Navier-Stokes-Fourier solutions and extending to bi-velocity models, with implications for understanding inviscid flows.

Contribution

It proposes a novel concept of dissipative measure-valued solutions based on energy and entropy, unifying limits of viscous models and Euler equations.

Findings

Weak solutions of Navier-Stokes-Fourier generate measure-valued solutions of Euler.

The new solution class includes vanishing dissipation limits.

Application to bi-velocity fluid models extends the framework.

Abstract

We consider the complete Euler system describing the time evolution of a general inviscid compressible fluid. We introduce a new concept of measure-valued solution based on the total energy balance and entropy inequality for the physical entropy without any renormalization. This class of so-called dissipative measure-valued solutions is large enough to include the vanishing dissipation limits of the Navier-Stokes-Fourier system. Our main result states that any sequence of weak solutions to the Navier-Stokes-Fourier system with vanishing viscosity and heat conductivity coefficients generates a dissipative measure-valued solution of the Euler system under some physically grounded constitutive relations. Finally, we discuss the same asymptotic limit for the bi-velocity fluid model introduced by H.Brenner.