[Paper Review] Existence of measure-valued solutions to a complete Euler system for a perfect gas

Jan Březina|arXiv (Cornell University)|May 15, 2018

Navier-Stokes equation solutions21 references3 citations

TL;DR

This paper establishes the existence of renormalized dissipative measure-valued (rDMV) solutions to the complete Euler system for a perfect gas in three dimensions using two distinct vanishing limit procedures: vanishing viscosity from the Navier-Stokes equations with entropy transport, and vanishing dissipation from Brenner's two-velocity model. The key contribution is proving the weak–strong uniqueness principle for rDMV solutions, ensuring they coincide with classical solutions when the latter exist.

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 existence 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].

Motivation & Objective

• To establish the existence of renormalized dissipative measure-valued (rDMV) solutions for the complete Euler system of a perfect gas.
• To address the lack of uniqueness in weak solutions due to oscillations and singularities in compressible Euler flows.
• To provide two distinct vanishing limit constructions—via Navier-Stokes with entropy transport and via Brenner’s two-velocity model—yielding rDMV solutions.
• To prove the weak–strong uniqueness principle for rDMV solutions, ensuring consistency with classical solutions when they exist.
• To extend the theoretical framework of measure-valued solutions by incorporating total energy balance and renormalized entropy inequality.

Proposed method

• Formulate the complete Euler system in conservative variables: mass, momentum, and total energy, with entropy governed by a renormalized inequality.
• Introduce rDMV solutions via the total energy balance and a regularized entropy inequality involving increasing concave functions χ.
• Construct rDMV solutions as limits of solutions to the Navier-Stokes equations with artificial viscosity and entropy transport.
• Construct rDMV solutions as limits of solutions to Brenner’s two-velocity model with vanishing dissipation.
• Use relative energy estimates and coercivity of the relative energy functional to derive the weak–strong uniqueness principle.
• Apply a Gronwall-type argument to the relative energy inequality to show that rDMV solutions must coincide with classical solutions when the latter exist.

Experimental results

Research questions

• RQ1 Can rDMV solutions be constructed for the complete Euler system of a perfect gas using vanishing viscosity limits of the Navier-Stokes equations with entropy transport?
• RQ2 Can rDMV solutions be generated via the vanishing dissipation limit of Brenner’s two-velocity model for a perfect gas?
• RQ3 Does the weak–strong uniqueness principle hold for rDMV solutions in the context of the complete Euler system for a perfect gas?
• RQ4 What is the role of the renormalized entropy inequality in ensuring the stability and uniqueness of rDMV solutions?
• RQ5 How do the total energy balance and renormalized entropy inequality together characterize physically consistent measure-valued solutions?

Key findings

• The existence of rDMV solutions to the complete Euler system for a perfect gas is established via two independent vanishing limit procedures: vanishing viscosity and vanishing dissipation.
• The rDMV solutions satisfy the total energy balance and a renormalized entropy inequality, ensuring thermodynamic consistency.
• The weak–strong uniqueness principle holds: if a classical solution exists, the rDMV solution must coincide with it almost everywhere in space-time.
• The relative energy inequality derived for rDMV solutions allows absorption of error terms via Gronwall’s inequality, leading to strong convergence of the limit solutions.
• The limit of the rDMV solution is represented by Dirac masses, implying strong L¹ convergence of the density, momentum, and total energy to the classical solution.
• The defect measure in the limit vanishes, confirming that the rDMV solution converges strongly to the classical solution in the L¹ norm.