Limit of a consistent approximation to the complete compressible Euler System

TL;DR

This paper proves that under minimal assumptions, weak limits of general consistent approximation schemes for the compressible Euler system in 2D and 3D converge strongly locally, even without entropy minimal principles.

Contribution

It establishes strong convergence of approximate solutions to the Euler system from weak limits under broad conditions, including vanishing viscosity and heat conductivity limits.

Findings

Weak limits of approximations are strong solutions locally.

Convergence holds under minimal initial data assumptions.

Includes schemes not satisfying entropy minimal principles.

Abstract

The goal of the present paper is to prove that if a weak limit of a consistent approximation scheme of compressible complete Euler system in the full space $ \mathbb{R}^d,\; d=2,3 $ is a weak solution of the system then eventually the approximate solutions converge strongly in suitable norms locally under a minimal assumption on the initial data of the approximate solutions. The class of consistent approximate solutions is quite general including the vanishing viscosity and heat conductivity limit. In particular, they may not satisfy the minimal principle for entropy.