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The Euler system of gas dynamics

Eduard Feireisl

Abstract

This is a survey highlighting several recent results concerning well/ill posedness of the Euler system of gas dynamics. Solutions of the system are identified as limits of consistent approximations generated either by physically more complex problems, notably the Navier- Stokes-Fourier system, or by the approximate schemes in numerical experiments. The role of the fundamental principles encoded in the First and Second law of thermodynamics in identifying a unique physically admissible solution is examined.

The Euler system of gas dynamics

Abstract

This is a survey highlighting several recent results concerning well/ill posedness of the Euler system of gas dynamics. Solutions of the system are identified as limits of consistent approximations generated either by physically more complex problems, notably the Navier- Stokes-Fourier system, or by the approximate schemes in numerical experiments. The role of the fundamental principles encoded in the First and Second law of thermodynamics in identifying a unique physically admissible solution is examined.

Paper Structure

This paper contains 26 sections, 9 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. The Second law of thermodynamics, entropy
  3. State--of--the art
  4. Strong/classical solutions
  5. Weak solutions
  6. Well posedness for the Euler system?
  7. Dissipative measure--valued (DMV) solutions
  8. Consistent approximations
  9. Asymptotic limits of consistent approximations
  10. Dissipative solutions--limits of turbulent consistent approximations
  11. Basic properties of DMV solutions
  12. Dissipative vs. weak solutions
  13. The second law of thermodynamics
  14. DiPerna's maximality criterion
  15. Dafermos' maximality criterion
  16. ...and 11 more sections

Key Result

Theorem 1.1

Suppose that Then there exists $T > 0$ such that the Euler system i1 -- i3, with the boundary condition i4, and the initial data admits a classical solution in $(0,T) \times \Omega$,

Theorems & Definitions (12)

  • Theorem 1.1: Short time existence
  • Theorem 1.2: Ill posedness for Riemann problem
  • Theorem 1.3: Density of wild data
  • Definition 2.1: DMV solution
  • Remark 2.2
  • Theorem 3.1: Asymptotic regularity of maximal solutions
  • Theorem 3.2: DMV solutions with small defect
  • Theorem 3.3: Regularity of maximal DMV solutions
  • Theorem 3.4
  • Theorem 4.1: Vanishing energy defect
  • ...and 2 more