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Maximal dissipation and well-posedness of the Euler system of gas dynamics

Eduard Feireisl, Ansgar Jüngel, Mária Lukáčová-Medvid'ová

TL;DR

This work addresses the persistent ill-posedness of the barotropic Euler system by adopting a dissipative, measure-valued framework that includes Reynolds stress and a nonincreasing total energy. The authors prove that any dissipative solution obeying Dafermos' maximal-dissipation criterion is in fact a weak solution, thus connecting maximal dissipation to classical admissibility. They introduce a practical two-step selection procedure to identify a unique semigroup of solutions from the dissipative class, and they define absolute energy minimizers to obtain a local, deterministic selection criterion with a potential for uniqueness. The results provide a rigorous mechanism to recover well-posedness and a deterministic data-to-solution map in a setting where classical weak solutions are not unique, with implications for numerical approximations and the zero-viscosity limit. Overall, the paper advances understanding of selection principles for Euler-type systems and offers concrete criteria for obtaining unique, energy-consistent solutions.

Abstract

We show that any dissipative (measure-valued) solution of the compressible Euler system that complies with Dafermos' criterion of maximal dissipation is necessarily an admissible weak solution. In addition, we propose a simple, at most two step, selection procedure to identify a unique semigroup solution in the class of dissipative solutions to the Euler system. Finally, we introduce a refined version of Dafermos' criterion yielding a unique solution of the problem for any finite energy initial data.

Maximal dissipation and well-posedness of the Euler system of gas dynamics

TL;DR

This work addresses the persistent ill-posedness of the barotropic Euler system by adopting a dissipative, measure-valued framework that includes Reynolds stress and a nonincreasing total energy. The authors prove that any dissipative solution obeying Dafermos' maximal-dissipation criterion is in fact a weak solution, thus connecting maximal dissipation to classical admissibility. They introduce a practical two-step selection procedure to identify a unique semigroup of solutions from the dissipative class, and they define absolute energy minimizers to obtain a local, deterministic selection criterion with a potential for uniqueness. The results provide a rigorous mechanism to recover well-posedness and a deterministic data-to-solution map in a setting where classical weak solutions are not unique, with implications for numerical approximations and the zero-viscosity limit. Overall, the paper advances understanding of selection principles for Euler-type systems and offers concrete criteria for obtaining unique, energy-consistent solutions.

Abstract

We show that any dissipative (measure-valued) solution of the compressible Euler system that complies with Dafermos' criterion of maximal dissipation is necessarily an admissible weak solution. In addition, we propose a simple, at most two step, selection procedure to identify a unique semigroup solution in the class of dissipative solutions to the Euler system. Finally, we introduce a refined version of Dafermos' criterion yielding a unique solution of the problem for any finite energy initial data.
Paper Structure (30 sections, 11 theorems, 147 equations, 1 figure)

This paper contains 30 sections, 11 theorems, 147 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Weak solutions
  3. Dissipative solutions
  4. Comparison with other concepts of generalized solutions
  5. Basic properties of dissipative solutions
  6. Maximal dissipation principle, DiPerna's conjecture, Dafermos' admissibility criterion
  7. Selection criteria
  8. Maximal dissipation principle revisited
  9. Dissipative solutions to the barotropic Euler system
  10. Definition of dissipative solutions
  11. Rate of energy dissipation, maximal dissipative solutions
  12. Properties of dissipative solutions
  13. Global existence
  14. Convexity of the solution set
  15. Topological properties of the solution set
  16. ...and 15 more sections

Key Result

Proposition 2.4

Let $(\varrho^i, {\bf m}^i, \mathcal{E}^i)$ be two admissible solutions emanating from the data $(\varrho^i_0, {\bf m}^i_0, \mathcal{E}^i_0)$, $i=1,2$. Suppose, Then

Figures (1)

  • Figure 1: Relations between the various solution concepts. The black dot denotes a solution set with a unique element. The gray color indicates solution sets with at least one element, while the existence of elements of the white sets is open. We prove that any maximal dissipative solution is in fact an admissible weak solution.

Theorems & Definitions (25)

  • Remark 1.1
  • Definition 2.1: Dissipative solution
  • Remark 2.2
  • Definition 2.3: Admissible dissipative solutions
  • Proposition 2.4
  • proof
  • Definition 2.5: Maximal dissipative solutions
  • Remark 2.6
  • Lemma 2.7: Convexity
  • proof
  • ...and 15 more