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.

• The paper establishes that the Euler system’s weak solutions are highly nonunique, prompting the need for dissipative measure-valued (DMV) frameworks.
• It rigorously develops selection criteria based on entropy maximization and convexity to construct measurable solution semigroups.
• The study connects theoretical analyses with numerical and statistical approaches, highlighting challenges in capturing turbulence and ensuring computational relevance.

Summary: The Euler System of Gas Dynamics

Introduction and Problematic Well-Posedness

The Euler system of gas dynamics represents the prototype hyperbolic system of conservation laws modeling the evolution of an ideal compressible gas. The paper provides an in-depth survey of the mathematical theory surrounding existence, uniqueness, and admissibility of solutions, with detailed attention to the role of the First and Second Laws of Thermodynamics and the challenges posed by the lack of well-posedness in multidimensional cases.

While the system is well-posed locally in time for smooth data (owing to the foundational work of Schochet), smooth solutions typically develop singularities in finite time, necessitating the extension to weak or measure-valued solutions. The entropy admissibility criterion, intended to restore physical relevance, is proven insufficient for uniqueness in several recent convex integration results. Wild initial data are shown to yield infinitely many entropy admissible weak solutions, and the set of such data is $L^q$-dense, establishing the ill-posed character of the Euler system in the weak solution framework.

Measure-Valued and Dissipative Solution Frameworks

To address non-uniqueness and possible nonexistence, the survey advocates for the dissipative measure-valued (DMV) solutions framework, which interprets physically relevant solutions as limits of consistent approximations (numerical discretizations or vanishing viscosity/dissipation limits of Navier–Stokes–Fourier systems). This framework is underpinned by convexity of the total energy and is able to capture oscillations and concentrations emerging in the solution sequence.

Key properties rigorously established for DMV solutions include:

• Global-in-time existence under minimal constraints on the initial data.
• Weak–strong uniqueness: DMV solutions coincide with strong solutions if these exist.
• Convexity and compactness of the set of DMV solutions for fixed initial data.
• Compatibility with consistent approximations and statistical limits, ensuring physical relevance.

Selection Principles and Admissibility Criteria

Given the excessive nonuniqueness, the paper discusses several maximality or selection principles grounded in the Second Law:

• DiPerna's Entropy Maximization Criterion: Select DMV solutions maximizing the total entropy for all times, inducing an order structure on the DMV solution set.
• Dafermos' Maximal Entropy Production: A local-in-time, "jump" maximality ordering. If a DMV solution is never outpaced in entropy by another that coincides up to some time, it is "Dafermos-maximal."

The survey highlights both positive and negative results. Asymptotic regularity theorems assert that DiPerna-maximal and Dafermos-maximal DMV solutions exhibit vanishing energy defect as $t\to\infty$; if an "absolute entropy maximizer" exists, it is a genuine weak solution. Nonetheless, the existence of maximal solutions in the Dafermos sense is an open problem, especially in the context of the possible existence of infinite ascending chains under the ordering.

Solution Semigroup and Selection Procedures

A principal goal elucidated is the construction of a measurable solution semigroup in the admissible solution class. Two main selection mechanisms are detailed:

1. Successive Minimization (Krylov–Cardona–Kapitanski): A countable family of cost functionals (including entropy and energy) is minimized successively to extract a unique semigroup trajectory, leveraging convexity and compactness of the DMV solution set.
2. Two-Step Convex Selection: First maximize a linear entropy functional, then, if nonunique, minimize a strictly convex energy functional (or equivalently, maximize turbulent energy) on the set of entropy maximizers. This process produces Borel-measurable selections and is theoretically compatible with the DMV framework.

Additionally, the possibility is raised to base selection on a single convex functional (relative entropy), which minimizes the Bregman divergence to the homogeneous equilibrium state, again leading to asymptotic vanishing of turbulent energy.

Statistical and Computable Solutions

The paper articulates the importance of computable (statistically robust) solutions by considering Cesàro averages and statistical limits of consistent approximations, connecting to Young measure and Monte Carlo methodologies. Convexity and closure with respect to the family of computable solutions are highlighted, and the coincidence between computable and DMV solution sets is posed as a significant open question.

Implications, Open Problems, and Perspectives

The analysis clarifies that both practical (numerical/statistical) and theoretical challenges remain:

• Convex integration results force recognition that entropy conditions alone cannot guarantee uniqueness, with nonuniqueness stemming from both oscillatory (turbulent) or connected selection mechanisms.
• The structure and reduction of the DMV solution set, especially targeting computable (approximable) solutions, are open directions.
• The thermodynamical admissibility criteria (absolute entropy maximization, maximal entropy production) provide selection principles that are mathematically rigorous and physically motivated but are still incomplete regarding computational practicability and general existence/uniqueness.

The theoretical implications span both deterministic and statistical (probabilistic/statistical solutions) approaches, bridging measure-theoretic frameworks with numerical approximation and analysis of turbulence.

Conclusion

The survey rigorously synthesizes the current mathematical understanding of the Euler system of gas dynamics, emphasizing the pathological nonuniqueness of weak solutions, the necessity of DMV frameworks, and the central role of thermodynamically motivated admissibility criteria. The construction of solution semigroups via selection mechanisms provides a structured pathway toward regaining well-posedness, but full resolution depends on ongoing investigation into statistical solution limits, closure and convexity properties of computable solutions, and the empirical validity of the selection criteria. The work directly informs the mathematical analysis and numerical simulation of compressible fluid flows and the statistical theory of turbulence, with broad relevance across PDE theory, numerical analysis, and applied mathematics.

Reference:

"The Euler system of gas dynamics" (2603.29619)