Variational formulation of hyperbolic conservation laws

Technical Summary: Variational Formulation of Hyperbolic Conservation Laws

Problem Statement
The paper addresses a foundational question posed by K. O. Friedrichs regarding the origin of symmetry in systems of hyperbolic conservation laws. While Friedrichs and Lax established that the existence of a strictly convex entropy function implies that the system can be symmetrized (i.e., the Jacobian matrices can be symmetrized by the Hessian of the entropy), Friedrichs questioned whether this symmetry is an intrinsic property that can be derived from the conservation laws themselves, rather than an assumption imposed on the system. This is particularly relevant for systems with $N \geq 3$ equations, where the symmetry condition constitutes an over-determined system for the entropy Hessian, and entropy functions do not always exist (e.g., "completely non-conservative" systems).

Furthermore, the paper investigates the selection of physically relevant weak solutions. While the entropy inequality ($\eta_t + \nabla \cdot q_\eta \leq 0$) serves as a selection criterion for scalar laws and 1D systems, it fails to guarantee uniqueness in multi-dimensional systems ($N \geq 2$). Dafermos proposed a "Maximum Entropy Production" (MaxEnt) principle as a stronger selection criterion, but counterexamples exist where this principle also fails to yield a unique solution. The paper seeks to reformulate the problem to derive the entropy function and the MaxEnt principle simultaneously from a variational framework.

Methodology
The author introduces a variational formulation based on an action functional $I_\eta(u)$ defined over a space of state functions $u$ and a set of observables $\eta$.

1. Variational Solutions: A pair $(\eta, u)$ is defined as a variational solution if $u$ is a minimizer of the action functional $I_\eta$ among all admissible states. The action functional is given by:
   $$I_\eta(u) = \int_{t_1}^{t_2} \int_\Omega \langle \nabla \eta(u), u_t + \sum_{j=1}^d f_j(u)_{x_j} \rangle_\phi \, dx \, dt$$
   Here, the integrand involves non-conservative products (due to potential discontinuities in $u$), interpreted in the sense of Dal Maso, LeFloch, and Murat (DLM) using a specific path $\phi$.

2. Derivation of Symmetry: The paper considers two types of perturbations:

   • Linear perturbations of the observable ($\eta_c = \eta + \langle c, u \rangle$): Requiring the action to be minimized with respect to these perturbations forces $u$ to satisfy the conservation laws in the weak sense.
   • Local smooth perturbations of the state ($u + \delta u$): Requiring the action to be stationary with respect to these perturbations leads to the condition:
     $$D^2\eta(u) A_j(u) = A_j(u)^\top D^2\eta(u)$$
     This is precisely the symmetrizability condition. Thus, the existence of an entropy function is derived as a necessary condition for the stationarity of the variational principle, rather than being assumed a priori.

3. Derivation of the Entropy Inequality and MaxEnt: By restricting the set of observables to those that satisfy the symmetrizability condition (i.e., entropy functions), the action functional simplifies to the total entropy change plus boundary flux. The minimization property of the variational solution then implies that the solution minimizes the entropy production rate (or maximizes entropy dissipation) relative to local perturbations. This yields a local version of Dafermos' MaxEnt principle directly from the variational formulation.

Key Contributions and Results

• Derivation of Entropy: The paper answers Friedrichs' question by showing that for systems admitting a variational solution, the entropy function is not an external assumption but an extremal object derived from the variational principle. The Hessian of the entropy function is shown to be the symmetrizer of the system's Jacobians.
• Classification of Systems: The formulation distinguishes between "entropic systems" (which admit a non-empty set of observables/entropies and thus variational solutions) and "non-entropic systems" (where the set of observables is empty, and no variational solution exists).
• Local MaxEnt Principle: The paper demonstrates that the variational principle implies a local version of Dafermos' MaxEnt criterion. Specifically, a variational solution minimizes the local entropy production rate among all weak solutions that agree with the solution up to a certain time.
• Limitations on Uniqueness: The paper acknowledges that while the variational formulation provides a rigorous framework for deriving entropy and the MaxEnt principle, it does not automatically resolve the uniqueness problem for multi-dimensional systems. It cites recent work (e.g., De Lellis, Székelyhidi, Chiodaroli, etc.) showing that even the MaxEnt principle fails to select a unique solution for certain initial data in the isentropic Euler equations. The variational formulation is noted to be more restrictive than the local MaxEnt criterion, but the existence of a suitable space of perturbations $W$ that guarantees uniqueness remains an open question.

Significance
The paper claims significance in providing a unified theoretical framework that:

1. Derives Symmetry: It shifts the status of entropy from an imposed physical constraint to a mathematical consequence of a variational principle, addressing Friedrichs' inquiry about the origin of symmetry.
2. Unifies Selection Principles: It connects the concept of weak solutions, the existence of entropy functions, and the MaxEnt selection criterion within a single variational structure.
3. Clarifies the Scope: It explicitly delineates the class of systems for which such a formulation is possible (entropic systems) and highlights the limitations of current selection principles in ensuring uniqueness for multi-dimensional hyperbolic conservation laws.

The work is presented as a step toward a deeper understanding of the mathematical structure of hyperbolic conservation laws, honoring the legacy of Peter Lax and the foundational work of Friedrichs, while acknowledging that the "scientific scandal" of non-uniqueness in multi-dimensional compressible flows remains an open challenge.