Weak Entropy-Energy Solution

• Weak entropy-energy solutions are a family of admissibility concepts for nonlinear PDEs that integrate thermodynamic constraints such as entropy production and energy balance, beyond the distributional form of the equations.
• They support rigorous analysis through methods like relative entropy, offering stability results and weak-strong uniqueness while accommodating nonsmooth states and defect measures.
• Applications span conservation laws, phase-field models, compressible fluid mechanics, and energy-reaction-diffusion systems, where energy inequalities and entropy measures ensure physical relevance and facilitate existence proofs.

Searching arXiv for recent and foundational papers on weak entropy-energy solutions and related entropy-based weak formulations. A weak entropy-energy solution is a thermodynamically structured weak solution concept for nonlinear PDEs in which admissibility is determined not only by the distributional form of the equations, but also by entropy production, energy balance, or a relative entropy principle. Across the literature, the expression does not denote a single universal definition. In first-order conservation laws, it is tied to companion entropy or energy laws and to the question whether weak solutions preserve them exactly (Gwiazda et al., 2017). In thermodynamically consistent phase-field, thermo-visco-elastic, and induction-hardening models, it typically means that the pointwise internal energy equation is replaced by a total energy inequality together with a weak entropy inequality or entropy equality (Lasarzik et al., 2019, Cieślak et al., 5 Oct 2025, Hömberg et al., 2020). In compressible fluid mechanics and energy-reaction-diffusion systems, the same idea appears through finite-energy or renormalized weak solutions satisfying relative entropy inequalities, which then support weak-strong uniqueness (Feireisl et al., 2011, Hopf, 2021).

1. General meaning and structural ingredients

In the cited works, weak entropy-energy formulations share a common architecture. First, the governing PDE is interpreted in a weak or renormalized sense, so that nonsmooth states, shocks, concentrations, or merely integrable fluxes can be accommodated. Second, one imposes an additional thermodynamic constraint: this may be an entropy inequality, an entropy equality written for a logarithmic temperature variable, a total energy inequality, a weak energy equation, or a relative entropy inequality against a smooth comparison state. Third, the admissible class is usually restricted by positivity or nonnegativity constraints on temperature, density, concentrations, or phase fractions, because the entropy variable itself often contains logarithms or convex thermodynamic potentials.

This structure appears in several distinct regimes. For conservation laws, the extra law is a companion conservation law, usually representing energy or entropy. For thermodynamically consistent evolution systems, the entropy balance is often more stable under weak convergence than the internal energy equation, so the latter is weakened or replaced. For renormalized formulations, entropy and energy dissipation supply the coercivity missing from the raw weak formulation. This suggests that “weak entropy-energy solution” is best understood as a family of admissibility concepts adapted to the thermodynamic structure of the model rather than as a single canonical definition.

A recurrent distinction is between formal equalities for smooth solutions and weakened inequalities for nonsmooth ones. Classical solutions satisfy the entropy or energy law as an identity. Weak solutions may fail to do so; the failure is then encoded either as dissipation, as a defect measure, or as a one-sided entropy inequality. In applications, that distinction is the mathematical expression of the second law of thermodynamics and of physical admissibility.

2. Conservation laws, companion laws, and the Onsager-type threshold

A general first-order conservation law can be written as

$\operatorname{div}_X(G(U(X)))=0,$

with unknown $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$ and flux $G:\mathcal O\to\mathbb R^{n\times(k+1)}$. A weak solution is defined distributionally by

$\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$

for all smooth compactly supported test functions. A companion law is determined by a map $Q:\mathcal O\to\mathbb R^{1\times(k+1)}$ for which there exists $\mathcal B:\mathcal O\to\mathbb R^{1\times n}$ satisfying

$$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$

For classical solutions this yields

$\operatorname{div}_X(Q(U(X)))=0,$

so $Q$ represents an exactly conserved energy or entropy flux. Weak solutions need not satisfy this additional law, and in applications the equality is often relaxed to an inequality selecting admissible dissipative states. The paper also recalls Godunov’s observation that symmetrizable systems are exactly those endowed with nontrivial companion laws (Gwiazda et al., 2017).

The central regularity result is an abstract Onsager-type criterion. If

$$U\in B^\alpha_{3,\infty}(\mathcal X;\mathcal O), \qquad \alpha>\frac13,$$

and the structural hypotheses on $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$0, $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$1, and $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$2 hold, then a weak solution of $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$3 also satisfies the companion law $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$4 as a weak equality. In the convex-range version one assumes, among other conditions, $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$5, $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$6, bounded second derivatives of $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$7, and cubic growth of $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$8; in the nonconvex variant convexity of $U:\mathcal X\to\mathcal O\subseteq\mathbb R^n$9 is replaced by compactness of the essential range of $G:\mathcal O\to\mathbb R^{n\times(k+1)}$0, which covers examples such as polyconvex elasticity (Gwiazda et al., 2017).

The exponent $G:\mathcal O\to\mathbb R^{n\times(k+1)}$1 comes from the commutator scaling

$G:\mathcal O\to\mathbb R^{n\times(k+1)}$2

Hence the error vanishes precisely when $G:\mathcal O\to\mathbb R^{n\times(k+1)}$3. Above the threshold, the entropy or energy companion law is recovered as an equality; at or below it, dissipation may occur. In this sense, weak entropy-energy admissibility in conservation-law theory is inseparable from regularity: it is not only the weak formulation that matters, but also whether nonlinear commutator defects disappear.

Shock waves provide the canonical hyperbolic example. They satisfy the conservation law distributionally, but not every distributional shock is admissible. The entropy inequality rules out nonphysical shocks. In incompressible Euler, the analogous phenomenon is subtler and is linked to Onsager’s conjecture. The conservation-law viewpoint therefore supplies the prototype for later weak entropy-energy formulations: one begins with a weak solution, then imposes the correct entropy or energy law, and finally asks when equality can still be justified.

3. Replacing the internal energy equation

In several thermodynamically consistent systems, the defining feature of a weak entropy-energy solution is that the pointwise internal energy balance is not retained as the primary weak equation. Instead, one combines a total energy inequality with a weak entropy inequality, or one rewrites the heat equation in entropy variables. This strategy is explicit for a non-isothermal phase-field model, where the system

$G:\mathcal O\to\mathbb R^{n\times(k+1)}$4

is treated on a bounded domain with homogeneous Neumann conditions. The weak solution concept requires, among other regularity properties,

$G:\mathcal O\to\mathbb R^{n\times(k+1)}$5

together with

$G:\mathcal O\to\mathbb R^{n\times(k+1)}$6

The pointwise internal energy balance is replaced by a total energy inequality

$G:\mathcal O\to\mathbb R^{n\times(k+1)}$7

supplemented by a weak entropy inequality for $G:\mathcal O\to\mathbb R^{n\times(k+1)}$8 (Lasarzik et al., 2019).

A more explicit use of the terminology appears in thermo-visco-elasticity with Mróz-type inelastic behavior. The model

$G:\mathcal O\to\mathbb R^{n\times(k+1)}$9

is reformulated via $\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$0, leading formally to

$\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$1

The weak entropy-energy solution is then defined through weak momentum balance, weak constitutive evolution, a weak entropy equation with positive defect measures $\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$2, and a total energy dissipation inequality. The temperature may appear as a measure,

$\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$3

with $\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$4 singular, and the entropy production may only be recoverable as a measure satisfying $\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$5 (Cieślak et al., 5 Oct 2025). This is a particularly strong illustration of why entropy variables can be more robust than the internal energy equation under weak convergence.

An analogous idea governs a model for induction hardening of steel coupling an energy balance, phase evolution, and Maxwell’s equations in potential form. There, the weak entropy solution is defined by an entropy production inequality involving $\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$6, a weak Maxwell equation, the pointwise phase law

$\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$7

and an energy inequality for

$\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$8

The authors emphasize that this formulation accommodates free energies with only $\int_{\mathcal X} G(U(X)) : D_X\psi(X)\, dX = 0$9 regularity, including phase-transition laws of the form

$Q:\mathcal O\to\mathbb R^{1\times(k+1)}$0

which were not naturally covered by earlier weak formulations of the temperature equation (Hömberg et al., 2020).

These examples show that the entropy-energy formulation is not merely a technical restatement. It is a change of state variables and of admissibility mechanism. Entropy controls positivity and dissipation through logarithmic quantities, while the energy inequality preserves global thermodynamic balance. The resulting weak solution concept is typically weaker than a full distributional internal-energy formulation, but stronger than a bare weak formulation of the remaining field equations.

4. Relative entropy, stability, and weak-strong uniqueness

A major development in the theory is the use of relative entropy as a nonlinear stability functional for weak entropy-energy solutions. For the compressible Navier–Stokes system, the pressure potential

$Q:\mathcal O\to\mathbb R^{1\times(k+1)}$1

induces the relative entropy

$Q:\mathcal O\to\mathbb R^{1\times(k+1)}$2

Every finite energy weak solution satisfies a relative entropy inequality with respect to any sufficiently smooth comparison pair $Q:\mathcal O\to\mathbb R^{1\times(k+1)}$3 obeying the boundary conditions. Choosing $Q:\mathcal O\to\mathbb R^{1\times(k+1)}$4 to be a strong solution yields a Grönwall estimate and hence weak-strong uniqueness. In this framework, finite-energy weak solutions are “suitable weak solutions” precisely because they satisfy a stronger comparison principle than the standard energy inequality alone (Feireisl et al., 2011).

For entropy-dissipating reaction-diffusion equations, the natural entropy is

$Q:\mathcal O\to\mathbb R^{1\times(k+1)}$5

and the formal relative entropy

$Q:\mathcal O\to\mathbb R^{1\times(k+1)}$6

is not strong enough to control reaction terms for weak or renormalized solutions when the reaction rates have no growth restriction. The key innovation is therefore the cutoff functional

$Q:\mathcal O\to\mathbb R^{1\times(k+1)}$7

where $Q:\mathcal O\to\mathbb R^{1\times(k+1)}$8 on $Q:\mathcal O\to\mathbb R^{1\times(k+1)}$9 and $\mathcal B:\mathcal O\to\mathbb R^{1\times n}$0 on $\mathcal B:\mathcal O\to\mathbb R^{1\times n}$1. This modified relative entropy leads to

$\mathcal B:\mathcal O\to\mathbb R^{1\times n}$2

and hence weak-strong uniqueness for renormalized solutions (Fischer, 2017).

Energy-reaction-diffusion systems provide a further refinement. There the natural weak class is that of renormalized solutions, but weak-strong uniqueness requires the stronger notion of a dissipative renormalised solution, characterized by an entropy dissipation inequality, a weak energy equation, and a strong energy inequality. Because neither the fluxes nor the reactions are necessarily integrable in the raw form, the classical relative entropy is replaced by a truncated variant

$\mathcal B:\mathcal O\to\mathbb R^{1\times n}$3

combined with an additional quadratic distance in the internal energy variable. The resulting functional controls the deviation from a strong solution and yields uniqueness as long as the strong solution exists (Hopf, 2021).

In the BV theory of hyperbolic conservation laws, strict convexity of the entropy can itself close the uniqueness problem. For a strictly hyperbolic $\mathcal B:\mathcal O\to\mathbb R^{1\times n}$4 system with each characteristic field genuinely nonlinear or linearly degenerate, every entropy weak solution taking values in the semigroup domain coincides with the Lipschitz semigroup trajectory

$\mathcal B:\mathcal O\to\mathbb R^{1\times n}$5

The proof uses relative entropy

$\mathcal B:\mathcal O\to\mathbb R^{1\times n}$6

finite propagation, and local comparison with Riemann and linearized solutions; the older auxiliary assumptions of “Tame Variation” or “Tame Oscillation” are not needed in the presence of a strictly convex entropy (Bressan et al., 2023).

Relative entropy thus plays two roles. It is an admissibility principle, because it encodes the correct thermodynamic convexity. It is also a stability mechanism, because it converts weak information into a quantitative comparison with smooth solutions. In modern usage, many weak entropy-energy solution classes are best understood through this comparison structure rather than through the underlying weak formulation alone.

5. Existence theories and compactness mechanisms

Existence proofs for weak entropy-energy solutions typically rely on entropy estimates that are stronger than the formal mechanical energy bounds. A clear example is Korteweg’s capillary fluid system with

$\mathcal B:\mathcal O\to\mathbb R^{1\times n}$7

for which the effective velocity

$\mathcal B:\mathcal O\to\mathbb R^{1\times n}$8

transforms the mass equation into

$\mathcal B:\mathcal O\to\mathbb R^{1\times n}$9

The resulting entropy estimate controls $$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$0, $$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$1, $$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$2, and dissipation terms such as $$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$3 and $$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$4. This new entropy is stronger than the standard energy estimate and supplies the compactness needed for global weak solutions and for a Prodi–Serrin-type continuation criterion (Haspot, 2011).

For the steady compressible heat-conducting chemically reacting mixture, the relevant weak class is the variational entropy solution. The continuity, momentum, and species equations are kept in weak form, but the total energy balance is not imposed locally; instead one uses an entropy inequality together with the global energy balance corresponding to the test function $$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$5. This relaxation is crucial for compactness. The paper proves existence of a variational entropy solution for any

$$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$6

while a genuine weak solution is obtained under the stronger threshold

$$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$7

The proof relies on improved density estimates, Bogovskiĭ-type testing, the effective viscous flux identity, and strong convergence of the density (Piasecki et al., 2016).

A related large-data existence theory appears for an alternative Navier–Stokes system for compressible viscous ideal gases with diffusion coefficient

$$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$8

The entropy function is

$$D_U Q_j(U)=\mathcal B(U)\, D_U G_j(U), \qquad j=0,\dots,k.$$9

and the entropy balance yields dissipation terms controlling $\operatorname{div}_X(Q(U(X)))=0,$0, $\operatorname{div}_X(Q(U(X)))=0,$1, the velocity gradient, and the radiation term. These estimates imply positivity of density and temperature almost everywhere, bounds on $\operatorname{div}_X(Q(U(X)))=0,$2, improved integrability of $\operatorname{div}_X(Q(U(X)))=0,$3 and $\operatorname{div}_X(Q(U(X)))=0,$4, and compactness sufficient for existence of weak entropy solutions for large initial data. The same structure is mirrored by a finite volume scheme, which converges to a weak entropy solution as the mesh is refined (Svärd, 2022).

A neighboring but distinct tradition is the Kružkov-type entropy theory for degenerate fractional convection-diffusion equations. There the solution concept is a weak entropy solution rather than a weak entropy-energy solution in the thermodynamic sense, and the central tool is an $\operatorname{div}_X(Q(U(X)))=0,$5-contraction principle rather than an energy inequality. This distinction is important: not every entropy solution theory is an entropy-energy theory, even when both serve as admissibility mechanisms (Cifani et al., 2010).

6. Long-time behavior, admissibility, and contemporary extensions

The entropy-energy framework also governs asymptotic behavior. For an electro-energy-reaction-diffusion system arising from a two-level semiconductor model of Shockley–Read–Hall type, the total charge

$\operatorname{div}_X(Q(U(X)))=0,$6

and total energy

$\operatorname{div}_X(Q(U(X)))=0,$7

are conserved, while the entropy is monotone. The relative entropy

$\operatorname{div}_X(Q(U(X)))=0,$8

acts as a Lyapunov functional, and the entropy production $\operatorname{div}_X(Q(U(X)))=0,$9 is nonnegative. Under the structural assumptions $Q$0, $Q$1, and $Q$2, the entropy-entropy production inequality

$Q$3

implies

$Q$4

hence exponential decay of the relative entropy and exponential convergence of the weak solution to equilibrium, supposing that global weak solutions exist (Kniely, 4 Apr 2025).

A contemporary computational extension appears in “Weak and Entropy PINNs.” For a conservation law

$Q$5

the method enforces weak conservation over space-time control volumes through

$Q$6

and entropy admissibility through

$Q$7

The resulting loss combines weak flux balance, entropy-violation penalization, and TVD regularization. For scalar conservation laws, the paper proves an explicit $Q$8 convergence rate toward the entropy solution via the Bouchut–Perthame framework, giving the first explicit $Q$9 convergence rate for a mesh-free control-volume PINN formulation (Oubarka et al., 25 Mar 2026). Although this is not an “entropy-energy solution” theory in the thermodynamic constitutive sense, it shows that weak entropy admissibility has become a computational design principle as well as an analytic one.

Several misconceptions are therefore ruled out by the literature. A weak entropy-energy solution is not merely a weak solution with an a priori bound. It is an admissible state selected by a thermodynamic law. Nor is the term rigidly universal: depending on the model, the decisive extra structure may be a companion conservation law, a weak entropy inequality, an entropy equality in logarithmic variables, a total energy inequality, a weak energy equation, a defect-measure formulation, or a relative entropy inequality. What remains invariant is the role of entropy and energy as selectors of physically meaningful weak dynamics.