Shinbrot Type Criteria for Energy Conservation of the Compressible Navier-Stokes Equations

Abstract: We prove that weak solutions to the compressible Navier-Stokes equations satisfy the energy equality under a Shinbrot-type regularity criterion. Our method applies to the fluids with both constant and degenerate viscosity and relies on a novel weak-type commutator estimate. These criterion are strictly weaker than those required in prior works [Arch. Ration. Mech. Anal., 225 (2017)] and [SIAM J. Math. Anal. 52 (2020)].

• The paper proves energy equality for weak isentropic compressible Navier–Stokes solutions in two and three dimensions when velocity integrability satisfies the Shinbrot-type regions, including 1/p + 3/q ≤ 1 for 3 ≤ q < 4 and 2/p + 2/q ≤ 1 for q ≥ 4.
• It introduces a weak-type temporal commutator estimate that treats the time derivative of density distributionally, avoiding unavailable pointwise integrability of ∂tρ and enabling spatiotemporal mollification at lower regularity.
• The results allow vacuum for constant viscosity with the weaker condition ∇√ρ ∈ L∞L3/2, while the degenerate-viscosity case reaches the endpoint u ∈ L2L∞ under a positive lower density bound; the constant-viscosity endpoint remains open.

Overview and context

This paper, by Chen, Zhang, Zhang, and Zheng (2602.13696), establishes energy equality for weak solutions of the isentropic compressible Navier–Stokes equations under Shinbrot-type integrability conditions on the velocity. The system under consideration is posed on either the torus $\mathbb{T}^N$ or a bounded domain $\Omega \subset \mathbb{R}^N$ ($N=2,3$) with $C^1$ boundary and no-slip condition:

$$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$

with $P = \rho^\gamma$, $\gamma > 1$, viscosities satisfying $\mu > 0$ and $2\mu + N\lambda \ge 0$, and vacuum allowed in the density.

The question of when weak solutions conserve energy originates with Onsager's conjecture for incompressible Euler flows, where Hölder regularity $1/3$ is the threshold between conservation and anomalous dissipation; sufficient Besov conditions above $\Omega \subset \mathbb{R}^N$0 were obtained by Constantin–E–Titi, Eyink, and Cheskidov et al., while non-conserving solutions below $\Omega \subset \mathbb{R}^N$1 were constructed by Isett and Buckmaster et al. For Navier–Stokes, Lions proved conservation for $\Omega \subset \mathbb{R}^N$2, Serrin gave the dimension-dependent condition $\Omega \subset \mathbb{R}^N$3 with $\Omega \subset \mathbb{R}^N$4, and Shinbrot removed the dimensional dependence via $\Omega \subset \mathbb{R}^N$5, $\Omega \subset \mathbb{R}^N$6. The extension to $\Omega \subset \mathbb{R}^N$7 through $\Omega \subset \mathbb{R}^N$8 follows from Sobolev embedding arguments.

For compressible systems, prior results required substantially stronger integrability. Yu [yu2017energy] proved energy equality on $\Omega \subset \mathbb{R}^N$9 assuming $N=2,3$0 with $N=2,3$1, $N=2,3$2, together with $N=2,3$3; Chen et al. [chen2020energy] handled bounded domains with no-slip boundary at the endpoint $N=2,3$4, $N=2,3$5; Ye et al. [ye2022energy] reached an $N=2,3$6 criterion but at the cost of requiring $N=2,3$7 rather than $N=2,3$8. A gap therefore persisted between these conditions and the genuine Shinbrot criteria, and this paper closes it.

Main results

The central theorem states that if a weak solution $N=2,3$9 (renormalized continuity equation, energy inequality, $C^1$0) satisfies

• $C^1$1 and $C^1$2,
• $C^1$3,
• $C^1$4 with $C^1$5 for $C^1$6, and $C^1$7 for $C^1$8,

then the full energy equality holds for every $C^1$9: the difference of initial and current total energies equals the cumulative viscous dissipation $$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$0.

This strictly enlarges the admissible exponent region relative to Yu's condition $$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$1 and Chen et al.'s $$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$2, $$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$3. Notably, the paper relaxes the density regularity to $$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$4 — weaker than the $$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$5 assumption used previously — while retaining compatibility with the Bresch–Desjardins entropy framework, which naturally produces such estimates via the existence theory of Vasseur–Yu [vasseur2016existence]. Vacuum is permitted in the constant-viscosity case.

A second theorem covers the degenerate viscosity system

$$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$6

under a density bounded away from vacuum, $$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$7, initial data $$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$8, and the same Shinbrot exponents — including the endpoint $$(\rho u)_t + \operatorname{div}(\rho u \otimes u) - \mu\Delta u - (\mu+\lambda)\nabla\operatorname{div}u + \nabla P = 0, \qquad \rho_t + \operatorname{div}(\rho u) = 0,$$9, which the constant-viscosity argument does not reach. This endpoint inclusion distinguishes the degenerate result from its constant-viscosity counterpart.

Methodology: spatiotemporal mollification and a weak-type commutator estimate

The principal obstruction to lowering the velocity integrability is the temporal derivative term $P = \rho^\gamma$0, which forces a spatiotemporal — not purely spatial — mollification of the momentum equation. In earlier proofs, the resulting commutator errors were controlled by first deriving $P = \rho^\gamma$1 bounds on $P = \rho^\gamma$2 through the mass equation, which itself requires at least Serrin-level regularity of $P = \rho^\gamma$3. Under Shinbrot-type assumptions such estimates are unavailable, so the previous strategy fails.

The paper's response has two components.

Structured test function and two-step framework. The local energy equality is obtained by testing the momentum equation against $P = \rho^\gamma$4, where superscript/subscript denote spatial/temporal mollifications respectively. This arrangement separates errors by direction: the temporal error is handled by the new commutator lemma, while convection, pressure, and diffusion terms are re-examined individually for convergence under weaker hypotheses. The global result follows by taking $P = \rho^\gamma$5 with a time cut-off $P = \rho^\gamma$6 and a boundary-adapted spatial cut-off $P = \rho^\gamma$7 satisfying $P = \rho^\gamma$8, then letting $P = \rho^\gamma$9; the boundary-layer terms vanish by absolute continuity of the integral, using the Hardy-type inequality $\gamma > 1$0.

Weak-type temporal commutator estimate. The key technical contribution is a distributional version of the classical Lions commutator estimate: for $\gamma > 1$1 with $\gamma > 1$2, $\gamma > 1$3, and test function $\gamma > 1$4,

$\gamma > 1$5

as $\gamma > 1$6, provided all exponents except possibly one are finite. Crucially, convergence is established directly without any $\gamma > 1$7 bound on $\gamma > 1$8; it suffices that $\gamma > 1$9 be a distribution in $\mu > 0$0, which the renormalized mass equation guarantees. The proof integrates $\mu > 0$1 against $\mu > 0$2 over the mollification window and controls the pairing via duality with $\mu > 0$3 and $\mu > 0$4, followed by a density argument in $\mu > 0$5. The authors note this lemma is of independent interest wherever commutators act on test functions with only distributional time derivatives.

With this tool, the error term from $\mu > 0$6 vanishes even though $\mu > 0$7 possesses no pointwise integrability, and the remaining inertial, pressure, and viscous fluxes converge using only $\mu > 0$8 plus the Shinbrot exponents, via interpolation embeddings of the form $\mu > 0$9. Continuity of $2\mu + N\lambda \ge 0$0 at $2\mu + N\lambda \ge 0$1 in $2\mu + N\lambda \ge 0$2 — needed to identify the initial energy — is obtained from compactness (an Aubin–Lions type lemma of Mellet–Vasseur type) applied to $2\mu + N\lambda \ge 0$3, whose time derivative is controlled through the identity $2\mu + N\lambda \ge 0$4.

For the degenerate viscosity system, the same scheme applies after modifying the diffusion limit to handle $2\mu + N\lambda \ge 0$5, following Yu's approach; the lower density bound supplies uniform $2\mu + N\lambda \ge 0$6 control of $2\mu + N\lambda \ge 0$7, which is what enables the $2\mu + N\lambda \ge 0$8 endpoint.

Limitations and open questions

Two restrictions are stated explicitly. First, for the constant-viscosity problem the method does not cover the endpoint $2\mu + N\lambda \ge 0$9, because $1/3$0 is non-separable and the density/mollification arguments rely on strong approximation; whether energy conservation holds at this endpoint remains open. Second, the degenerate-viscosity theorem requires the density to be bounded away from vacuum, so the degenerate case with vacuum is not addressed. Additionally, the constant-viscosity result requires the auxiliary assumption $1/3$1 and $1/3$2; these are consistent with known existence theory but are additional hypotheses beyond the velocity criterion alone.

Conclusion

This paper proves energy equality for weak solutions of both constant- and degenerate-viscosity compressible Navier–Stokes systems under Shinbrot-type velocity criteria, thereby closing the gap left by Yu's $1/3$3 condition and Chen et al.'s $1/3$4, $1/3$5 requirement. The enabling device is a weak-type temporal commutator estimate that requires $1/3$6 only as a $1/3$7-valued distribution, eliminating the need for pointwise-in-time integrability of the density derivative. The main open question left by the analysis is whether the endpoint $1/3$8 can be reached for the constant-viscosity case, and whether the degenerate-viscosity result extends to densities admitting vacuum.