Oleinik Entropy for the Leroux System

Definitions and mathematical statement

Problem. Establish the Oleinik entropy condition for the Riemann invariants arising in the hydrodynamic limit of the Leroux system. The supplied assessment identifies the setting as a stochastic microscopic model with two conserved components; it does not supply its generator, flux normalization, scaling, initial data, boundary conditions, or the precise Oleinik inequality. Stochastic microscopic dynamics must not be replaced without justification by a macroscopically noise-driven PDE.

In a conditional notation for the limiting conservation law, write $\partial_tU+\partial_xF(U)=0$, with $U=(U_1,U_2)$, and let $w_1,w_2$ be the specified Riemann invariants. In a smooth strictly hyperbolic region, their gradients can be normalized as left characteristic eigenvectors, so $\nabla w_i\,DF=\lambda_i\nabla w_i$, and smooth solutions obey $\partial_tw_i(U)+\lambda_i(U)\partial_xw_i(U)=0$. The requested result concerns the appropriate one-sided Oleinik admissibility control for these invariants in the weak hydrodynamic limit, with the source's signs, constants, and sense of inequality. Those details are absent, so no particular scalar inequality or characteristic normalization is silently assigned to this system.

For comparison of terminology, a convex entropy pair $(\eta,q)$ satisfies $\nabla q=\nabla\eta\,DF$, and its distributional entropy inequality is $\partial_t\eta(U)+\partial_xq(U)\le0$. Merely writing this generic entropy inequality is not a verification of the requested Oleinik condition, nor does a deterministic uniqueness theorem alone prove that microscopic limit points satisfy that condition. The model-specific limiting objects, admissibility assumptions, and exact one-sided estimate must be recovered to obtain a fully quantified theorem.

The supplied literature triage checked 2026-08-21 labels the problem open: the Oleinik-type verification is described as a missing step for the stochastic two-component Leroux setting, while deterministic-system Oleinik uniqueness theory does not resolve that microscopic problem. This dated assessment requires independent verification.

Short English statement

Prove that the Riemann invariants of the hydrodynamic limit satisfy the intended one-sided Oleinik entropy condition. The exact microscopic model and inequality are missing; deterministic uniqueness results alone do not supply the required stochastic-limit verification.

Sources

• Local UnsolvedMath registry, permanent ID 30000127, code catalogue entry; Oleinik Entropy for the Leroux System, Oberwolfach Reports Open Problems.
• Large Scale Stochastic Dynamics (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/43, pp. 2229--2310. https://doi.org/10.4171/owr/2004/43. Proposed year and year tag: 2004; no proposer named.
• Supplied literature label Original OWR report: https://doi.org/10.4171/owr/2004/43.
• Supplied literature label Bressan--Goatin, Oleinik estimates and uniqueness: https://www.math.ntnu.no/conservation/1997/035.html.
• Supplied literature label Related hydrodynamic renormalization literature: https://arxiv.org/abs/math/0610645.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field reports source inspection, not independently checked here.