arxiv: 2605.10478 · v1 · submitted 2026-05-11 · 🧮 math.AP

Recognition: no theorem link

Nonexistence of vanishing-viscosity limits for mechanical Hamiltonian ergodic problems

Hung V. Tran, Yifeng Yu, Ziran Liu

Pith reviewed 2026-05-12 04:39 UTC · model grok-4.3

classification 🧮 math.AP

keywords vanishing viscosityergodic problemHamilton-Jacobi equationcounterexampleone-dimensionalviscous approximationmechanical Hamiltonian

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The pith

A one-dimensional example with C^3 potential shows the vanishing-viscosity limit does not exist for the ergodic problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an explicit one-dimensional counterexample with a C^3 function F on the torus where the solutions to the viscous ergodic equation fail to converge as the viscosity parameter ε tends to zero. This directly answers negatively the question posed by Jauslin, Kreiss, and Moser on whether such limits always exist. A reader would care because vanishing-viscosity approximations are a standard tool for selecting solutions and computing ergodic constants in Hamilton-Jacobi equations arising in control and homogenization. The result demonstrates that C^3 regularity on the potential is not enough to guarantee convergence in even the simplest mechanical case. If the claim holds, then additional conditions on F or the Hamiltonian are needed for the limit to be well-defined.

Core claim

The authors construct a function F belonging to C^3 on the one-dimensional torus such that the normalized solutions φ^ε to the equation 1/2 |Dφ^ε|^2 + F(x) - ε Δφ^ε = c(ε) do not converge as ε approaches 0. This shows that the vanishing-viscosity limit need not exist for mechanical Hamiltonian ergodic problems.

What carries the argument

The family of solutions φ^ε to the viscous ergodic problem on the torus, with a specially chosen C^3 potential F that forces non-convergence as viscosity vanishes.

Load-bearing premise

The specific choice of F in C^3 admits solutions φ^ε for every ε > 0 whose behavior can be analyzed to prove that no limit exists.

What would settle it

Numerical computation of φ^ε on a fine grid for a decreasing sequence of ε values using the paper's explicit F, checking whether the profiles remain bounded and approach one function or instead oscillate without settling.

Figures

Figures reproduced from arXiv: 2605.10478 by Hung V. Tran, Yifeng Yu, Ziran Liu.

read the original abstract

For $\varepsilon>0$, let $\phi^\varepsilon$ be the solution of the ergodic problem \[ \frac12 |D\phi^\varepsilon|^2+F(x)-\varepsilon\Delta\phi^\varepsilon=c(\varepsilon) \qquad \text{on } \mathbb{T}^n, \] normalized by $\phi^\varepsilon(0)=0$. We construct a one-dimensional example with $F\in C^3$ for which the vanishing-viscosity limit $\lim_{\varepsilon\to0}\phi^\varepsilon$ does not exist. This gives a negative answer to a problem proposed by Jauslin, Kreiss, and Moser [10].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs an explicit one-dimensional counterexample: a function F ∈ C³ on the circle such that the normalized solutions φ^ε to the viscous ergodic problem (1/2)|Dφ^ε|² + F(x) − εΔφ^ε = c(ε) on T¹, with φ^ε(0)=0, fail to converge in any reasonable topology as ε→0. This supplies a negative answer to the question posed by Jauslin, Kreiss and Moser.

Significance. The result is significant because it furnishes a concrete, low-dimensional mechanical Hamiltonian for which the vanishing-viscosity limit does not exist, even though existence and uniqueness of φ^ε for each fixed ε>0 are guaranteed by standard viscous Hamilton–Jacobi theory. The construction therefore separates the question of solvability for positive viscosity from the question of passage to the limit, with direct implications for homogenization and weak KAM theory.

major comments (2)

[§3] §3 (construction of F): the explicit C³ function F is chosen so that the effective Hamiltonian produces at least two distinct accumulation points for {φ^ε}; the verification that these accumulation points are indeed distinct relies on a careful asymptotic analysis of the viscous solutions. The argument appears load-bearing for the non-convergence claim and should be checked line-by-line against the ODE reduction available in one dimension.
[Proof of Theorem 1.1] Proof of Theorem 1.1: the passage from the viscous equation to the claimed non-convergence uses the normalization φ^ε(0)=0 together with the specific form of c(ε). It is not immediately clear whether the same F also yields non-convergence in the uniform topology or only in weaker senses; a precise statement of the topology in which the limit fails would strengthen the result.

minor comments (2)

The abstract states F∈C³; it would be useful to record in the introduction whether the construction can be made C^∞ or whether C³ is sharp.
Notation: the dependence of c(ε) on ε is introduced without an explicit formula; a short remark on how c(ε) is determined (e.g., by the solvability condition ∫(½|Dφ|²+F)dx = c(ε)) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the result, and the recommendation for minor revision. We address the two major comments below.

read point-by-point responses

Referee: [§3] §3 (construction of F): the explicit C³ function F is chosen so that the effective Hamiltonian produces at least two distinct accumulation points for {φ^ε}; the verification that these accumulation points are indeed distinct relies on a careful asymptotic analysis of the viscous solutions. The argument appears load-bearing for the non-convergence claim and should be checked line-by-line against the ODE reduction available in one dimension.

Authors: We appreciate the referee drawing attention to the verification step. The construction and asymptotic analysis in Section 3 are performed via the one-dimensional ODE reduction of the viscous equation. We have re-checked these calculations line-by-line and confirm that the two accumulation points remain distinct. To make the argument more transparent, we will add a short remark in the revised Section 3 that explicitly outlines the key steps of the ODE analysis used to distinguish the accumulation points. revision: yes
Referee: [Proof of Theorem 1.1] Proof of Theorem 1.1: the passage from the viscous equation to the claimed non-convergence uses the normalization φ^ε(0)=0 together with the specific form of c(ε). It is not immediately clear whether the same F also yields non-convergence in the uniform topology or only in weaker senses; a precise statement of the topology in which the limit fails would strengthen the result.

Authors: We agree that an explicit statement of the topology strengthens the result. The non-convergence established in the proof of Theorem 1.1 holds in the uniform (C^0) topology: the two accumulation points differ by a fixed positive constant at certain points on the circle, while the normalization φ^ε(0)=0 together with the form of c(ε) is used only to fix the additive constant. We will revise the statement of Theorem 1.1 to make this C^0 non-convergence explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct counterexample construction

full rationale

The paper's central result is an explicit one-dimensional construction of F ∈ C³ on the torus for which the family of normalized viscous solutions φ^ε to the ergodic problem fails to converge as ε → 0. Standard existence theory for the viscous Hamilton–Jacobi equation guarantees solvability for each ε > 0 once the ergodic constant c(ε) is chosen appropriately; the argument then exhibits two distinct limit points for this specific F without fitting parameters, renaming known results, or relying on load-bearing self-citations. The cited problem statement [10] is external and the derivation chain consists of direct analysis of the constructed example rather than any reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a suitable F in C^3 and the analysis of the limit behavior of its solutions.

axioms (1)

domain assumption Existence and uniqueness of solutions to the viscous ergodic problem for each ε>0.
The paper assumes the PDE has solutions φ^ε for the given F.

pith-pipeline@v0.9.0 · 5405 in / 1073 out tokens · 32438 ms · 2026-05-12T04:39:41.924361+00:00 · methodology

discussion (0)

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