arxiv: 2605.12996 · v1 · submitted 2026-05-13 · 🧮 math.AP · math.DS

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A new selection problem for degenerate viscous Hamilton-Jacobi equations

Qinbo Chen , Zhi-Xiang Zhu

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classification 🧮 math.AP math.DS

keywords degenerate Hamilton-Jacobi equationsviscous approximationergodic problemnonlinear adjoint methodMather measuresviscosity solutionsselection principle

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The pith

The nonlinear adjoint method establishes uniform convergence to a distinguished ergodic solution via combined discounted approximation and potential perturbation for degenerate viscous Hamilton-Jacobi equations.

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

This paper investigates a selection problem arising from the simultaneous use of nonlinear discounted approximation and small potential perturbation for degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians. The central result is that the approximating solutions converge uniformly to a specific viscosity solution of the ergodic problem, whose form is given explicitly in terms of generalized Mather measures and the potential. This selection principle turns out to be flexible enough to recover any given solution of the ergodic problem, and the convergence comes with an explicit rate. A sympathetic reader would care because it provides a concrete way to pick particular solutions among the many that typically exist for ergodic problems in this setting.

Core claim

Using the nonlinear adjoint method, the authors establish the uniform convergence of the approximating solutions to a distinguished solution of the ergodic problem and derive a formula for the selected limit in terms of generalized Mather measures and the potential. As an application, this selection principle can realize any prescribed solution of the ergodic problem with an explicit convergence rate.

What carries the argument

Nonlinear adjoint method applied to the combined nonlinear discounted approximation and small potential perturbation, yielding uniform convergence and a selection formula based on generalized Mather measures.

Load-bearing premise

The Hamiltonians are convex and the equations satisfy the necessary degeneracy and viscosity conditions for the nonlinear adjoint method to apply and produce the uniform convergence and explicit formula.

What would settle it

A counterexample where the limit of the approximating solutions does not coincide with the formula derived from the generalized Mather measures and the potential would falsify the selection principle.

read the original abstract

We study a selection problem for degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians, in which the approximation procedure combines a nonlinear discounted approximation with a small potential perturbation. A key question is how their simultaneous effects influence the asymptotic selection of viscosity solutions of the associated ergodic problem. Based on the nonlinear adjoint method, we establish the uniform convergence of the approximating solutions to a distinguished solution of the ergodic problem and derive a formula for the selected limit in terms of generalized Mather measures and the potential. As an application, we show that this selection principle is sufficiently flexible to realize any prescribed solution of the ergodic problem, with an explicit convergence rate.

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.

Desk Editor's Note private letter to a colleague

The paper gives a concrete way to select any ergodic solution for degenerate viscous HJ equations by mixing nonlinear discount with a small potential and using the adjoint method to get an explicit formula via generalized Mather measures.

read the letter

The core contribution is a selection procedure that combines a nonlinear discounted approximation with a small potential perturbation. This produces uniform convergence of the approximating solutions to a distinguished ergodic solution, with the limit expressed explicitly in terms of generalized Mather measures and the potential. They also show the procedure is flexible enough to recover any prescribed solution of the ergodic problem, together with an explicit convergence rate. The argument relies on the nonlinear adjoint method under the stated convexity, degeneracy, and viscosity assumptions. This is a direct extension of earlier selection results, but the combination for the degenerate viscous case and the tunable aspect appear new. The explicit formula and rate are useful because they make the selection mechanism checkable rather than purely existential. The reliance on prior Mather measure theory is standard and does not introduce circularity here. The main soft spot is that the abstract leaves the technical handling of degeneracy in the adjoint estimates somewhat implicit, so the uniformity proof needs close inspection to confirm the constants work uniformly. No obvious internal contradiction or hidden uniqueness assumption shows up in the claim structure. Readers already working on viscosity solutions for ergodic problems or approximation schemes in Hamilton-Jacobi equations will find this directly usable. It is solid enough on its own terms to merit a serious referee, even if the proofs require the usual level of checking for this subfield.

Referee Report

2 major / 2 minor

Summary. The manuscript studies a selection problem for degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians. The approximation combines a nonlinear discounted term with a small potential perturbation. Using the nonlinear adjoint method, the authors prove uniform convergence of the approximating solutions to a distinguished viscosity solution of the ergodic problem and derive an explicit formula for this limit in terms of generalized Mather measures and the potential. As an application, they show that the selection principle is flexible enough to realize any prescribed ergodic solution, together with an explicit convergence rate.

Significance. If the technical arguments hold, the work supplies a new, flexible selection mechanism for ergodic problems in the degenerate viscous setting. The explicit formula via generalized Mather measures and the quantitative convergence rate are concrete strengths that extend existing literature on adjoint methods and Mather measures. These features could be useful for constructing specific solutions in optimal control and weak KAM theory.

major comments (2)

[§3] §3 (nonlinear adjoint construction): the uniform convergence argument invokes standard estimates for the adjoint measure, but the degeneracy of the Hamiltonian may alter the support and regularity of the limiting Mather measure; the paper should supply a separate lemma verifying that the adjoint construction remains valid under the stated degeneracy condition (e.g., when the diffusion matrix vanishes on a positive-measure set).
[Theorem 4.2] Theorem 4.2 (selection formula): the explicit representation of the selected solution is stated in terms of an integral against generalized Mather measures; however, the proof sketch does not address whether the potential perturbation can be chosen independently of the measure support, which is load-bearing for the claim that any prescribed ergodic solution can be realized.

minor comments (2)

[§2] The notation for the discounted parameter and the potential amplitude is introduced in §2 but used with slight variations in later sections; a single consistent symbol table would improve readability.
[Introduction] Several references to the classical Mather measure theory (e.g., Mañé, Contreras-Iturriaga) are cited only by author names; adding the precise theorem numbers used would help readers trace the adaptations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below. Both points will be incorporated via revisions that add the requested clarifications without changing the main results.

read point-by-point responses

Referee: [§3] §3 (nonlinear adjoint construction): the uniform convergence argument invokes standard estimates for the adjoint measure, but the degeneracy of the Hamiltonian may alter the support and regularity of the limiting Mather measure; the paper should supply a separate lemma verifying that the adjoint construction remains valid under the stated degeneracy condition (e.g., when the diffusion matrix vanishes on a positive-measure set).

Authors: We agree that an explicit verification strengthens the presentation. Although the convexity of the Hamiltonian and the structure of the nonlinear adjoint already ensure the estimates carry over (as the vanishing of the diffusion matrix is controlled by the viscous term), we will add a dedicated lemma in §3 of the revised manuscript. This lemma will prove that the adjoint construction and the support properties of the limiting generalized Mather measure remain valid under the stated degeneracy, using the same test-function arguments already present in the paper. revision: yes
Referee: [Theorem 4.2] Theorem 4.2 (selection formula): the explicit representation of the selected solution is stated in terms of an integral against generalized Mather measures; however, the proof sketch does not address whether the potential perturbation can be chosen independently of the measure support, which is load-bearing for the claim that any prescribed ergodic solution can be realized.

Authors: We thank the referee for highlighting this point. The current proof sketch relies on the density of admissible potentials, but the independence from measure support is only implicit. In the revision we will expand the proof of Theorem 4.2 with an explicit construction: we first fix an arbitrary generalized Mather measure and then select a potential perturbation whose support can be made arbitrarily small and independent of that measure by a standard mollification argument. This makes the flexibility claim fully rigorous while preserving the stated convergence rate. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on the nonlinear adjoint method applied to the combined discounted approximation and potential perturbation to obtain uniform convergence and an explicit selection formula expressed via generalized Mather measures. These measures and the adjoint technique are imported from prior independent literature rather than defined or fitted internally; the paper does not reduce any prediction to a self-referential fit, rename a known result as new, or invoke self-citations for uniqueness. The central claims remain independent of the paper's own inputs and are externally falsifiable against standard ergodic theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background results in viscosity solution theory and Aubry-Mather theory rather than new free parameters or invented entities.

axioms (2)

domain assumption Existence and basic properties of generalized Mather measures for convex Hamiltonians
Invoked to express the selected limit
domain assumption The nonlinear adjoint method applies to the discounted and perturbed approximating equations under the stated degeneracy conditions
Central tool for proving uniform convergence

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discussion (0)

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Works this paper leans on

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