arxiv: 2604.24329 · v1 · submitted 2026-04-27 · 🧮 math.AP

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A PDE formulation of Lyapunov stability for contact-type Hamilton-Jacobi equations

Panrui Ni , Jun Yan

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Pith reviewed 2026-05-08 02:18 UTC · model grok-4.3

classification 🧮 math.AP

keywords Lyapunov stabilityHamilton-Jacobi equationsviscosity solutionscritical valuePDE criteriacontact-type equationsstability analysis

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The pith

Lyapunov stability for contact-type Hamilton-Jacobi equations is characterized by PDE conditions on the Hamiltonian critical value and viscosity subsolutions, for merely continuous convex coercive Hamiltonians.

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

The paper seeks to replace dynamical conditions based on Mather measures with purely PDE criteria to determine Lyapunov stability and instability of stationary solutions to contact-type Hamilton-Jacobi equations on compact manifolds. It shows this works when the Hamiltonian is only continuous, convex, and coercive rather than smooth and Tonelli. A reader would care because the new criteria make stability verifiable directly from PDE quantities and connect it to asymptotic behaviors of viscosity solutions. If the claim holds, stability analysis no longer requires advanced dynamical tools and applies to a larger class of Hamiltonians.

Core claim

We study the Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on a compact manifold. Previous works typically assume C^3 Tonelli Hamiltonians and characterize stability in terms of Mather measures. In this paper, we consider continuous, convex and coercive Hamiltonians and establish verifiable PDE-type criteria for both stability and instability. In particular, the dynamical conditions involving Mather measures are replaced by conditions expressed in terms of the critical value of the Hamiltonian and viscosity subsolutions.

What carries the argument

PDE-type criteria for stability and instability expressed via the critical value of the Hamiltonian and viscosity subsolutions, which replace Mather-measure conditions.

If this is right

Stability and instability become checkable by PDE methods for a broader class of Hamiltonians.
The framework directly links stability properties to asymptotic behaviors of viscosity solutions.
Both positive and negative stability results are obtained from the same PDE quantities.
Stability analysis no longer requires the dynamical machinery of Mather measures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The PDE criteria may allow direct numerical verification of stability in applications by solving auxiliary Hamilton-Jacobi problems.
This replacement of measure-theoretic conditions could extend to related first-order PDE problems where similar critical values appear.
The connection to asymptotic behaviors suggests the criteria might classify long-time limits of solutions in a uniform way.

Load-bearing premise

That Lyapunov stability and instability conditions for these equations can be fully and equivalently captured by the critical value of the Hamiltonian together with properties of viscosity subsolutions, without any reference to Mather measures.

What would settle it

A concrete continuous, convex, coercive Hamiltonian on a compact manifold together with a stationary solution where the PDE criterion predicts stability yet the solution fails to be Lyapunov stable (or the reverse case).

read the original abstract

We study the Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on a compact manifold. Previous works typically assume $C^3$ Tonelli Hamiltonians and characterize stability in terms of Mather measures. In this paper, we consider continuous, convex and coercive Hamiltonians and establish verifiable PDE-type criteria for both stability and instability. In particular, the dynamical conditions involving Mather measures are replaced by conditions expressed in terms of the critical value of the Hamiltonian and viscosity subsolutions. This provides a PDE-based framework for stability analysis and reveals connections with various asymptotic behaviors of viscosity solutions.

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 PDE criteria for Lyapunov stability of stationary solutions to contact Hamilton-Jacobi equations by swapping Mather measures for conditions on the critical value and viscosity subsolutions, under weaker continuous convex coercive assumptions.

read the letter

The core advance here is a shift from dynamical characterizations using Mather measures to purely PDE ones based on the critical value of the Hamiltonian and suitable viscosity subsolutions. This works for continuous, convex, coercive Hamiltonians on compact manifolds rather than the usual C^3 Tonelli class. The abstract and stress-test note indicate the proofs rely on standard viscosity comparison principles and the usual sup-inf definition of the critical value, which hold under the stated regularity. That is a genuine broadening of the setting and a cleaner viewpoint for people who prefer to stay inside viscosity solution theory instead of invoking ergodic measures or Aubry-Mather theory. The instability criteria are presented as part of the same package, which is useful if the goal is to handle both directions without switching frameworks. The connections to asymptotic behavior of solutions are mentioned but not over-claimed in the abstract. On the soft side, the exact statements of the new criteria are not visible in the abstract, so one would want to check in the full text whether the PDE conditions are fully equivalent to the old dynamical ones or only sufficient, and whether any edge cases on the manifold require extra care. The circularity burden looks low since the reformulation avoids self-referential fitting. This is a specialized note aimed at researchers already working on Hamilton-Jacobi equations in optimal control or calculus of variations. It is not a broad breakthrough but it is a clean technical step that deserves referee time to verify the details and see how far the criteria extend. I would send it to peer review rather than desk reject.

Referee Report

0 major / 3 minor

Summary. The paper studies Lyapunov stability of stationary solutions to contact-type Hamilton-Jacobi equations on compact manifolds. For continuous, convex, and coercive Hamiltonians (weaker than the usual C^3 Tonelli assumption), it claims to replace dynamical stability criteria based on Mather measures with verifiable PDE-type conditions involving the critical value of the Hamiltonian and the existence of suitable viscosity subsolutions. The work also connects these criteria to various asymptotic behaviors of viscosity solutions.

Significance. If the claimed criteria are rigorously established, the result offers a PDE-centric framework for stability analysis that avoids explicit dynamical objects like Mather measures. This extends the setting to merely continuous Hamiltonians and may simplify verification in applications, while linking stability to standard viscosity-solution properties and critical-value formulas.

minor comments (3)

The abstract states that 'verifiable PDE-type criteria' are established, but the introduction or §2 should include an explicit statement of the main stability/instability theorems (e.g., Theorem A or B) with the precise conditions on the critical value and subsolutions, to make the claim immediately checkable.
Notation for the critical value (likely denoted c(H) or similar) and the contact-type equation should be fixed early and used consistently; any variation between the dynamical and PDE formulations risks confusion for readers familiar with weak KAM theory.
The manuscript would benefit from a short comparison table or paragraph contrasting the new PDE conditions with the prior Mather-measure criteria, even if only at the level of assumptions and conclusions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary, and recommendation of minor revision. We appreciate the recognition that the PDE-based criteria provide a verifiable alternative to dynamical conditions involving Mather measures, and we will incorporate the suggested minor changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper replaces Mather-measure dynamical criteria with PDE conditions based on the critical value (defined via the standard sup-inf formula) and viscosity subsolutions, using only continuous convex coercive Hamiltonians and standard viscosity comparison principles. No step reduces by construction to a fitted input, self-citation chain, or renamed ansatz; the central stability/instability criteria are derived from independent properties of the Hamiltonian and subsolutions without load-bearing self-references or definitional loops.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard setting of viscosity solutions for Hamilton-Jacobi equations together with the stated regularity assumptions on the Hamiltonian.

axioms (2)

domain assumption Hamiltonians are continuous, convex, and coercive on a compact manifold
Explicitly stated in the abstract as the class of equations considered.
domain assumption Contact-type Hamilton-Jacobi equations admit a well-defined critical value
Invoked when the criteria are expressed in terms of the critical value.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

On the inhomogeneous discounted Hamilton-Jacobi equations
math.AP 2026-05 unverdicted novelty 6.0

For inhomogeneous discounted HJ equations on closed manifolds, viscosity solutions exist and are asymptotically stable precisely when the constant exceeds a critical value c0, with convergence rates determined by inte...

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