The existence and stability of viscosity solutions to perturbed contact Hamilton-Jacobi equations

Huan Wu; Shiqing Zhang

REVIEW 1 major objections 4 minor 28 references

The existence and stability of viscosity solutions to perturbed contact Hamilton-Jacobi equations

T0 review · 1 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Small perturbations of contact Hamilton–Jacobi equations preserve stable viscosity solutions.

desk verdict Theorem 1.1 is basically right; Theorem 1.2 has a wrong-semigroup typo that is load-bearing but fixable. read the letter →

arxiv 2501.04998 v2 pith:4TNV6QCJ submitted 2025-01-09 math.AP

Huan Wu , Shiqing Zhang This is my paper · ORCID

classification math.AP MSC 35F2135D4037J51

keywords contactHamilton-JacobiequationsviscositysolutionsLyapunovstabilityperturbationtheorybackwardsemigroupasymptotic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that a Lyapunov asymptotically stable viscosity solution of a contact Hamilton–Jacobi equation persists under small smooth perturbations. If u_- solves H(x,D_xu,u)=0 and is locally Lyapunov asymptotically stable, then for every tolerance δ>0 there is an ε_δ>0 such that for all ε∈[0,ε_δ] the perturbed equation H(x,D_xu,u)+εP(x,D_xu,u)=0 has a viscosity solution u^ε_- within δ of u_-. The proof identifies viscosity solutions with fixed points of a backward evolution semigroup and shows the perturbed semigroup has a fixed point near u_-. It then shows that when ∂H/∂u>0 along the lifted solution set, this nearby solution is unique and itself locally Lyapunov asymptotically stable. The significance is that dissipative contact equations, which are not covered by classical conservative Hamilton-Jacobi theory, are shown to be structurally stable around stable stationary states.

What carries the argument

The central mechanism is the backward solution semigroup {T^ε_t}_{t≥0} acting on C(M,ℝ), whose value T^ε_t φ(x) is the unique viscosity solution of the evolutionary equation ∂_t u + H(x,D_xu,u)+εP(x,D_xu,u)=0 with initial data φ. A theorem from the variational theory of contact Hamiltonians says that a function is a stationary viscosity solution of the time-independent equation exactly when it is a fixed point of every T^ε_t. The paper compares the unperturbed and perturbed semigroups through the bound ‖T^ε_t φ - T^-_t φ‖_∞ ≤ (ε/λ)($e^{{λt}}$-1), obtained from the Legendre transforms and the Lipschitz-in-u property. Lyapunov asymptotic stability of u_- is then used to trap the iterates T^ε_t u_- in a shrinking interval, and a liminf construction produces a fixed point, hence a viscosity solution, uniformly close to u_-. The strict sign condition ∂H/∂u>0 on Λ_{u_-} transfers to the perturbed solution by a compactness and Arzelà-Ascoli argument, yielding uniqueness and asymptotic stability.

What would settle it

Fix a simple contact Hamiltonian on the circle with a known locally Lyapunov asymptotically stable viscosity solution u_- and a compactly supported P with |P|≤1; if direct computation shows the perturbed equation H+εP=0 has no viscosity solution within δ of u_- for arbitrarily small ε, Theorem 1.1 fails, and if two distinct such solutions coexist for arbitrarily small ε while ∂H/∂u>0 on Λ_{u_-}, Theorem 1.2 fails.

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Extended reading notes

Core claim

On a closed Riemannian manifold, for a $C^{3}$ contact Hamiltonian H(x,p,u) satisfying positive definiteness, superlinearity, and uniform Lipschitz dependence on u, the authors establish the following. If u_-∈S^- is a viscosity solution of H(x,D_xu,u)=0 that is locally Lyapunov asymptotically stable with respect to the backward semigroup {T^-_t}, then for any δ>0 there exists ε_δ>0 such that for every ε∈[0,ε_δ] the perturbed equation admits a viscosity solution u^ε_- with ‖u^ε_- - u_-‖_∞≤δ. If in addition ∂H/∂u>0 on Λ_{u_-}, the closure of the set of points (x,D_xu_-(x),u_-(x)) at differentiability points of u_-, then for δ,ε small the solution is unique in the δ-neighbourhood of u_- and is itself locally Lyapunov asymptotically stable.

Load-bearing premise

The load-bearing premise is that viscosity solutions of the contact Hamilton-Jacobi equation are exactly the fixed points of the backward solution semigroup {T^-_t}; if that equivalence fails for contact Hamiltonians satisfying (H1)-(H3), the construction of a fixed point of {T^ε_t} would not produce a viscosity solution of the perturbed equation.

Editorial extensions

If this is right

For every small ε, the perturbed Hamiltonian H+εP is admissible at zero: equation (HJε) admits a viscosity solution, not merely for specially chosen perturbations P.
The solution u^ε_- converges uniformly to u_- as ε→0, so stable stationary states of the unperturbed dissipative system have nearby stationary states after perturbation.
Under the strict monotonicity condition ∂H/∂u>0, the nearby stationary state is the unique viscosity solution within the specified neighbourhood, so the perturbation does not create alternative solutions there.
The perturbed stationary state inherits local Lyapunov asymptotic stability, so the qualitative dynamics near u_- is preserved for small perturbations.
The explicit choice ε_δ := λ min{δ,δ_0}/(2(e^{λ t_δ}-1)) makes the allowed perturbation size depend on the unperturbed contraction time t_δ and the Lipschitz constant λ.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural testable extension, not pursued in the paper, is to allow unbounded C^3 perturbations P with bounded u-derivative; the comparison estimate behind Proposition 2.10 may still hold with modified constants, so the persistence theorem could extend beyond compact support.
Because the proof constructs u^ε_- as a liminf of the semigroup trajectory, one could ask whether the limsup construction gives a different fixed point when uniqueness fails, and whether the interval of viscosity solutions shrinks to a point as ε→0.
The result suggests a selection principle: among several unperturbed stable solutions, a small perturbation keeps a nearby solution near each one, leaving open the global question of which stable branch is selected as ε varies.
In applications to dissipative contact Hamiltonians, such as thermodynamic models, the theorem means small parameter changes do not destroy stationary regimes that are already asymptotically stable; this is a structural-stability statement in the C^0 topology of solutions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper proves that a locally Lyapunov asymptotically stable viscosity solution of a contact Hamilton–Jacobi equation persists under small perturbations; the perturbed equation has a nearby viscosity solution, and under a strict monotonicity condition that solution is locally unique and asymptotically stable. That is a legitimate, useful extension of the Wang–Yan–Xu–Zhao framework to parametric perturbations, and it is not in the cited literature. The main existence argument is sound in strategy: compare the perturbed and unperturbed semigroups, control the difference explicitly (Proposition 2.10 is a clean estimate), construct a fixed point by an inf–limiting procedure, and identify it as a viscosity solution. No circularity; the paper relies on external theorems from [21], [24], and [27], not on its own claims.

The soft spots are real but mostly cosmetic. In Theorem 1.1, the sub/supersolutions are chosen at distance min{δ,δ0} from u−; when δ ≥ δ0 that puts them on the boundary of the Lyapunov basin, not strictly inside it, so the claimed convergence of T^-_t uδ is not exactly justified. That is a minor boundary issue, fixed by taking min{δ,δ0/2}. More serious: the final paragraph of Theorem 1.2 uses the unperturbed semigroup T^-_t to conclude convergence to u^ε_−, the perturbed solution. That is the wrong semigroup. The correct step is to apply Lemma 4.1 to the perturbed Hamiltonian H^ε, using Lemma 4.2’s conclusion that ∂H^ε/∂u > 0 on Λ_{u^ε_−}; then T^ε_t φ → u^ε_−, which is exactly what proves both local asymptotic stability and uniqueness among nearby stationary solutions. As written, the stability and uniqueness conclusions of Theorem 1.2 are not proved. I agree with the stress-test note that this looks like a correctable typo rather than a broken strategy, but it must be fixed before the theorem is accepted.

The paper is for researchers in weak KAM theory and contact Hamiltonian dynamics. It deserves a serious referee: the problem is meaningful, the main existence theorem is plausible and carefully argued, and the gap in Theorem 1.2 is localized and repairable. I would send it to review and ask for a revised version that corrects the semigroup in Theorem 1.2 and tightens the δ0 boundary point in Theorem 1.1.

Referee Report

1 major / 4 minor

Summary. The paper studies contact Hamilton-Jacobi equations of the form H(x,Du,u)=0 on a closed manifold and their small perturbations H(x,Du,u)+εP(x,Du,u)=0. The main results are Theorem 1.1, which asserts that a locally Lyapunov asymptotically stable viscosity solution u_- of the unperturbed equation persists as a viscosity solution u^ε_- of the perturbed equation with ||u^ε_- - u_-|| ≤ δ for any prescribed δ>0 once ε is sufficiently small, and Theorem 1.2, which adds that if ∂H/∂u>0 on the set Λ_{u_-}, then the nearby perturbed solution is unique and locally asymptotically stable. The proofs rely on weak KAM theory for contact Hamiltonians, the identification of viscosity solutions with fixed points of the backward semigroup, and a new comparison estimate between the perturbed and unperturbed semigroups.

Significance. If the results are correct, they provide a natural robustness theorem for weakly stable solutions of contact Hamilton-Jacobi equations under perturbations, a question that has not been systematically treated before. The main novel technical ingredient is Proposition 2.10, the explicit estimate |T^ε_t φ - T^-_t φ| ≤ (ε/λ)(e^{λt}-1), which is clearly useful beyond this paper. The authors also give a careful compactness argument in Lemma 4.2 to transfer positivity of ∂H/∂u from Λ_{u_-} to Λ_{u^ε_-}. However, the paper as written contains two gaps in the proofs of the main theorems, one in the handling of the stability basin in Theorem 1.1 and one in the use of the unperturbed semigroup in Theorem 1.2; both are local and appear correctable, but they currently leave the central claims not fully proved.

major comments (1)

[Section 4, proof of Theorem 1.2, final paragraph] The final paragraph asserts that Lemma 4.2 provides a δ′0 > 0 such that for every φ near u^ε_- one has lim_{t→∞} T^-_t φ = u^ε_-, and then concludes that u^ε_- is locally asymptotically stable and unique for (HJε). This is not correct as written. First, Lemma 4.2 does not state a basin-of-attraction property for u^ε_-; it only states positivity of ∂H^ε/∂u on Λ_{u^ε_-}. Second, the stability notion for a solution of (HJε) uses the perturbed semigroup T^ε_t, not T^-_t, and convergence under T^-_t does not imply convergence under T^ε_t. The intended step is to apply the perturbed analogue of Lemma 4.1 to the pair (H^ε, u^ε_-), using the positivity from Lemma 4.2; this would give lim_{t→∞} T^ε_t φ = u^ε_- and then uniqueness among nearby fixed points of T^ε_t. This load-bearing step is missing, so Theorem 1.2 is not proved as written.

minor comments (4)

[Section 4, proof of Lemma 4.2, Step 2] The text says 'By Proposition 2.8, we can find (u^{ε_n}_-, L, 0)-calibrated curve' but this should be '(u^{ε_n}_-, L^{ε_n}, 0)-calibrated curve', since the relevant Lagrangian for the perturbed equation is L^{ε_n}.
[Section 3, proof of Theorem 1.1] The step 'u^ε_-(x) = lim_{t→∞} inf_{s≥t} T^ε_s u_-(x) uniformly' is stated as easy to prove; the authors should spell out that this follows from the equi-Lipschitz estimate established just above together with Dini's theorem, since the infimum over s≥t is monotone in t.
[Section 2.2, after definition of H^ε] The authors state that there is a θ such that H^ε satisfies (H1)-(H3) for ε∈[0,θ]; this should be justified by noting that positive definiteness in p is uniform on the compact support of P, rather than merely pointwise.
[Title and Abstract] There are several English and typographical errors, for example 'pertu rbed' in the title and 'does exist viscosity solution' in the abstract; the manuscript should be carefully proofread.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation chain rests on external weak KAM results and independent comparison estimates, with no input equivalent to the conclusions.

full rationale

The paper's claims do not reduce to their hypotheses by construction. Theorem 1.1 is proved by comparing the perturbed and unperturbed backward semigroups via Proposition 2.10 and then constructing a fixed point of T^ε_t as a uniform liminf limit; the characterization of viscosity solutions as fixed points of the semigroup is imported from Proposition 2.6, citing the external work [24] by Wang, Wang and Yan, and is extended to H^ε in Section 2.2 using the observation that H^ε satisfies (H1)-(H3) for small ε. This is independent external support, not a self-citation chain. Theorem 1.2 relies on Lemma 4.1, cited from the external paper [27] by Xu, Yan and Zhao, and on Lemma 4.2, which is proved in the present paper through a compactness argument; the condition ∂H/∂u>0 on Λ_{u_-} is not treated as equivalent to stability by definition. No fitted parameter is later renamed as a prediction, no uniqueness theorem is imported from the present authors' own prior work, and no ansatz is smuggled in through citation. The final paragraph of the proof of Theorem 1.2 writes lim_{t→∞} T^-_t φ = u^ε_- where the intended perturbed statement would use T^ε_t; this appears to be a typographical or notational gap and is a correctness concern rather than a circular reduction. Overall, the central derivation is self-contained relative to its stated external hypotheses, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no invented entities. It relies on the standard weak KAM theory for contact Hamiltonians, the fixed-point characterization of viscosity solutions, and the stability criterion of Xu-Yan-Zhao. The central contribution is a perturbation argument built on these existing tools.

assumptions (8)

domain assumption H satisfies (H1)-(H3): positive definiteness, superlinearity, and uniform Lipschitz in u.
These standard Tonelli-type hypotheses are stated in Section 1.2 and ensure existence of the semigroup and weak KAM solutions.
domain assumption Admissibility: the set of constants c for which H(x, Du, u)=c has a viscosity solution contains 0.
Stated in Section 1.1; it guarantees the unperturbed equation (HJs) has a viscosity solution.
domain assumption P ∈ C^3_0(T*M × R, R) with |P|≤1.
Assumed in Theorems 1.1 and 1.2; used to ensure H^ε satisfies (H1)-(H3) for small ε.
domain assumption u_- is locally Lyapunov asymptotically stable.
Central hypothesis of Theorem 1.1; it provides the attracting tube around u_- used in the sandwich argument.
domain assumption ∂H/∂u > 0 on Λ_{u_-}.
Hypothesis of Theorem 1.2; via Lemma 4.1 ([27]) it yields local asymptotic stability of u_-.
domain assumption Viscosity solutions coincide with fixed points of the backward semigroup (Proposition 2.6, from [24]).
This external theorem is the backbone of the existence proof: finding a fixed point of T^ε_t gives a viscosity solution of (HJε).
domain assumption The weak KAM representation formula and semigroup properties of {T^-_t} (Propositions 2.3 and 2.4, from [23]).
Used throughout to compare perturbed and unperturbed semigroups and to build the fixed point.
domain assumption Lemma 4.1 from [27]: if ∂H/∂u > 0 on Λ_{u_-}, then u_- is locally asymptotically stable.
Used in the proof of Theorem 1.2 to transfer positivity to stability for the perturbed Hamiltonian.

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Pith. "Pith review of The existence and stability of viscosity solutions to perturbed contact Hamilton-Jacobi equations." pith.science (2026). https://pith.science/paper/4TNV6QCJ

@misc{pith2026250104998,
  author       = {Pith},
  title        = {Pith review of: The existence and stability of viscosity solutions to perturbed contact Hamilton-Jacobi equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TNV6QCJ}},
  note         = {Machine review of arXiv:2501.04998}
}

abstract

We consider a contact Hamiltonian $H(x,p,u)$ with certain dependence on the contact variable $u$. If $u_{-}$ is a viscosity solution of the contact Hamilton-Jacobi equation \[H(x,D_{x}u(x),u(x))=0,\quad x\in M,\] and $u_{-}$ is locally Lyapunov asymptotically stable, we will prove that the perturbed equation \[H(x,D_{x}u(x),u(x))+\varepsilon P(x,D_{x}u(x),u(x))=0,\quad x\in M,\] does exist viscosity solution $u_{-}^{\varepsilon}$ which converges uniformly to $u_{-}$, as perturbation parameter $\varepsilon$ converges to 0. Moreover, we give a case that in a neighborhood of viscosity solution $u_-$, the perturbed equation has an unique viscosity solution $u_{-}^{\varepsilon}$. Furthermore, $u_{-}^{\varepsilon}$ keeps locally Lyapunov asymptotically stability.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [21]

Y . Ruan, K. Wang and J. Y an, Lyapunov stability and uniqu eness problems for Hamilton- Jacobi equations without monotonicity, https://doi.org/ 10.48550/arXiv.2501.08556

work page Pith review arXiv doi:10.48550/arxiv.2501.08556

[2] [24]

K. Wang, L. Wang and J. Y an, Aubry-Mather theory for cont act Hamiltonian systems, Comm. Math. Phys., 366 (2019), 981-1023. 20

work page 2019

[3] [27]

Y . Xu, J. Y an and K. Zhao, Dynamics of globally minimizing orbits in contact Hamiltonian systems, https://arxiv.org/abs/2412.20658

work page Pith review arXiv

[4] [1]

Arnold, Mathematical Methods of Classical Mechanics , 2nd edition, Graduate Texts in Mathematics, 60

V . Arnold, Mathematical Methods of Classical Mechanics , 2nd edition, Graduate Texts in Mathematics, 60. Springer-V erlag, New Y ork, (1989)

work page 1989

[5] [2]

Bravetti, H

A. Bravetti, H. Cruz and D. Tapias, Contact Hamiltonian m echanics, Ann. Physics, 376 (2017) 17-39

work page 2017

[6] [3]

Crandall and P

M. Crandall and P . L. Lions, Viscosity solutions of Hamil ton-Jacobi equations, Trans. Amer. Math.Soc., 277 (1983), 1-42

work page 1983

[7] [4]

Crandall, L

M. Crandall, L. Evans and P . L. Lions, Some properties of v iscosity solutions of Hamilton Jacobi equations, Trans. Amer. Math. Soc., 277 (1984), 487- 502

work page 1984

[8] [5]

Contreras, R

G. Contreras, R. Iturriaga, G. P . Paternain and M. Patern ain, Lagrangian graphs, minimiz- ing measures and Mañé’s critical values. Geom. Funct. Anal. , 8 (1998), 788–809

work page 1998