Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations

Junkai Qi

Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations

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Pith's one-line read Vanishing viscosity for degenerate Hamilton–Jacobi equations converges at rate O(ε|log ε|) pointwise and O(ε) on average.

desk verdict Solid, fully written extension of the O(ε|log ε|) vanishing-viscosity rate to a(x)≥0 that may vanish; the Da-control via entropy is the real technical step. read the letter →

arxiv 2607.05186 v2 pith:W53HYGYT submitted 2026-07-06 math.AP

Junkai Qi This is my paper · ORCID

classification math.AP MSC 35F2135K6535B4049L25

keywords Hamilton–JacobiequationsvanishingviscositydegeneratediffusionnonlinearadjointmethodconvergenceratesentropyestimatesweightedHessian

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

This paper quantifies how fast viscous approximations approach the viscosity solution of a Hamilton–Jacobi equation when the diffusion coefficient a(x) may vanish. On the torus, under uniform convexity of the Hamiltonian in momentum and standard growth, the solutions of the ε-viscous equation differ from the limiting solution by at most Cε(1+|log ε|) uniformly in space and time. When the same error is averaged against any smooth probability density, the log factor disappears and the rate improves to O(ε) times a mild factor depending only on the density. The argument adapts the nonlinear adjoint method to the spatially dependent, possibly degenerate diffusion: entropy and Fisher-information bounds on the adjoint density, combined with a weighted Hessian estimate that absorbs the extra derivatives of a(x), produce the improved rates. A sympathetic reader cares because previous sharp rates were known mainly for non-degenerate or first-order problems; the paper shows the same quantitative picture survives when diffusion can shut off.

What carries the argument

The nonlinear adjoint method with a weighted Hessian estimate: the error is expressed via the adjoint density σ^ε; entropy and Fisher-information bounds on σ^ε, together with the structural inequality |Da|^{2}≤ Ca, control the extra Da terms that appear after differentiation and yield the sharp weighted L^{2} estimate on D^{2}u^ε.

What would settle it

Construct a uniformly convex Hamiltonian and a nonnegative C^{2} diffusion a that vanishes somewhere such that either the L^∞ difference ∥u^ε−u∥ is larger than Cε|log ε| for a sequence ε o0, or the averaged error against some fixed smooth density r grows faster than O(ε).

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Extended reading notes

Core claim

Under the structural assumptions (uniform convexity of H in p, C² regularity of a and H, and quadratic growth of DxH), the vanishing-viscosity solutions u^ε of the possibly degenerate equation converge to the viscosity solution u at the pointwise rate O(ε|log ε|) on the torus, and the averaged error against any smooth probability density r is O(ε) times (1+∥Dr∥_L1+∫r|log r|dx)^{1/2}.

Load-bearing premise

The Hamiltonian must be uniformly convex in the momentum variable; without that strict convexity the second-derivative absorption step fails and the logarithmic rate is lost.

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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 punchline is simple: Qi gets the Cirant–Goffi / Chaintron–Daudin logarithmic rate for viscous HJ when the diffusion is a(x)+ε with a≥0 allowed to vanish, plus a clean averaged O(ε) statement against smooth densities. Both theorems are new for this degenerate setting.

What works is the technical handling of the Da terms. After differentiating, those terms produce D(Δu^ε) and derivatives of the adjoint density. The paper controls them with the elementary inequality |Da|^{2} ≤ C a (Lemma 2.1) together with entropy/Fisher estimates on the adjoint (Lemmas 2.4 and 3.1) and a weighted Hessian bound (Prop. 2.5). The time-cutoff χ_ au and the endpoint splitting are standard but carefully executed; the averaged argument (Section 3) drops the cutoff cleanly once the terminal datum is smooth. Proofs are complete, constants come only from the structural assumptions and the Bernstein bound (1.5), and the citation pattern is honest about the non-degenerate predecessors.

The soft spot is exactly the one the reader flags: uniform convexity D^{2}_pp H ≥ heta I is load-bearing for absorbing the quadratic Hessian after applying Δ. That is the same hypothesis used in the sharp-rate literature being extended, so it is not a hidden flaw, but the result does not reach merely semi-convex Hamiltonians. Everything else (growth of H, C^{2} regularity of a and g, torus setting) is standard and used transparently. No circularity, no free parameters.

This is for people already working on quantitative viscosity theory, selection, or homogenization who need rates under degenerate diffusion. It is not a conceptual leap, but it is a usable, self-contained extension. I would send it to a serious referee without hesitation; the math is there to check.

Referee Report

0 major / 4 minor

Summary. The paper studies quantitative rates at which solutions u^ε of the viscous approximation (a(x)+ε)Δu^ε = u_t^ε + H(x,Du^ε) converge to the viscosity solution u of the possibly degenerate equation a(x)Δu = u_t + H(x,Du) on the torus. Under C^{2} regularity of a,g and H, uniform convexity D^{2}_pp H ≥ θ I > 0, and the quadratic growth |D_x H| ≤ C(1+|p|^{2}), Theorem 1.1 establishes the pointwise bound ∥u^ε − u∥_∞ ≤ C ε(1+|log ε|). Theorem 1.2 removes the logarithm when the error at time T is tested against a smooth probability density r, yielding an averaged O(ε) estimate controlled by ∥Dr∥_1 and the entropy of r. The proofs use the nonlinear adjoint method, a weighted Hessian estimate for u^ε, and entropy/Fisher-information bounds on the adjoint density that absorb the extra Da terms arising from spatial degeneracy.

Significance. If correct, the results extend the sharp O(ε|log ε|) vanishing-viscosity rates of Cirant–Goffi and Chaintron–Daudin from the uniformly parabolic or first-order settings to equations with a spatially varying, possibly vanishing diffusion coefficient a(x) ≥ 0. The averaged O(ε) statement is new even in this literature. The technical core—control of Da-coupled third-order terms via the elementary inequality |Da|^{2} ≤ C a together with entropy production for the adjoint density—is self-contained, uses only standard tools (Young, Hölder, integration by parts, and a density bound from Bogachev–Krylov–Röckner–Shaposhnikov), and produces constants independent of ε from the structural hypotheses alone. Full proofs are written out, making the contribution immediately usable for related quantitative problems in degenerate viscous Hamilton–Jacobi theory.

minor comments (4)

In the introduction (p. 3) it would help the reader to recall briefly that the Bernstein bound (1.5) is taken from Cagnetti–Gomes–Mitake–Tran [7] and that the growth (1.4) is precisely the condition needed for that estimate; a one-sentence pointer would make the dependence transparent.
Lemma 2.4 invokes [18, Corollary 7.2.3] for the pointwise bound on the adjoint density. Adding a short remark that the constants depend only on n, the L^∞ bounds on b^ε and Da, and the ellipticity constant ε would clarify that no hidden ε-dependence enters the subsequent entropy integrals.
In the statement of Theorem 1.2 the factor (1 + ∥Dr∥_1 + ∫ r|log r|)^{1/2} multiplies Cε; it is already clear that C is independent of r, but a parenthetical note that the estimate remains O(ε) for any fixed smooth r would prevent a possible misreading that the rate deteriorates with r.
Typographical consistency: the manuscript mixes “vanishing viscosity” and “vanishing-viscosity” (hyphenation) and occasionally writes “Hamilton–Jacobi” versus “Hamilton-Jacobi”; a uniform choice throughout would improve polish.

Circularity Check

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No significant circularity: pure a-priori estimates derived from stated structural assumptions

full rationale

The paper is a self-contained analytic derivation of convergence rates. Theorems 1.1–1.2 follow from the nonlinear adjoint identity (2.4)/(3.3), the elementary degeneracy bound |Da|^{2} ≤ C a (Lemma 2.1), entropy/Fisher-information controls on the adjoint density (Lemmas 2.4, 3.1), and weighted Hessian estimates (Proposition 2.5, Proposition 3.2) that absorb the quadratic second-derivative terms via the uniform-convexity hypothesis D^{2}_pp H ≥ θ I in (1.3). All constants are produced from the a-priori L∞ bounds (1.5) and the structural assumptions (1.3)–(1.4); none are fitted to data or defined in terms of the target rate. Citations to Cagnetti–Gomes–Mitake–Tran and Cirant–Goffi supply independently established background estimates (Bernstein bounds, method template) that are not restated as the present claims. No self-definitional loop, fitted-input-as-prediction, load-bearing self-citation chain, uniqueness import, ansatz smuggling, or renaming of a known result appears. The derivation is therefore free of circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper rests on standard structural hypotheses for viscous Hamilton–Jacobi theory (uniform convexity in p, quadratic growth of D_x H, C² regularity of a and g) plus the elementary consequence |Da|² ≤ C a that follows from a ≥ 0 and a ∈ C². No free parameters are fitted; no new physical entities are postulated. All constants C absorb only norms of the given data on the compact set determined by the Bernstein bound.

assumptions (5)

domain assumption D²_pp H(x,p) ≥ θ I_n for some θ > 0 (uniform convexity in momentum)
Assumption (1.3); used to obtain the coercive term θ|D²u^ε|² after applying the Laplacian (equation (2.26)).
domain assumption |D_x H(x,p)| ≤ C(1+|p|²)
Growth condition (1.4); guarantees the Bernstein bound (1.5) remains uniform in ε.
standard math a ∈ C²(T^n), a ≥ 0 implies |Da|² ≤ C a
Lemma 2.1; elementary consequence of a^{1/2} being Lipschitz; used repeatedly to absorb Da-terms by Young’s inequality against A_ε = a+ε.
domain assumption Bernstein estimate ∥Du^ε∥_∞ + ∥u_t^ε∥_∞ ≤ C independent of ε
Cited from Cagnetti–Gomes–Mitake–Tran [7]; supplies the compact set on which all derivatives of H are bounded.
standard math Existence and uniqueness of viscosity solutions and of the adjoint density σ^ε
Standard theory for uniformly parabolic (A_ε ≥ ε) linear equations and for convex HJ equations; used for the duality identity (2.4).

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Pith. "Pith review of Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations." pith.science (2026). https://pith.science/paper/W53HYGYT

@misc{pith2026260705186,
  author       = {Pith},
  title        = {Pith review of: Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W53HYGYT}},
  note         = {Machine review of arXiv:2607.05186}
}

read the original abstract

We study quantitative convergence rates for vanishing viscosity approximations of possibly degenerate viscous Hamilton--Jacobi equations on the flat torus. The limiting equation contains a spatially dependent diffusion coefficient a(x) >= 0, which is allowed to vanish. Under standard structural assumptions on the Hamiltonian, we first prove a pointwise convergence rate of order O(epsilon |log epsilon|). We then show that, when the error is tested against a smooth probability density, the logarithmic loss can be removed and an averaged O(epsilon) rate holds. The proof is based on the nonlinear adjoint method, weighted Hessian estimates, and entropy estimates for the adjoint density.

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Works this paper leans on

18 extracted references · 5 linked inside Pith

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Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.

[1] [1]

Lions,Generalized solutions of Hamilton–Jacobi equations, Research Notes in Mathe- matics, vol

P.-L. Lions,Generalized solutions of Hamilton–Jacobi equations, Research Notes in Mathe- matics, vol. 69, Pitman, Boston–London–Melbourne, 1982

1982

[2] [2]

M. G. Crandall and P.-L. Lions,Two approximations of solutions of Hamilton–Jacobi equa- tions, Math. Comp.43(1984), no. 167, 1–19

1984

[3] [3]

H. V. Tran,Hamilton–Jacobi equations: theory and applications, Graduate Studies in Math- ematics, vol. 213, American Mathematical Society, Providence, RI, 2021

2021

[4] [4]

Han and S

Y. Han and S. N. T. Tu,Remarks on the vanishing viscosity process of state-constraint Hamilton–Jacobi equations, Appl. Math. Optim.86(2022), article no. 3

2022

[5] [5]

L. C. Evans,Adjoint and compensated compactness methods for Hamilton–Jacobi PDE, Arch. Ration. Mech. Anal.197(2010), no. 3, 1053–1088

2010

[6] [6]

H. V. Tran,Adjoint methods for static Hamilton–Jacobi equations, Calc. Var. Partial Differ- ential Equations41(2011), no. 3–4, 301–319

2011

[7] [7]

Cagnetti, D

F. Cagnetti, D. Gomes, H. Mitake, and H. V. Tran,A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire32(2015), no. 1, 183–200

2015

[8] [8]

Davini, A

A. Davini, A. Fathi, R. Iturriaga, and M. Zavidovique,Convergence of the solutions of the discounted Hamilton–Jacobi equation, Invent. Math.206(2016), no. 1, 29–55

2016