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Representation formulas and large time behavior for solutions to some nonconvex Hamilton-Jacobi equations

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Pith's one-line read This paper proves a new representation formula for nonconvex Hamilton–Jacobi equations on the torus and uses it to show that, under a one-sided growth condition, every solution converges uniformly to a stationary solution as time grows.

desk verdict The representation formula in Theorem 1.1 is a genuine and clean contribution, but the advertised large-time result, Theorem 1.4, has a real gap in its final comparison step and should not be accepted as is. read the letter →

arxiv 2505.01377 v1 pith:H5EZCRPJ submitted 2025-05-02 math.AP

Hung Vinh Tran This is my paper · ORCID
classification math.AP MSC 35B4035F2149L25
keywords representationformulaslargetimebehaviornonconvexHamilton-JacobiequationsviscositysolutionsnonlinearadjointmethodMathermeasuresvanishing
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 establishes a new representation formula for viscosity solutions of first-order Hamilton–Jacobi equations on the torus in the nonconvex case, and uses it to prove a large-time behavior result. The formula writes $u(z,T)$ as the integral of the initial data against a probability measure built from the adjoint equation, plus an integral of $D_pH(x,p)\cdot p - H(x,p)$ over a measure that encodes all characteristics passing through $(z,T)$. Under a one-sided growth condition on the Hamiltonian, with $H(x,0)\le 0$ and the cell-problem value $H(0)=0$, the paper proves that every solution converges uniformly to a Lipschitz stationary solution of $H(x,Dv)=0$ as $t\to\infty$. This extends earlier large-time results that required convexity or strict convexity near the zero sublevel set. The same adjoint measures also yield Mather measures in the nonconvex setting as large-time averages.

What carries the argument

The load-bearing object is the nonlinear adjoint method. For the vanishing-viscosity equation solved by $u^\varepsilon$, the linearized operator has an adjoint equation whose backward solution $\sigma^{\varepsilon,z}$ is a probability density transported from the point $(z,T)$; pushing $Du^\varepsilon$ forward by this density gives the measure $\mu_{\varepsilon,z}$, and its limits as $\varepsilon\to0$ are $\sigma_z$ and $\mu_z$ in the representation formula. The second engine is assumption (A2), equivalently (A2'): the map $s\mapsto s^{-(\theta+1)}H(x,sp)$ is nondecreasing for $s>0$, a directional growth condition along rays in momentum space. This converts the representation formula into a one-sided bound $u_t^\varepsilon \ge -C/T-C\varepsilon^{1/2}$ and then $u_t\ge -C/T$ in the viscosity sense, which yields the subsolution property of the large-time limits and drives the comparison argument.

What would settle it

One concrete test is numerical: solve $u_t + |u_x|^4 - |u_x|^2 = 0$ on the one-dimensional torus with $u(x,0)=\sin(2\pi x)$; Theorem 1.4 predicts uniform convergence of $u(\cdot,t)$ to a stationary function with $|v_x|^4-|v_x|^2=0$ a.e., so any persistent time-periodic oscillation or traveling wave in the computed solution would refute the claim.

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Extended reading notes

Core claim

The paper's central discovery is that the viscosity solution of the nonconvex Cauchy problem on the torus admits an exact adjoint-measure representation even after characteristics cross: for each $(z,T)$, $u(z,T) = \int_{\mathbb{T}^n} g\,d\sigma_z + \int_{\mathbb{T}^n\times\mathbb{R}^n\times[0,T]} (D_pH(x,p)\cdot p - H(x,p))\,d\mu_z$, where $\sigma_z$ is a probability measure on the torus and $\mu_z$ a measure on $(x,p,t)$-space obtained as limits of vanishing-viscosity adjoint solutions. This formula generalizes the method of characteristics, which fails once shocks form. The paper then uses the formula, together with the one-sided homogeneity assumption (A2), to prove Theorem 1.4: if $H(0)=0$, there exists a Lipschitz solution $v$ of the cell problem $H(x,Dv)=0$ such that $\|u(\cdot,t)-v\|_{L^\infty(\mathbb{T}^n)}\to 0$ as $t\to\infty$. The large-time limit is obtained by comparing the solution with stationary subsolutions, and the one-sided estimate on $u_t$ is the crucial control.

Load-bearing premise

The final comparison step assumes that every cluster point of $u(\cdot,t)$ as $t\to\infty$ obeys the same subsolution inequality $H(x,Dw)\le 0$ that was verified for the specially constructed limit $v$, so that two cluster points can be ordered in both directions; without that inheritance, the symmetric argument that rules out different limits does not close.

Editorial extensions

If this is right

  • For Hamiltonians satisfying (A1)–(A2) with $H(0)=0$ and any $C^2$ periodic initial data, $u(\cdot,t)$ converges uniformly to a Lipschitz stationary solution $v$ of the cell problem as $t\to\infty$.
  • The representation formula assigns a precise measure-theoretic meaning to 'all characteristics through $(z,T)$' even after shocks form, generalizing the method of characteristics.
  • Under (A2) with nonnegative initial data, $T^{1/\theta}u(z,T)$ is nondecreasing in $T$, giving a scaling monotonicity law for the solution.
  • Mather measures exist in the nonconvex setting, and every weak limit of the time-averaged adjoint measures is a Mather measure.
  • The vanishing-viscosity convergence rate $|u^\varepsilon-u|\le C(1+T)\varepsilon^{1/2}$ is optimal for general Hamiltonians under (A1).

Reading between the lines

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

  • If the mechanism behind Theorem 1.4 is right, adding the symmetric upper-bound condition the paper calls (A3) would likely turn the one-sided estimate into a two-sided one and could yield explicit convergence rates for $u(\cdot,t)\to v$, a question the paper leaves open.
  • A direct route to the paper's Questions 1–2 is to prove that the adjoint measures $\mu_{\varepsilon,z}$ converge as $\varepsilon\to0$; if they do, the large-time profile $v$ might be characterized explicitly in terms of the initial data $g$.
  • For separable Hamiltonians $H(x,p)=c(x)K(p)$, the vanishing of the dissipative measures in Lemma 3.7 suggests that this class may be the natural testbed for the paper's Question 6 on whether Mather measures form a uniqueness set for the cell problem.
  • The monotonicity law $T^{1/\theta}u(z,T)$ nondecreasing for nonnegative $g$ is directly checkable by experiment on the model Hamiltonian $|p|^4-|p|^2$, and a violation would pinpoint the sharp range of $\theta$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies viscosity solutions of nonconvex first-order Hamilton-Jacobi equations on the torus. It gives a new representation formula (Theorem 1.1) obtained via the nonlinear adjoint method, recovers the convex setting formula (Theorem 1.2), proves existence of Mather measures (Theorem 1.3), and states a large-time convergence result (Theorem 1.4) under assumptions (A1)-(A2) with H(0)=0. The proof of Theorem 1.4 relies on a one-sided estimate on u_t obtained from (A2), combined with a comparison argument against a stationary subsolution.

Significance. If Theorem 1.4 is correct, it is a meaningful extension of large-time behavior results to a class of nonconvex Hamiltonians, complementing the work of Barles-Ishii-Mitake and Barles-Souganidis. The representation formula (1.5) is new and may have independent applications. The paper also presents clean, self-contained proofs of the convex representation formula and the existence of Mather measures. However, the proof of the central large-time theorem has a critical gap, so the main advertised novelty is not established as written.

major comments (2)
  1. [Section 4.1, proof of Theorem 1.4, paragraph beginning 'Assume by contradiction'] The step 'By a similar logic, w≤v' is not justified. To run the comparison argument in the reverse direction, the arbitrary omega-limit w must be a viscosity subsolution of H(x,Dw)≤0. The only estimate available, (4.4), is the one-sided bound u_t ≥ -C/T; it does not imply that the time slices u(·,t_k) are approximate stationary subsolutions. A standard viscosity argument showing that a limit of time slices is a subsolution would require control of u_t from above as well, exactly what (A3) in Remark 7(ii) would supply. In fact the proof never establishes H(x,Dv)=0; it only gives H(x,Dv)≤0 for the specially constructed limit v. Thus even if the uniqueness of the subsequential limit were proved, v would not be shown to be a solution of (1.9) as claimed in the theorem.
  2. [Section 4.1, Lemma 4.3] The proof of Lemma 4.3 bounds the integral ∫_0^T ∫ (D_p H·p−H) dν^ε using the boundedness of u in (4.1), but the representation formula in Theorem 1.1 applies to u^ε, not u. The comparison (4.1) is for the inviscid solution; u^ε is not known to be bounded uniformly in T and ε. Consequently the estimate |∫_0^T∫(D_p H·p−H)dν^ε| ≤ C is not established for all T. At the specific times T_ε=ε^{-1/4} used in Theorem 1.4 the bound can be recovered from Lemma 4.6, but the lemma as stated and proved is incorrect and needs either a corrected proof or a restricted statement.
minor comments (4)
  1. [Abstract and Introduction] The phrase 'large time behavior' is used without noting that the proof of Theorem 1.4 is incomplete; the abstract should reflect the actual status of the result.
  2. [Section 4.2, Lemma 4.5] The notation σ_{z,T} and μ_{z,T} is used without recalling that these measures depend on the subsequence of ε→0; this dependence should be acknowledged in the statement.
  3. [Examples 2 and 5] The sentence 'H(0)=0 as H(0)=0' is redundant and should be rephrased.
  4. [Throughout] There are occasional typos and grammatical issues (e.g., 'vεxixj', 'H(0)=H(0)'), and some displays are hard to parse; a careful proofreading is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the new theorems are derived from the PDE and adjoint calculations, and the flagged gap in Theorem 1.4 is an unproved comparison step, not a circular one.

full rationale

The paper's central derivations are self-contained. Theorem 1.1 is obtained by direct integration of d/dt ∫ u^ε σ^{ε,z} dx against the nonlinear adjoint state, with no fitted parameter or assumed conclusion. Theorem 1.2 is explicitly labeled as not new and as already appearing in [20], and its proof is carried out in the paper. Theorem 1.3 is likewise acknowledged as previously proved in [6], with the paper offering a different proof via Theorem 1.1 and time averages. Theorem 1.4's main estimates, Lemmas 4.1-4.3 and Corollary 4.4, are proved from (A1)-(A2) and the representation formula; the cited [5] is used only as inspiration, not as a black-box premise. The self-citations are not load-bearing: they point to earlier results or to methods, while the new claims are argued from equations established in the paper. The one substantive problem is the step in the proof of Theorem 1.4 where, after showing v ≤ w, the paper states 'By a similar logic, w≤v' (Section 4.1). This would require an arbitrary time-slice limit w to satisfy H(x,Dw) ≤ 0, but Corollary 4.4 only provides the one-sided estimate u_t ≥ -C/T; the reverse control u_t ≤ C/T is not available under (A2), and Remark 7(ii) notes that an upper bound would require (A3). That is a missing proof argument, not a circular reduction: the conclusion is not assumed in the hypotheses, nor is it obtained by renaming an input. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted. The assumptions (A1) and (A2) are structural conditions on the Hamiltonian that define the class of problems studied. The measures sigma_z and mu_z are constructed from the adjoint method, not postulated. The paper relies on standard viscosity solution theory and the Lions-Papanicolaou-Varadhan cell problem theory.

assumptions (4)
  • domain assumption Assumption (A1): lim_{|p|->infty} min_{x} (1/2 H(x,p)^2 + D_x H(x,p) dot p) = +infty.
    Used in Section 1.1 to guarantee global Lipschitz bounds for u^epsilon and u via the Bernstein method, and to ensure compactness of the adjoint measures.
  • domain assumption Assumption (A2): there exists theta > 0 such that D_p H(x,p) dot p >= (theta+1)H(x,p) for all (x,p) in T^n x R^n.
    Imposed for the large time behavior results (Section 4). It is a one-sided radial monotonicity condition, equivalent to s^{-(theta+1)}H(x,sp) nondecreasing in s, and is not derived from (A1).
  • standard math Lions-Papanicolaou-Varadhan theory: existence of the effective Hamiltonian Hbar(P) and viscosity solutions v_P to the cell problem (1.8).
    Invoked in Sections 3 and 4 for the cell problem (1.9) as a background result from the literature.
  • standard math Standard comparison principle and stability for viscosity solutions of first-order Hamilton-Jacobi equations.
    Used throughout, especially in the proof of Theorem 1.4 (comparison with stationary subsolutions) and in Lemma 4.5.

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Pith. "Pith review of Representation formulas and large time behavior for solutions to some nonconvex Hamilton-Jacobi equations." pith.science (2026). https://pith.science/paper/H5EZCRPJ

@misc{pith2026250501377,
  author       = {Pith},
  title        = {Pith review of: Representation formulas and large time behavior for solutions to some nonconvex Hamilton-Jacobi equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H5EZCRPJ}},
  note         = {Machine review of arXiv:2505.01377}
}
read the original abstract

We give a new representation formula for solutions to nonconvex first-order Hamilton--Jacobi equations in the periodic setting and present some applications. We then prove the large time behavior for solutions under some additional assumptions.

Figures

Figures reproduced from arXiv: 2505.01377 by the authors.

Figure 4.1
Figure 4.1. Graph of H(p) = |p| 4 − |p| 2 in one dimension [PITH_FULL_IMAGE:figures/full_fig_p023_4_1.png] view at source ↗

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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. Full citation record

  1. Vanishing discount limits for first-order fully nonlinear Hamilton-Jacobi equations on noncompact domains

    math.AP 2025-07 conditional novelty 6.0 of 10

    Vanishing discount limits for fully nonlinear contact Hamilton-Jacobi equations on R^n converge locally uniformly to the maximal solution selected by a Mather-measure criterion.

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