{"id":"f4bfb9d3-853d-4d07-ac57-e2809fd334ae","arxiv_id":"2505.01377","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves a new adjoint-measure representation for nonconvex Hamilton-Jacobi solutions and claims a large time convergence result under a new growth condition, but the convergence proof has a gap.","lead":"This paper derives a new representation formula for nonconvex Hamilton-Jacobi equations on the torus using nonlinear adjoint measures. It also claims a large time convergence theorem, but that theorem's proof has an unjustified comparison step.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4's final comparison step assumes arbitrary time-slice limits are subsolutions; the one-sided estimate (4.4) does not supply this, so the claimed uniqueness of the limit is unproved.","rationale":"The paper's advertised new large time behavior result is Theorem 1.4. The proof constructs one subsequential limit v of u(·,T_{ε_k}) and shows v is a viscosity subsolution of H(x,Dv)≤0. To pass from this single limit to full convergence, the proof must rule out other subsequential limits w. It attempts this by asserting 'By a similar logic, w≤v'. That assertion requires w to be a subsolution, so the comparison principle can be applied from the times t_k. The only estimate available for arbitrary time slices is Corollary 4.4's lower bound u_t ≥ -C/T. This bound gives control in the past direction, but the viscosity subsolution test at time t_k also needs control on the future side s>t_k; without an upper bound on u_t, the constructed time-slice limit need not be a subsolution. Remark 7(ii) underscores the issue by noting that an upper bound would come from the separate assumption (A3). Thus the proof has a genuine gap at the final comparison step, exactly as the reader identified. The representation formula and Mather measure results appear to be independent and likely correct, but the central large time behavior claim is not proved. The proposed concrete test isolates the missing estimate: if u_t ≤ C/T cannot be derived from (A1)-(A2), then the gap is real; if it can, the proof would be repairable. Either way, the current manuscript does not establish Theorem 1.4, so the reader's REJECT verdict is appropriate and unchanged.","tokens_in":22809,"tokens_out":39632,"duration_ms":426110,"concrete_test":"Check whether the time slices u(·,t_k) are approximate viscosity subsolutions of H(x,·)≤0 under (A1)-(A2) alone. Concretely, take a smooth φ and a strict maximizer x_k of u(·,t_k)-φ, and attempt to complete the C^1 space-time test ψ(x,s)=φ(x)+A(s-t_k)^2 in the viscosity subsolution test for (1.1) at (x_k,t_k). For s>t_k this requires an upper bound on u_t (or on u(x_k,s)-u(x_k,t_k)) that is o(1) as t_k→∞; derive this from Corollary 4.4. If the derivation fails, the 'similar logic' step in Theorem 1.4 is invalid. Alternatively, add (A3) and verify the proof goes through, confirming that the upper bound is what is missing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's objection is on target. In the proof of Theorem 1.4 (Section 4.1), the specially constructed limit v is a subsolution H(x,Dv)≤0 because (4.5) gives an approximate elliptic subsolution inequality for u^{ε_k}(·,T_{ε_k}). To conclude that any other limit w = lim_k u(·,t_k) also satisfies H(x,Dw)≤0 ('By a similar logic, w≤v'), one needs the time slices u(·,t_k) to be approximate subsolutions of H≤o(1). The only time estimate available is Corollary 4.4, (4.4): u_t ≥ -C/T. This inequality controls the past of each time slice; the viscosity subsolution test at time t_k also requires controlling s>t_k, where u_t may be positive and no bound of the form u_t ≤ C/T is proved. (A2) yields only the lower bound in Lemma 4.3; Remark 7(ii) explicitly notes that (A3) would give upper bounds, confirming the missing ingredient. Without an upper bound on u_t, arbitrary time slices need not be approximate subsolutions of the stationary equation, so the step 'By a similar logic, w≤v' is unsupported. Consequently, uniqueness of the subsequential limit, and therefore the claimed convergence to a solution of (1.9), is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23020,"tokens_out":26578,"duration_ms":261039,"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":[{"comment":"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.","section":"Section 4.1, proof of Theorem 1.4, paragraph beginning 'Assume by contradiction'"},{"comment":"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.","section":"Section 4.1, Lemma 4.3"}],"minor_comments":[{"comment":"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.","section":"Abstract and Introduction"},{"comment":"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.","section":"Section 4.2, Lemma 4.5"},{"comment":"The sentence 'H(0)=0 as H(0)=0' is redundant and should be rephrased.","section":"Examples 2 and 5"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The representation formula Theorem 1.1 is solid and new, and the exposition is generally clear. However, the main new theorem (Theorem 1.4) is not proved: the 'similar logic' comparison step requires a property of arbitrary omega-limits that is not established by the one-sided estimate (4.4). The flaw in Lemma 4.3 is also real but may be repairable at the specific times T_ε. Should the author provide a correct proof of the missing subsolution property (or of the converse comparison), the paper would likely be a valuable contribution. At present, the central claim is unproved, so the paper cannot be accepted in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth your time, but with a clear caveat. The paper has two advertised results: a new representation formula for nonconvex Hamilton–Jacobi equations, and a large-time behavior theorem. The first is in good shape. Theorem 1.1, derived via the nonlinear adjoint method, is clean, correct, and genuinely new in this time-dependent nonconvex setting. The paper is also honest about what is not new: Theorems 1.2 and 1.3 are explicitly labeled as already known, and the discussion of prior work by Barles–Ishii–Mitake, Barles–Souganidis, and the author's own earlier papers is fair. The examples in Section 4.3 are useful and do a good job separating assumption (A2) from earlier convexity-type conditions.\n\nThe problem is Theorem 1.4. The proof's final step says, after deriving v ≤ w for an arbitrary omega-limit w, that \"by a similar logic, w ≤ v.\" That symmetry is not available. The one-sided estimate (4.4), u_t ≥ -C/T, only gives an approximate subsolution inequality for the specially selected subsequence at times T_epsilon. It does not give the same property for arbitrary time slices t_k going to infinity, and it certainly does not give the supersolution side. The author's own Remark 7(ii) confirms the missing ingredient: an upper bound on u_t would come from an assumption like (A3), which is not made. So uniqueness of the subsequential limit, and hence convergence to a solution of (1.9), is not established. This is a load-bearing gap, not a cosmetic one.\n\nThe rest of the paper is mostly solid. The auxiliary estimates in Section 4 are derived carefully, and the representation formula is used in an interesting way to get the one-sided control. The citation pattern is reasonable; the self-citations are to closely related work and are not inflated. There is no data or code involved, and no parameter fitting, so the usual reproducibility concerns do not apply.\n\nWho is this for? Anyone working on nonconvex Hamilton–Jacobi equations and the nonlinear adjoint method. The representation formula alone is worth reading and citing. The large-time theorem should not be taken off the shelf until the comparison argument is repaired.\n\nMy recommendation: send it to a serious referee, but tell the referee to focus on Section 4.1 first. If Theorem 1.4 cannot be fixed, the paper still has a publishable core in Theorem 1.1 and the surrounding analysis, but the advertised headline needs to change.","headline":"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.","tokens_in":23630,"tokens_out":1562,"would_cite":true,"duration_ms":17436,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B40","35F21","49L25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["representation formulas","large time behavior","nonconvex Hamilton-Jacobi equations","viscosity solutions","nonlinear adjoint method","Mather measures","vanishing viscosity"],"falsifier":"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.","tokens_in":22531,"feed_emoji":"⏳","tokens_out":17855,"duration_ms":160114,"temperature":0.7,"pith_summary":"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.","feed_headline":"Large-time convergence proved for nonconvex Hamilton-Jacobi equations","feed_subtitle":"Adjoint-measure formula works past shocks and nails the large-time limit beyond convex cases.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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 $\\theta$."],"forward_implications":["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)."],"supporting_citations":[{"why":"Introduces the nonlinear adjoint method for Hamilton–Jacobi PDEs, the basis of the representation formula in Theorem 1.1.","marker":"[13]"},{"why":"Establishes the static adjoint method whose measure estimates control the convergence of the vanishing-viscosity solutions.","marker":"[36]"},{"why":"Defines Mather measures in the nonconvex setting and supplies the vanishing-viscosity estimates used throughout the paper.","marker":"[6]"},{"why":"Provides the adjoint-based strategy for large-time behavior from the uniformly convex case that Theorem 1.4 adapts.","marker":"[5]"},{"why":"Earlier large-time behavior result for some nonconvex Hamiltonians that the new theorem complements.","marker":"[3]"},{"why":"Earlier PDE approach to large-time asymptotics whose conditions on the zero sublevel set are replaced here by assumption (A2).","marker":"[2]"},{"why":"Gives the convex-setting representation formula that the paper recovers as a special case via Legendre transform.","marker":"[20]"},{"why":"Establishes the cell problem and the existence of the effective Hamiltonian whose stationary solutions are the convergence targets.","marker":"[29]"}],"fun_headline_variants":["Adjoint measures crack nonconvex HJ large-time limits","Nonconvex HJ: exact formula plus large-time convergence","Beyond convexity: adjoint formula yields HJ large-time limit","Shock-proof formula proves HJ convergence for nonconvex case","New representation formula for nonconvex HJ, large-time limit proven"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Adjoint measures crack nonconvex HJ large-time limits","Nonconvex HJ: exact formula plus large-time convergence","Beyond convexity: adjoint formula yields HJ large-time limit","Shock-proof formula proves HJ convergence for nonconvex case","New representation formula for nonconvex HJ, large-time limit proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000517,"raw_usage":{"total_tokens":2444,"prompt_tokens":822,"completion_tokens":1622,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":1537}},"tokens_in":438,"tokens_out":1622,"duration_ms":10970,"temperature":1.0,"reasoning_tokens":1537,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:20:48.421396+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the nonlinear adjoint method for Hamilton–Jacobi PDEs, the basis of the representation formula in Theorem 1.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the static adjoint method whose measure estimates control the convergence of the vanishing-viscosity solutions."},{"cited_title":"Cagnetti, D","cited_arxiv_id":null,"evidence_quote":"Defines Mather measures in the nonconvex setting and supplies the vanishing-viscosity estimates used throughout the paper."},{"cited_title":"Cagnetti, D","cited_arxiv_id":null,"evidence_quote":"Provides the adjoint-based strategy for large-time behavior from the uniformly convex case that Theorem 1.4 adapts."},{"cited_title":"Barles, P","cited_arxiv_id":null,"evidence_quote":"Earlier large-time behavior result for some nonconvex Hamiltonians that the new theorem complements."},{"cited_title":"Barles, H","cited_arxiv_id":null,"evidence_quote":"Earlier PDE approach to large-time asymptotics whose conditions on the zero sublevel set are replaced here by assumption (A2)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the convex-setting representation formula that the paper recovers as a special case via Legendre transform."},{"cited_title":"Lions, G","cited_arxiv_id":null,"evidence_quote":"Establishes the cell problem and the existence of the effective Hamiltonian whose stationary solutions are the convergence targets."}],"review_version":1}