{"id":"34a61069-8ba7-4a01-b759-7ceb86777331","arxiv_id":"2506.13255","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Vanishing viscosity for uniformly convex Hamilton-Jacobi equations converges at the optimal rate O(epsilon log epsilon), improving the old O(sqrt(epsilon)) bound.","lead":"This paper proves that solutions of viscous Hamilton-Jacobi equations converge to their inviscid limit at the optimal rate of order epsilon times log epsilon, instead of the previously known square root of epsilon. The result resolves a long-standing question for uniformly convex Hamiltonians and gives a matching example showing the rate cannot be improved.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.6 drops the 1/θ factor from Proposition 2.4(ii), so the printed proof establishes the dεlogε lower bound only when θ≥1; the θ<1 case requires an additional rescaling argument that the manuscript does not give.","rationale":"The paper proves a plausible and probably correct optimal-rate theorem. The main proof ingredients, the entropy estimate Proposition 2.4(i), the semiconcavity propagation in (6), and the sup-convolution regularization, are standard and are used in a coherent way. However, the proof of Theorem 2.6 as printed contains a concrete algebraic inconsistency: Proposition 2.4(ii) carries a 1/θ factor, and that factor disappears without comment when the estimate is inserted into the lower-bound computation. Restoring the factor shows the leading coefficient is d/θ, which yields the stated dεlogε lower bound only in the regime θ≥1. The missing argument is likely a rescaling, and the theorem is probably repairable, which supports the reader's CONDITIONAL verdict rather than REJECT. The reader's formal weakest assumption was semiconcavity propagation, but their rationale already identifies the θ-factor discrepancy; my concern is the same concrete proof gap, so I mark partial agreement. A direct symbolic recomputation of the Theorem 2.6 display, or an explicit check in the quadratic case, would settle whether the gap is merely typographical or requires a substantive extra argument.","tokens_in":17340,"tokens_out":19360,"duration_ms":223513,"concrete_test":"Independently re-derive Theorem 2.6 retaining the 1/θ factor from Proposition 2.4(ii). For H(x,p)=θ|p|²/2, d=2, g(x)=−|x|, and θ=1/2, compute φ^ε−φ^0 at (t,x)=(0,0) from the explicit Cole-Hopf formula (14) for ε=10^{−k}, k=2,...,6, and compare with (a) the advertised bound dεlogε−Cε, and (b) the bound obtained from the printed proof with the θ factor restored. If the explicit asymptotic difference has leading coefficient (d−1)/2, as Proposition 3.4 gives for θ=1, the theorem may still be true, but the printed proof must either show the θ<1 implication or add the rescaling step that reduces to the θ=1 case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central lower-bound proof has a coefficient error. Proposition 2.4(ii) states (1/2)∫∫ Δφ^{0,δ} dμ ds ≥ (d/(2θ)) log(2πετ) − τL²/(2εθ) − C. Multiplying by ε as in the proof of Theorem 2.6 gives ε d/(2θ) log(2πετ) − τL²/(2θ) − Cε. The displayed proof instead writes ε d/2 log(2πετ) − τL²/2 − Cε, silently deleting the θ. With δ=τ=ε, the restored term yields a lower bound whose leading coefficient is d/θ, not d. Since logε<0, this proves the advertised inequality φ^ε−φ^0 ≥ dεlogε−Cε only when θ≥1 (where d/θ≤d makes the proved bound stronger than the claim); for θ<1 the coefficient d/θ gives a more negative right-hand side, so the advertised bound is not a consequence of the stated estimates. This is not a refutation of Theorem 1.2: the quadratic case suggests a rescaling ψ=θφ, which sends H to a Hamiltonian with convexity constant at least 1 and viscosity coefficient θε, and would recover dεlogε with the θ-dependence absorbed into C. But the manuscript does not supply this rescaling or any other argument covering θ<1. The factor inconsistency is therefore a genuine gap in the proof as written, and it is located exactly in the step from Proposition 2.4(ii) to Theorem 2.6.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the vanishing-viscosity approximation of uniformly convex Hamilton-Jacobi equations with Lipschitz, semiconcave terminal data. It claims an optimal L∞ rate of order ε log ε, improving on the classical √ε rate, and it provides an explicit example showing that the rate cannot be improved in dimension d ≥ 2. The proof combines the semiconcavity of the viscous solutions with a sup-convolution regularization of the first-order solution and entropy estimates for the Fokker-Planck flow associated with the difference of the equations. A separate section treats the purely quadratic case, where the Cole-Hopf formula yields the same rate for merely Lipschitz terminal data.","tokens_in":17626,"tokens_out":16433,"duration_ms":195274,"significance":"If the proof is completed in the form stated, the paper settles a long-standing open question on the optimal vanishing-viscosity rate in all dimensions for uniformly convex Hamiltonians. The entropy-estimate method in Proposition 2.4 is a genuine technical contribution, and the explicit example in Section 3.2 provides a concrete witness that the rate cannot be sharpened. I found no circularity: the entropy bound is derived from Girsanov and standard estimates, not assumed. The simultaneous independent work [CG25b] is acknowledged, and Section 3 usefully extends the result to non-semiconcave data in the quadratic case.","major_comments":[{"comment":"The transition from Proposition 2.4(ii) to Theorem 2.6 drops the factor 1/θ. Proposition 2.4(ii) states (1/2)∫∫Δφ^{0,δ} dμ ds ≥ d/(2θ) log(2πετ) − τL²/(2εθ) − C. Multiplying by ε and inserting this into Theorem 2.6 gives εd/(2θ) log(2πετ) − τL²/(2θ) − Cε, but the displayed estimate in Theorem 2.6 has εd/2 log(2πετ) − τL²/2 − Cε. With δ=τ=ε, the corrected term yields a leading coefficient d/θ, not d. Since log ε < 0, the advertised lower bound φ^ε−φ^0 ≥ dε log ε − Cε follows from the stated estimates only when θ ≥ 1. For θ < 1, the coefficient d/θ gives a more negative right-hand side, so the claimed inequality is not a consequence of the proof. The order ε log ε is still consistent with the corrected coefficient, but the specific constant d in Theorem 1.2 is not established for all θ>0 as written. This gap needs to be repaired, either by an additional argument that recovers the constant d for θ<1 or by modifying the statement of the lower bound.","section":"Theorem 2.6 and Proposition 2.4(ii)"}],"minor_comments":[{"comment":"The sentence 'Since ∆ϕ^{ε,δ}_t ≥ −λd using the semiconcavity (6)' is not correct as written: semiconcavity gives an upper bound on the Hessian, not a lower bound. The surrounding algebra needs an upper bound on Δϕ^ε_s, namely Δϕ^ε_s ≤ dλ, in order to control the term −(θ/2)∫Δϕ^ε_s dμ. I assume this is a typographical slip, but the sign and the superscript should be corrected.","section":"Proposition 2.4(ii), proof"},{"comment":"There is a repeated typo 'semiconvavity' where 'semiconcavity' is meant, including near the display following Assumption (A.1).","section":"Section 2.2"},{"comment":"The abstract contains a line-break in 'conve rgence' and Section 1.1 has 'direclty'; a proofreading pass is needed.","section":"Abstract and Section 1.1"},{"comment":"The expansion for k=1 has no ε log ε term, consistent with the restriction of the optimality example to d≥2, but the statement 'for 1≤k≤d' could briefly note this so the reader is not misled.","section":"Proposition 3.4"},{"comment":"Reference [CG25b] is listed as 'Incoming article'; if a preprint or journal version is available, it should be updated.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The coefficient error around θ is the only substantive obstacle I see. It does not appear to invalidate the optimal-order claim, since the corrected lower bound still has order ε log ε, but the statement of Theorem 1.2 with the universal constant d is not proved as written. This should be fixable by either a more careful tracking of θ or a revision of the constant, so I would not reject; major revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a real step forward, but the lower bound in Theorem 1.2 is mis-stated. For the uniformly convex setting they prove the right rate, O(epsilon log epsilon), which is a longstanding open improvement over O(sqrt(epsilon)). The entropy method is nice: regularize by sup-convolution and integrate against a Fokker-Planck flow to trade the sqrt(epsilon) bound for epsilon log epsilon. The optimality example is also clean.\n\nThe problem is the theta dependence. Proposition 2.4(ii) has a factor 1/theta in the key estimate, and the proof of Theorem 2.6 drops it: the displayed inequality writes epsilon d/2 log(2 pi epsilon tau) instead of epsilon d/(2 theta) log(...). Restoring the factor and choosing delta = tau = epsilon gives a lower bound proportional to (d/theta) epsilon log epsilon, not d epsilon log epsilon. This is not a harmless typo. For theta < 1, the claimed bound with constant d is actually false. Take H(p) = (theta/2)|p|^2 and g(x) = -|x| in R^2. A direct Laplace expansion of the Cole-Hopf formula gives phi^epsilon - phi^0 = (1/(2 theta)) epsilon log epsilon + O(epsilon) at the origin. For theta = 0.1 this is 5 epsilon log epsilon, which is far more negative than the 2 epsilon log epsilon the theorem promises, so the inequality fails for small epsilon. The stress-test note is right about the dropped factor but too charitable in saying it is not a refutation; the stated lower bound is false as written.\n\nThe fix is straightforward: the lower bound should have a theta-dependent prefactor, likely d/theta (or something like (d-1)/(2 theta) in the quadratic example), with the constant C depending on theta. The upper bound and the O(epsilon log epsilon) rate are unaffected. The authors also need to check the rescaling claim in the stress-test note, since psi = theta phi keeps the viscosity coefficient epsilon, not theta epsilon, so it does not recover a universal d.\n\nApart from this, the paper is well written, the literature is handled honestly, and the concurrent torus result [CG25b] is acknowledged. The proof outline is clear and the technical core (Proposition 2.4) appears sound after restoring the theta.\n\nRecommendation: send to peer review, but only after the lower bound statement is corrected and the theta dependence is worked out. A serious referee will catch this immediately; the paper should not be accepted as is.","headline":"Strong result with a real coefficient error: the lower bound in Theorem 1.2 is false for small theta, though the O(epsilon log epsilon) rate is correct.","tokens_in":18202,"tokens_out":16018,"would_cite":false,"duration_ms":154922,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35F21","35B25","49L25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Vanishing viscosity converges at $\\epsilon \\log \\epsilon$, not $\\sqrt{\\epsilon}$.","keywords":["vanishing viscosity","Hamilton-Jacobi equations","optimal convergence rate","uniformly convex Hamiltonian","semiconcavity","Fokker-Planck equation","entropy estimates","viscosity solutions"],"falsifier":"For the explicit datum $g(y)=-|y|$ in dimension $d=2$, quadratic Hamiltonian $H(p)=|p|^2/2$, and $x=0$, Proposition 3.4 predicts $\\phi^\\epsilon_t(0)-\\phi^0_t(0)=\\frac12\\epsilon\\log\\epsilon+O(\\epsilon)$. Evaluating the explicit Cole–Hopf integral (14) at small $\\epsilon$ and computing the ratio $(\\phi^\\epsilon-\\phi^0)/(\\epsilon\\log\\epsilon)$ as $\\epsilon\\to0$ should give $1/2$; any other limit would refute the claimed optimal rate and constant.","tokens_in":17099,"feed_emoji":"🧮","tokens_out":15165,"duration_ms":161658,"temperature":0.7,"pith_summary":"Adding a small viscosity term to a first-order Hamilton–Jacobi equation smooths its solutions; the paper asks how fast those smoothed solutions return to the inviscid limit as the viscosity $\\epsilon$ vanishes. Under the standard assumptions that the Hamiltonian is uniformly convex and the terminal condition is Lipschitz and semiconcave, it proves that the convergence rate is of order $\\epsilon \\log \\epsilon$, improving the previously known $\\sqrt{\\epsilon}$ rate that was widely believed optimal. It also proves a matching lower bound with a universal factor $d$ (the space dimension), and constructs an explicit terminal datum for which the expansion has leading term $\\frac{k-1}{2}\\epsilon \\log \\epsilon$, showing the rate cannot be sharpened. The interest is that convexity genuinely accelerates convergence, and the logarithmic loss is the true cost of singularities in the limiting solution.","feed_headline":"Vanishing viscosity converges at ε log ε, not √ε","feed_subtitle":"For convex Hamilton–Jacobi equations the old √ε bound is off by a log factor, and the new rate is provably sharp.","key_machinery":"The proof is carried by two objects: the sup-convolution regularisation $\\phi^{0,\\delta}(x)=\\sup_y(\\phi^0(y)-|x-y|^2/(2\\delta))$, which is $-1/\\delta$-semiconvex and therefore has Laplacian bounded below by $-d/\\delta$, and the Fokker–Planck flow $\\mu^{\\epsilon,\\delta}$ driven by the averaged gradient field built from $\\nabla\\phi^\\epsilon$ and $\\nabla\\phi^{0,\\delta}$. Integrating the equation for $\\phi^\\epsilon-\\phi^{0,\\delta}$ against this flow turns the error into an integral of $-\\frac{\\epsilon}{2}\\Delta\\phi^{0,\\delta}$ against $\\mu$. The key estimate (Proposition 2.4-(ii)) bounds this integral from below by the entropy of $\\mu$ at the exit time, and the entropy itself is at most $-\\frac{d}{2}\\log(2\\pi\\epsilon\\tau)+\\frac{\\tau}{2\\epsilon}L^2$; that logarithm is the source of the $\\epsilon\\log\\epsilon$ term. Choosing the regularisation scale $\\delta=\\tau=\\epsilon$ balances the $-1/\\delta$-semiconvexity bound, the $O(\\delta)$ regularisation error, and the entropy gain, producing $d\\epsilon\\log\\epsilon$ as the leading-order error.","core_discovery":"The central result (Theorem 1.2) is a two-sided estimate: for every $\\epsilon\\in(0,1/2]$, $\\sup_{t,x}|\\phi^\\epsilon_t(x)-\\phi^0_t(x)| \\le -C_{\\log}\\epsilon\\log\\epsilon$, and $\\phi^\\epsilon_t(x)-\\phi^0_t(x)\\ge d\\epsilon\\log\\epsilon-C\\epsilon$ for all $(t,x)$. Here $\\phi^\\epsilon$ solves the viscous Hamilton–Jacobi–Bellman equation with viscosity $\\epsilon/2$ and $\\phi^0$ solves the first-order equation; the constant $d$ is the spatial dimension. The lower bound carries the content: because $\\log\\epsilon<0$, the leading term is positive and has a universal dimensional coefficient, so convergence is slower than linear by a logarithm. Section 3.2 proves optimality: for terminal data $g_k(y)=-|P_k(y)|$ in the quadratic case, the explicit Cole–Hopf and Hopf–Lax formulas give $\\phi^{k,\\epsilon}_t(0)-\\phi^{k,0}_t(0)=\\frac{k-1}{2}\\epsilon\\log\\epsilon+O(\\epsilon)$, which for $k\\ge2$ has exactly order $\\epsilon\\log\\epsilon$ and prevents any sharper global rate. The paper also extends the lower bound to merely Lipschitz terminal data when the Hamiltonian is purely quadratic, using the semiconcavity that the equation itself generates.","pith_inferences":["Beyond the paper, the appearance of the dimension $d$ in the leading term suggests that the prefactor counts the dimension of the set of minimisers in the Hopf–Lax representation: in the sharp example the minimising set is a $(d-k)$-flat and the prefactor is $(k-1)/2$, so a conjectural general formula would express the leading constant through the geometry of the argmin set.","A natural stress test is to keep the equation quadratic but replace the identity diffusion by a non-degenerate matrix with bounded derivatives; the entropy mechanism should still yield $\\epsilon\\log\\epsilon$ with the trace of the diffusion matrix replacing $d$, although the paper does not carry out this extension.","Because the proof uses uniform convexity only through the lower bound $\\theta I\\le\\nabla^2_{pp}H$, a merely convex Hamiltonian such as $H(p)=|p|$ should lie on the other side of the divide and revert to the slower $\\sqrt{\\epsilon}$ rate; a numerical experiment on the explicit flat-Hamiltonian formulas would test whether uniform convexity is the true threshold."],"forward_implications":["The global convergence rate for uniformly convex Hamilton–Jacobi equations in every dimension is $\\epsilon|\\log\\epsilon|$, so the $\\sqrt{\\epsilon}$ rate known from the early differential-game approach is not optimal under uniform convexity.","The lower bound $\\phi^\\epsilon_t(x)-\\phi^0_t(x)\\ge d\\epsilon\\log\\epsilon-C\\epsilon$ identifies the logarithmic term as the leading, universal cost of the singularities of the inviscid solution; inside the strong-regularity region the paper's comparison argument recovers the faster linear rate.","The explicit datum $g_k(y)=-|P_k(y)|$ shows the rate is sharp in dimensions $d\\ge2$, with prefactor $(k-1)/2$ at the origin; no global bound better than order $\\epsilon\\log\\epsilon$ can hold.","For the purely quadratic Hamiltonian with only Lipschitz terminal data, the same rate remains valid because the equation itself generates $(T-t)^{-1}$-semiconcavity, so the semiconcavity assumption on $g$ can be dropped in that case.","In the mean-field control analogue, the paper's result implies that the convergence of value functions as $N\\to\\infty$ is faster than $N^{-1/2}$ but no faster than $N^{-1}\\log N$, giving a quantitative limitation in the non-smooth regime."],"supporting_citations":[{"why":"First derived the $\\sqrt{\\epsilon}$ convergence rate by a differential-game argument; this is the sub-optimal bound that the paper improves.","marker":"[Fle64b]"},{"why":"Provides the well-posedness, gradient bound, and semiconcavity estimates for the viscous equation that enter Assumption (A.1) and inequality (6).","marker":"[Lio84]"},{"why":"Introduces sup-convolution with the $O(\\delta)$ approximation and $-1/\\delta$-semiconvexity properties used to regularise $\\phi^0$.","marker":"[LL86]"},{"why":"Supplies existence, positivity, and regularity of the Green function for the linear parabolic flow used in Proposition 2.4.","marker":"[Aro68]"},{"why":"Gives the entropy-derivative and Fisher-information identities used to differentiate the entropy of $\\mu^{\\epsilon,\\delta}$ along the Fokker–Planck flow.","marker":"[Bog+22]"},{"why":"Provides the semiconcavity theory and the concavifying effect used to handle merely Lipschitz terminal data in the quadratic case.","marker":"[CS04]"},{"why":"Established the optimal $O(\\epsilon\\log\\epsilon)$ rate in dimension one for quadratic Hamiltonians via the Cole–Hopf formula; the present paper extends the optimal rate to every dimension.","marker":"[Qia+24]"},{"why":"Proves the linear-algebra estimate relating the Hessian of $H$ to the trace of the solution's Hessian, used in Proposition 2.4-(ii).","marker":"[Cha23]"},{"why":"Shows in a related stochastic-control setting that a global rate better than order $\\epsilon\\log\\epsilon$ cannot be expected because of singularities, motivating the sharpness result.","marker":"[Fle71]"}],"fun_headline_variants":["Sharp rate: ε log ε for convex Hamilton–Jacobi","Optimal viscosity rate: ε log ε, not √ε","For convex HJ, convergence is ε log ε, not √ε","Vanishing viscosity: sharp rate is ε log ε","Proven optimal: ε log ε beats √ε for convex HJ"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof depends on the terminal datum being semiconcave — curvature bounded above by a constant — and on that bound being inherited by every solution $\\phi^\\epsilon$ uniformly in $\\epsilon$; if this propagation fails, the entropy estimate no longer controls the Laplacian term and the argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Sharp rate: ε log ε for convex Hamilton–Jacobi","Optimal viscosity rate: ε log ε, not √ε","For convex HJ, convergence is ε log ε, not √ε","Vanishing viscosity: sharp rate is ε log ε","Proven optimal: ε log ε beats √ε for convex HJ"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000899,"raw_usage":{"total_tokens":3897,"prompt_tokens":999,"completion_tokens":2898,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":2812}},"tokens_in":615,"tokens_out":2898,"duration_ms":24581,"temperature":1.0,"reasoning_tokens":2812,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:39:25.745664+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the explicit datum $g(y)=-|y|$ in dimension $d=2$, quadratic Hamiltonian $H(p)=|p|^2/2$, and $x=0$, Proposition 3.4 predicts $\\phi^\\epsilon_t(0)-\\phi^0_t(0)=\\frac12\\epsilon\\log\\epsilon+O(\\epsilon)$. Evaluating the explicit Cole–Hopf integral (14) at small $\\epsilon$ and computing the ratio $(\\phi^\\epsilon-\\phi^0)/(\\epsilon\\log\\epsilon)$ as $\\epsilon\\to0$ should give $1/2$; any other limit would refute the claimed optimal rate and constant.","supporting_citations":[],"review_version":1}