{"id":"e22c0e11-b761-44ac-ad1f-e1b838dec362","arxiv_id":"2505.07015","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"In one dimension, the 2-Wasserstein distance between the nonlocal and local porous medium solutions is bounded by a constant times the square root of the smoothing parameter.","lead":"This paper proves a rate bound for how fast a nonlocal version of the porous medium equation approaches the standard one in one dimension. The proof is short, uses gradient flow inequalities, and works with much weaker assumptions than earlier work.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's core step, combining two EVIs into a differential inequality for W2^2(uε,u), is asserted without the required comparison/chain-rule lemma; the gap is fillable but must be made explicit.","rationale":"The central estimate is the differential inequality for W2^2(uε(t),u(t)); everything after it is a straightforward integration using estimate (D). The single most load-bearing point is therefore the derivation of that differential inequality from the two EVIs, and the paper compresses the necessary comparison/chain-rule lemma into the phrase 'Combining (2.3) and (2.4) we get'. I checked the algebra: with H(t,s)=W2^2(uε(t),u(s)), the first EVI bounds the t-derivative and the second bounds the s-derivative, so the displayed inequality is formally correct. What is missing is the justification that the EVI permits time-dependent test measures and that the chain rule is legitimate. This is a genuine gap in presentation, not a demonstrated counterexample, and it is very likely fixable. Estimate (D) is also load-bearing, but the paper's entropy-identity sketch supports it and it is imported from prior work; I do not see a concrete failure there. The numerical sharpness claims are overstated, but they do not bear on the theorem's validity. Hence my concern matches the reader's weakest_assumption and does not move the verdict.","tokens_in":10789,"tokens_out":21594,"duration_ms":215056,"concrete_test":"Write out the one-page comparison lemma: define H(t,s)=W2^2(uε(t),u(s)), use (2.3) to bound ∂_t H, use the analogous EVI for u (with the roles of the two time variables swapped) to bound ∂_s H, and apply the chain rule along the diagonal s=t. Verify whether the EVI in [3, Theorem 11.1.4] has the null set independent of σ; if not, prove the diagonal inequality by first testing against a countable dense set {σ_k}⊂D(F) and passing to the limit σ_k→u(t) using the absolute continuity of both flows. If this chain-rule argument goes through, Theorem 1.1 is proved; if a term is missing or the admissible test measures must be time-independent, then the stated √ε rate would not follow from the displayed argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 hinges on the unstated comparison step between (2.3) and (2.4). For H(t,s)=W2^2(uε(t),u(s)), the EVI for uε gives (1/2)∂_t H(t,s) ≤ Eε(u(s))-Eε(uε(t)), while the EVI for u gives (1/2)∂_s H(t,s) ≤ E(uε(t))-E(u(s)). Adding these and setting s=t yields the inequality used in the paper: (1/2)d/dt W2^2(uε(t),u(t)) ≤ Eε(u(t))-Eε(uε(t))+E(uε(t))-E(u(t)). The paper does not state this lemma, nor does it justify that the EVI from [3, Theorem 11.1.4] allows test measures depending on t: as usually stated, the inequality holds for every fixed σ∈D(F) for a.e. t, with the exceptional set possibly depending on σ. Without such a justification the central bound is not established. The omitted step is likely repairable, since both flows are absolutely continuous in W2 and one can argue through a countable dense subset of the domain, but as written the proof is incomplete. A secondary premise is estimate (D), imported from [23,24]; the paper's log-entropy sketch is credible and I do not see a problem there. The numerical overclaim about sharpness does not affect the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a quantitative convergence rate for the nonlocal-to-local limit in one dimension between the nonlocal porous-medium-type equation ∂t uε = ∂x(uε ∂x(Wε ∗ uε)), with Wε the exponentially decaying Laplace kernel, and the local quadratic porous medium equation ∂t u = ∂xx(u²/2). The main result, Theorem 1.1, asserts that W2(uε(t,·), u(t,·)) ≤ C√ε for all t ∈ [0,T] under only u0 ∈ P2(R) ∩ L∞(R). The proof combines the Evolutionary Variational Inequality characterization of both gradient flows with uniform a priori estimates imported from [23,24]; a finite-volume numerical section illustrates the rate and suggests possible improvements for small ε. The paper is short, clearly written, and the overall strategy is attractive, but the central EVI comparison step is not fully justified as written.","tokens_in":11128,"tokens_out":19269,"duration_ms":187670,"significance":"If the gap described below is repaired, this is a clean and useful result: it obtains the √ε rate in the Wasserstein distance under weaker regularity assumptions than the concurrent work [1] (no W^{1,∞} condition and no torus), and the EVI strategy is transparent. The imported a priori estimate (D) of Proposition 2.1 is adequately supported by the log-entropy computation sketched there, and the finite-volume experiments, while not a proof, are consistent with the theorem. The main missing piece is a standard but unstated comparison lemma for two EVI gradient flows; once supplied, the proof should be complete.","major_comments":[{"comment":"The step 'Combining (2.3) and (2.4)' is the central step of the proof and is not justified as written. First, Eq. (2.4) differentiates W2^2(u(t,·), uε(s,·)) with respect to t, whereas the needed inequality differentiates H(t,s)=W2^2(uε(t,·),u(s,·)) with respect to the second argument s; the displayed derivative variable appears to be a typo. Second, even after that correction, applying Theorem 2.3 with the time-dependent test measures u(s) and uε(t) requires a comparison lemma for two EVI gradient flows: for a.e. t, (1/2)(d/dt)W2^2(μ_t,ν_t) ≤ F(ν_t)-F(μ_t)+G(μ_t)-G(ν_t), with F,G the two energy functionals. This lemma is not stated or cited, and the exceptional-null-set issue for test measures depending on the evolution variable must be addressed, for example by using absolute continuity and approximation by a countable dense subset of the domain. Since the final bound relies entirely on this inequality, Theorem 1.1 is not fully proved as written; the gap is standard and likely repairable.","section":"Section 2.3, Eqs. (2.3)-(2.4)"}],"minor_comments":[{"comment":"The displayed identity for the energy dissipation is missing the right-hand side; it should be written as an equality to 0.","section":"Section 2.1, Eq. (2.1)"},{"comment":"The paper does not explicitly verify the hypotheses of the quoted characterization, namely properness, lower semicontinuity and λ-geodesic convexity of E and Eε, and density of their domains. These are standard for the two functionals considered, but a brief verification would make the application self-contained.","section":"Section 2.2, Theorem 2.3"},{"comment":"The numerical upwind flux appears to contain an index error: the second term uses v^-_{i-1/2} and (u_i^ε)_L at the interface i+1/2; the standard upwind reconstruction would use v^-_{i+1/2} and the left reconstructed value of the neighboring cell. Please check and correct.","section":"Section 3, Eq. (3.1b)"},{"comment":"The statement that the numerical experiments demonstrate that (1.4) is 'the best, global in ε estimate' goes beyond what finite-volume experiments can establish; it should be phrased as an observation or conjecture, particularly since the improved small-ε rate is deferred to the forthcoming paper [6]. The comparison of exponents in the sentence containing 'ε<√ε<ε^{1/4}' is also confusing for ε>1 and should be clarified.","section":"Section 4.1"},{"comment":"The geodesic-convexity proof is written for the case where the optimal transport is given by a map T, i.e. for absolutely continuous ρ0; since Theorem 2.3 is applied on all of P2(R), a sentence on approximation of general measures by absolutely continuous ones is needed.","section":"Proposition 2.4"}],"recommendation":"major_revision","confidential_remarks":"The omitted comparison lemma is the only substantive obstacle; I expect the authors can repair it quickly with a standard argument, so I recommend major revision rather than rejection. The numerical overclaim in Section 4.1 should be softened before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this is a nice, compact paper proving a √ε Wasserstein rate for a particular nonlocal-to-local limit in 1D via EVI gradient flow ideas. The rate itself was already available from Amassad–Zhou for a broader kernel class, so the novelty is not the rate. What is genuinely new is the relaxation of the initial datum to P2∩L∞ on the whole line (no W^{1,∞} on the torus) and a much simpler proof technique. The numerical section adds some evidence about sharpness, but it overstates what it shows. The analytic proof is essentially correct. The stress-test note is right that the combination of the two EVIs is not fully justified as written. The EVI holds for every fixed test measure but with an exceptional set that can depend on the test; to set s=t one needs an explicit comparison/chain-rule argument, or a justification via absolute continuity and approximation by a countable dense set. That is a genuine but small gap, and it is repairable. The referee should ask for it to be stated. Everything else checks out: estimate (D) imported from Perthame–Vauchelet et al. supplies the ε factor, and the log-entropy sketch in Proposition 2.1 is plausible and supported by the cited work. Minor issues: the paper says the numerics 'demonstrate' that the estimate is the best global-in-ε estimate. That is too strong. The experiments show linear behavior for small ε and slower behavior for large ε, which suggests sharpness, but a few finite-volume runs on bounded domains with specific initial data do not demonstrate it. Also, the forthcoming work [6] is cited as confirmation, but it is not evidence yet. The paper is honest about the comparison to Amassad–Zhou, which it cites clearly. Who is this for? People working on blob methods, nonlocal approximations of degenerate diffusion, or gradient flows in Wasserstein space. It is a clean proof-of-concept that the EVI route gives rates under weak assumptions. It deserves a serious referee. I would recommend acceptance after minor revision: state the missing lemma, and temper the numerics language.","headline":"Clean EVI proof of the √ε rate on the whole line with L1∩L∞ data; the rate is pre-existing, but the method and assumptions are new, and the paper is sound once the EVI comparison lemma is stated explicitly.","tokens_in":639,"tokens_out":3089,"would_cite":true,"duration_ms":50882,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A15","35Q70","35B40","65M08"],"pacs":[],"model":"deepseek-v4-flash","headline":"In one dimension, the nonlocal porous medium equation converges to the local equation at rate $O(\\sqrt{\\varepsilon})$ in 2-Wasserstein distance for initial data in $\\mathcal{P}_2(\\mathbb{R})\\cap L^\\infty(\\mathbb{R})$.","keywords":["nonlocal-to-local limit","porous medium equation","Wasserstein distance","gradient flows","evolutionary variational inequality","blob method","finite volume method","rate of convergence"],"falsifier":"Solve (1.1) and (1.3) numerically with $\\Delta x\\le\\varepsilon$ for a Gaussian initial datum and plot $W_2(u^\\varepsilon,u)/\\sqrt{\\varepsilon}$; if this ratio is unbounded as $\\varepsilon\\to0$ for any fixed time $t>0$, the claimed constant $C$ in Theorem 1.1 cannot exist.","tokens_in":10630,"feed_emoji":"📐","tokens_out":13173,"duration_ms":111957,"temperature":0.7,"pith_summary":"This paper proves that in one spatial dimension the solution of the nonlocal porous-medium equation $\\partial_t u^\\varepsilon-\\partial_x(u^\\varepsilon\\partial_x W^\\varepsilon*u^\\varepsilon)=0$ converges to the solution of the quadratic porous medium equation $\\partial_t u-\\partial_{xx}(u^2/2)=0$ at a quantified rate: the $2$-Wasserstein distance satisfies $W_2(u^\\varepsilon(t,\\cdot),u(t,\\cdot))\\le C\\sqrt{\\varepsilon}$ for every $t\\in[0,T]$, whenever the initial datum lies in $\\mathcal{P}_2(\\mathbb{R})\\cap L^\\infty(\\mathbb{R})$. The constant $C$ depends only on the initial datum, the time horizon, and the kernel, never on $\\varepsilon$. The proof is short, combining the evolutionary variational inequality characterization of both gradient flows with uniform a priori estimates. The result matters because the nonlocal equation underlies deterministic particle or blob approximations of the porous medium equation, and the rate makes the approximation error explicit without any differentiability assumption on the initial datum.","feed_headline":"Nonlocal diffusion to porous medium: rate √ε proved","feed_subtitle":"Two variational inequalities and one regularity estimate give a uniform error bound in the 2-Wasserstein metric.","key_machinery":"The machinery is the evolutionary variational inequality (EVI), a gradient-flow characterization asserting that for every competitor $\\sigma$ in the domain, $F(\\mu_t)+\\frac12\\frac{d}{dt}W_2^2(\\mu_t,\\sigma)\\le F(\\sigma)$ for almost every $t$. Both the nonlocal and local equations are EVI gradient flows for their respective energies, and the proof combines the two EVIs with the elliptic identity $-\\varepsilon^2\\partial_{xx}W^\\varepsilon+W^\\varepsilon=\\delta_0$ and the imported estimate that $\\varepsilon\\partial_{xx}(W^\\varepsilon*u^\\varepsilon)$ is uniformly bounded in $L^2((0,T)\\times\\mathbb{R})$. The identity translates the $L^2$ distance between $u^\\varepsilon$ and its mollification into $\\varepsilon^2$ times a second derivative, and the boundedness of that second derivative is what converts the energy error into an $O(\\varepsilon)$ term, ultimately an $O(\\sqrt{\\varepsilon})$ Wasserstein rate.","core_discovery":"The paper's central claim is that the nonlocal-to-local limit has order $\\sqrt{\\varepsilon}$ in $W_2$, under minimal integrability assumptions. The mechanism is a comparison of two Wasserstein gradient flows: the nonlocal energy $E^\\varepsilon[\\rho]=\\frac12\\int_\\mathbb{R}\\rho\\,W^\\varepsilon*\\rho\\,dx$ and the local energy $E[\\rho]=\\frac12\\int_\\mathbb{R}\\rho^2\\,dx$ are both $\\lambda$-geodesically convex with $\\lambda=0$, so their flows are characterized by the same evolutionary variational inequality. Inserting one flow as the competitor in the other's inequality and adding the two inequalities bounds $\\frac12\\frac{d}{dt}W_2^2(u^\\varepsilon,u)$ by the energy differences $E^\\varepsilon(u)-E(u)$ and $E(u^\\varepsilon)-E^\\varepsilon(u^\\varepsilon)$. The first difference is nonpositive by the elementary inequality $u(x)u(y)\\le \\frac12u(x)^2+\\frac12u(y)^2$; the second is bounded by $\\frac12\\|u^\\varepsilon\\|_{L^2}\\|u^\\varepsilon-u^\\varepsilon*W^\\varepsilon\\|_{L^2}$, which via the identity $u^\\varepsilon-u^\\varepsilon*W^\\varepsilon=-\\varepsilon^2\\partial_{xx}(W^\\varepsilon*u^\\varepsilon)$ and the a priori estimate $\\|\\varepsilon\\partial_{xx}(W^\\varepsilon*u^\\varepsilon)\\|_{L^2}\\le C$ becomes $C\\varepsilon$. Integrating in time yields the $\\sqrt{\\varepsilon}$ rate.","pith_inferences":["The unstated comparison lemma for $\\lambda=0$ EVI gradient flows is the real workhorse; stated explicitly, the same two-line energy-difference argument would give rates for any pair of nonlocal/local gradient flows that admit a matching second-order estimate.","The numerical transition from $\\varepsilon$ to $\\sqrt{\\varepsilon}$ under no-flux boundaries hints at a boundary-layer mechanism: one could test whether the $W_2$ error is dominated by a layer of width $\\varepsilon$ near the boundary, in which case interior estimates might still be linear in $\\varepsilon$.","Estimate (D) is imported rather than proved; deriving it from scratch for more general kernels, or finding a kernel for which it fails, would delimit exactly how far the EVI-comparison method extends.","The one-dimensional proof uses monotonicity of optimal transport maps to show geodesic convexity of $E^\\varepsilon$; a higher-dimensional analogue would need a different convexity argument, consistent with the alternative route used in the broader recent treatment of the problem."],"forward_implications":["For every initial datum in $\\mathcal{P}_2(\\mathbb{R})\\cap L^\\infty(\\mathbb{R})$, the blob/particle approximation to the quadratic porous medium equation converges in $W_2$ at rate $O(\\sqrt{\\varepsilon})$, uniformly on any finite time interval.","The rate is global in $\\varepsilon$: the bound $C\\sqrt{\\varepsilon}$ holds for all $\\varepsilon>0$, not only in an asymptotic regime, because no restriction on $\\varepsilon$ is imposed in the proof.","No derivative or higher regularity of the initial datum is needed; finite second moment, $L^\\infty$ boundedness, and integrability suffice.","The numerical simulations indicate that for small $\\varepsilon$ the observed rate is closer to $\\varepsilon$ than to $\\sqrt{\\varepsilon}$, so the theorem's bound may not be sharp in the small-$\\varepsilon$ regime while remaining the best global-in-$\\varepsilon$ estimate.","Under no-flux boundary conditions, the numerical experiments suggest the rate can degrade to $\\sqrt{\\varepsilon}$ once the solution touches the boundary, showing that boundary interaction can erode the faster rate seen on the whole line."],"supporting_citations":[{"why":"Supplies the EVI characterization of Wasserstein gradient flows (Theorem 2.3), which is the core comparison tool of Section 2.3.","marker":"[3]"},{"why":"Provides the uniform a priori estimates (A)-(E) for solutions of (1.1), including the estimate (D) that produces the $\\varepsilon$ factor.","marker":"[23]"},{"why":"Companion source for the same uniform bounds used in Proposition 2.1.","marker":"[24]"},{"why":"Gives the one-dimensional geodesic-convexity argument for the nonlocal energy $E^\\varepsilon$ used in Proposition 2.4.","marker":"[11]"},{"why":"Identifies the EVI gradient flows with the PDE solutions of (1.1) and (1.3), letting the proof apply the EVIs to $u^\\varepsilon$ and $u$.","marker":"[2]"},{"why":"Supplies the maximum-principle argument for the uniform $L^\\infty$ bound within Proposition 2.1.","marker":"[26]"},{"why":"Provides the standard geodesic convexity of the local quadratic energy $E$ used in Proposition 2.4.","marker":"[28]"}],"fun_headline_variants":["√ε rate for nonlocal-to-local limit via EVI","Nonlocal diffusion converges at √ε in Wasserstein metric","Simple proof: nonlocal-to-local convergence rate √ε","Wasserstein gradient flow comparison gives √ε rate","One-dimensional porous medium limit: √ε quantified"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes an unstated comparison principle that turns the two variational inequalities into a bound on how fast the two flows separate, and it imports a second-order regularity estimate for the nonlocal solutions; if either ingredient gives way, the proof collapses.","fun_headline_variants_meta":{"raw":{"variants":["√ε rate for nonlocal-to-local limit via EVI","Nonlocal diffusion converges at √ε in Wasserstein metric","Simple proof: nonlocal-to-local convergence rate √ε","Wasserstein gradient flow comparison gives √ε rate","One-dimensional porous medium limit: √ε quantified"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000255,"raw_usage":{"total_tokens":1591,"prompt_tokens":984,"completion_tokens":607,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":528}},"tokens_in":600,"tokens_out":607,"duration_ms":6479,"temperature":1.0,"reasoning_tokens":528,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:28:09.792849+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve (1.1) and (1.3) numerically with $\\Delta x\\le\\varepsilon$ for a Gaussian initial datum and plot $W_2(u^\\varepsilon,u)/\\sqrt{\\varepsilon}$; if this ratio is unbounded as $\\varepsilon\\to0$ for any fixed time $t>0$, the claimed constant $C$ in Theorem 1.1 cannot exist.","supporting_citations":[{"cited_title":"Perthame, M","cited_arxiv_id":null,"evidence_quote":"Provides the uniform a priori estimates (A)-(E) for solutions of (1.1), including the estimate (D) that produces the $\\varepsilon$ factor."},{"cited_title":"Perthame and N","cited_arxiv_id":null,"evidence_quote":"Companion source for the same uniform bounds used in Proposition 2.1."},{"cited_title":"Math., 231(1):306–327, 2012","cited_arxiv_id":null,"evidence_quote":"Gives the one-dimensional geodesic-convexity argument for the nonlocal energy $E^\\varepsilon$ used in Proposition 2.4."},{"cited_title":"Ambrosio and N","cited_arxiv_id":null,"evidence_quote":"Identifies the EVI gradient flows with the PDE solutions of (1.1) and (1.3), letting the proof apply the EVIs to $u^\\varepsilon$ and $u$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the maximum-principle argument for the uniform $L^\\infty$ bound within Proposition 2.1."},{"cited_title":"Villani.Topics in optimal transportation, volume 58 ofGraduate Studies in Mathematics","cited_arxiv_id":null,"evidence_quote":"Provides the standard geodesic convexity of the local quadratic energy $E$ used in Proposition 2.4."}],"review_version":1}