{"id":"7e0a792e-7f7c-42bb-b81a-0854ca80d175","arxiv_id":"2501.04311","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"This paper proves contraction and t^{-1/4} time decay, up to a shift, for large perturbations of planar viscous shocks in multi-D scalar conservation laws with strictly convex flux.","lead":"A planar viscous shock in a multi-dimensional scalar conservation law is shown to attract large L2 perturbations up to a time-dependent shift, with an explicit t^{-1/4} decay rate when the perturbation is also in L1. The result extends known one-dimensional and Burgers-type estimates to general strictly convex fluxes with periodic transverse directions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"I read the paper in good faith and tried to locate the weakest point in the proof of the central decay claim. The argument is long and technically intricate, but the main steps are internally consistent: the shift ODE in Section 3.2 is designed so that the relative-entropy derivative is non-positive on both regions |Y| >= epsilon^2 and |Y| <= epsilon^2; Proposition 3.1 is proved from Proposition 4.1 and Proposition 4.2; and the decay in Appendix B follows from the contraction estimate together with the L1-contraction property and the Gagliardo-Nirenberg inequality of [7]. The proof of Proposition 2.1 is the new technical heart, and I checked the constant-W case, the Cauchy-Schwarz step in Lemma 2.4, and the Case I/Case II splitting; I did not find a falsifying example or a clearly invalid inequality. The reader's weakest assumption points to Proposition 2.1 and to Lemma 2.2; I agree most strongly about Lemma 2.2 because the multi-dimensional extension of the time-derivative identity is not shown in the text and is essential. The noted sign and denominator typos, such as epsilon^2 versus epsilon^{3/2} in Proposition 2.1, are real presentation issues but do not by themselves undermine the argument. Since the residual risk is self-containment rather than a demonstrated flaw, I do not move the verdict: CONDITIONAL remains appropriate, and an independent derivation of Lemma 2.2 would settle the main remaining uncertainty.","tokens_in":29650,"tokens_out":34991,"duration_ms":314218,"concrete_test":"Independently re-derive Lemma 2.2 for R x T^2: expand d/dt integral a(xi) eta(u(t, xi+X(t), x') | u_tilde(xi)) dxi dx' using (2.2), explicitly tracking q2, q3 and the x' components of div(mu(u) grad eta'(u)); verify that all transversal boundary terms vanish by periodicity and that the surviving terms equal Y(u_X) dot X + B(u_X) - G(u_X). As a regression check, set X=0 and take u independent of x' to confirm that the formula reduces to [8, Lemma 2.2]. If the identity holds, the contraction and decay estimates follow as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I could not identify a concrete mathematical error that would falsify Theorem 1.2. The overall structure is coherent: Proposition 2.1 supplies the multi-dimensional nonlinear Poincare inequality, Proposition 3.1 converts it into a differential inequality, and Appendix B obtains the t^{-1/4} decay from L1 contraction and the Gagliardo-Nirenberg interpolation. The most load-bearing unverified input is Lemma 2.2, imported without proof from the 1D paper [8]. In R x T^2, the derivation of the weighted relative-entropy time derivative involves transversal flux terms q2, q3 and the full gradient in the diffusion term; if a transversal boundary term survives or a sign is wrong, the key estimate (3.3) fails at the first step. Several other statements are also deferred with 'proof omitted' or 'essentially same as in [8]' (Lemma 4.2, Lemma 4.3, parts of Proposition 4.2), so the manuscript is not self-contained. Proposition 2.1 has minor notational inconsistencies, such as epsilon^2 versus epsilon^{3/2} in the transversal diffusion denominator, but these appear fixable and the Case I/Case II estimates are plausible; simple constant-W checks do not produce a counterexample. The central claim is therefore conditionally acceptable, with the residual risk concentrated in the unproved multi-dimensional identity and the imported localization lemmas.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies multi-dimensional scalar viscous conservation laws with strictly convex flux on R x T^2, around a planar viscous shock of small strength. The main results are Theorem 1.1, a contraction property for large L^2 perturbations up to a dynamical shift, and Theorem 1.2, a quantitative t^{-1/4} decay rate in L^2 when the initial perturbation is also in L^1. The proof reduces Theorem 1.1 to Proposition 3.1 via a shift ODE, then splits the analysis into a tail/truncation regime and a small-|η'(u)-η'(ũ)| regime. The latter uses a multi-dimensional nonlinear Poincaré inequality (Proposition 2.1). Theorem 1.2 is obtained in Appendix B by combining the contraction estimate with L^1 contraction and a Gagliardo-Nirenberg interpolation inequality. The manuscript is clearly written and the overall architecture is coherent, but several load-bearing ingredients are imported from earlier papers without full verification, and there is a gap between the assumptions of Theorem 1.2 and the quantities used in its proof.","tokens_in":29907,"tokens_out":6708,"duration_ms":67792,"significance":"If the main results are correct, this is the first quantitative large-perturbation decay estimate for planar viscous shocks in a multi-dimensional scalar viscous conservation law, extending the one-dimensional results of Kang [8] and the multi-D Burgers result in [9]. The proof introduces a genuinely new ingredient: a multi-dimensional nonlinear Poincaré inequality with L^∞ constraint on R x T^2, used to control cubic terms by the diffusion along both the shock and transversal directions. The paper also provides a detailed proof of the algebraic lemma behind that inequality (Appendix A). These are substantial contributions. However, the manuscript currently depends critically on unproved or only sketched imported lemmas, so the significance will be fully realized only after those gaps are closed.","major_comments":[{"comment":"Lemma 2.2 is the central relative-entropy identity on which Proposition 3.1 and Theorem 1.1 rest, but its proof is omitted with only the statement 'the proof is essentially the same as in [8, Lemma 2.2]'. This identity contains the transversal flux components q2 and q3 and the full gradient in the diffusion term, so it is not a purely one-dimensional computation. The time derivative of ∫ a η(uX|ũ) must be verified in detail for the R x T^2 setting; if any transversal boundary term survives or a sign is incorrect, the key estimate (3.3) fails at the first step. This is a load-bearing gap and should be addressed before the contraction claim is accepted.","section":"§2.2, Lemma 2.2"},{"comment":"Theorem 1.2 assumes only u0 - ũ ∈ L^1 ∩ L^∞(Ω), but the proof in Appendix B explicitly uses ‖u0 - ũ‖_{L^2(Ω)} and the L^2 contraction estimate (1.7), which requires the L^2 hypothesis of Theorem 1.1. On R x T^2, L^1 ∩ L^∞ does not imply L^2, so the stated class of initial data is insufficient. Either Theorem 1.2 should include u0 - ũ ∈ L^2, or the proof must establish finite relative entropy and finite L^2 size directly from L^1 ∩ L^∞; the latter seems unlikely without additional assumptions. This is a load-bearing issue for the theorem as stated.","section":"Theorem 1.2 and Appendix B"},{"comment":"There are notational inconsistencies in the statement and proof of the central nonlinear Poincaré inequality. In Lemma 2.4 the domain is written as (0,1) x T^2, but the estimate is stated at (z,y) with y ∈ T; the proof then integrates over T^2. In the proof of Proposition 2.1, the line applying Lemma 2.4 writes √(L(z)-L(1-z)) although Lemma 2.4 gives √(L(z)+L(1-z)). Also, the transversal diffusion coefficient appears as (1-δ)/ε^{3/2} in the statement, as (1-δ)/ε^2 in one displayed formula inside the proof, and later is compared with ε^{3/2}. These inconsistencies are likely fixable, but because Proposition 2.1 is a key new result, the final version must give a clean, self-consistent statement and proof.","section":"§2.5, Proposition 2.1 and Lemma 2.4"},{"comment":"Several steps that control the region |η'(u)-η'(ũ)| ≥ δ1 are deferred: Lemma 4.2 is stated with 'we omit the proof', Lemma 4.3 is said to follow 'easily' with details in [8,12], and the proof of Proposition 4.2 repeatedly refers to [8, Proposition 4.5] for the main estimates, including (4.8) and (4.10). These lemmas are not cosmetic; they provide the uniform bound on Y and the comparison between the truncated and original functionals that are essential for Case I and Case II of the conclusion. The manuscript should either include complete proofs or give a detailed verification that each one-dimensional argument extends to R x T^2, including the transversal diffusion and the periodic directions.","section":"§4.2, Lemmas 4.2 and 4.3; §4.2, proof of Proposition 4.2"}],"minor_comments":[{"comment":"There is a typographical error: 'Theroem' should be 'Theorem'.","section":"§1.1, Remark 1.2"},{"comment":"In the display for B(u), the term `(η'(u)-η(ũ))` should presumably be `(η'(u)-η'(ũ))`; the current text is inconsistent with the surrounding factors.","section":"§2.2, Lemma 2.2"},{"comment":"The notation `T2` is sometimes used for T^2, and the proof of Lemma 2.4 uses variables (z,y) with y ∈ T while integrating over T^2; this should be harmonized for readability.","section":"§2.5, proof of Proposition 2.1"},{"comment":"The word 'Propotision' appears in 'Propotision 4.5'; it should be 'Proposition'.","section":"§4.2, after Lemma 4.2"},{"comment":"In the interpolation step, the exponent `B^{2/4}` is written where `B^{1/2}` is meant; also the constants `C0` and `C∗` are conflated in several inequalities, which makes the dependence of constants hard to track.","section":"Appendix B, proof of Theorem 1.2"},{"comment":"The estimate in the statement is only proved for f ∈ C^1, but it is applied to W that is merely L^2 with √(z(1-z))∂zW ∈ L^2; the approximation step should be mentioned explicitly so the reader knows the lemma applies to the relevant functions.","section":"§2.5, Lemma 2.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is a natural continuation of the authors' earlier work and the method is appropriate. My main concern is not novelty but rigor: the proof outsources several critical identities and estimates to [8] and [9], and the statement of Theorem 1.2 has an assumption/premise mismatch that needs to be fixed. I would be willing to accept after a careful revision that addresses the unproved multi-dimensional identity and the L^2 assumption issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: this is a serious, technically demanding extension of the a-contraction with shifts program to multi-D scalar viscous conservation laws on R x T^2. The genuinely new result is Theorem 1.2: for strictly convex flux, large L1 intersect L∞ perturbations of a small planar viscous shock converge in L2 at rate t^{-1/4}, up to a uniformly bounded shift. That rate is not in the cited literature; the Burgers analogue is the authors' companion paper and the 1D analogue is Kang's 2021 paper. The proof machinery is the established framework, so the novelty is incremental but real.\n\nWhat the paper does well: the central reduction is coherent. Theorem 1.1 is reduced to Proposition 3.1, which is split into a Taylor expansion regime and a truncation regime. Proposition 2.1, the multi-D nonlinear Poincare inequality, is proved in detail and is the genuine technical heart: the change of variables to z, Lemma 2.4's pointwise estimate, and the control of the cubic term by diffusion look sound. Appendix A supplies the needed algebra and Appendix B sketches the Gagliardo-Nirenberg interpolation; the overall structure is checkable.\n\nSoft spots, in proportion: the paper is not self-contained at several load-bearing points. Lemma 2.2, the weighted relative entropy time-derivative identity, is imported without proof from [8]. In R x T^2 it involves transversal flux terms and the full gradient in the diffusion term, so this is not a trivial transcription; the stress-test worry about a missing transversal boundary term is reasonable, and if the identity fails or a sign is off, (3.3) fails at the first step. I could not find an actual error, and simple constant-W checks do not produce a counterexample. Lemma 4.2 and Lemma 4.3 are likewise deferred with 'essentially same as in [8]'. There is a minor notational inconsistency in Proposition 2.1 (epsilon^2 versus epsilon^{3/2} in the transversal diffusion denominator), and the reader's sign typo in that proposition matches a discrepancy in the proof; both appear fixable. The relation to [14] (Kang-Vasseur-Wang) should be clarified: the contraction part of Theorem 1.1 may overlap with prior work, and the authors should state precisely what is new. The L1 contraction is used as a black box, which is fine for scalar laws, but the case split in Appendix B.2 is sketched rather than fully justified; that is minor.\n\nWho this is for: researchers in shock stability, especially users of a-contraction methods. It deserves a serious referee: the main theorem is new, the proof is plausible and largely checkable, and the gaps are concentrated and likely fillable. I would not desk reject; I would send it to a knowledgeable PDE analyst with instructions to verify Lemma 2.2 and Proposition 2.1 carefully.","headline":"A credible multi-D extension of the a-contraction framework giving a new t^{-1/4} decay for large perturbations of planar viscous shocks; worth refereeing despite the imported lemmas.","tokens_in":30436,"tokens_out":1844,"would_cite":true,"duration_ms":16753,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35L65","35L67","35B35","35B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"Large perturbations of multi-D viscous shocks decay at rate $t^{-1/4}$.","keywords":["viscous conservation law","planar viscous shock","large perturbation","contraction","relative entropy","Poincaré inequality","time decay","a-contraction with shifts"],"falsifier":"Take the Burgers flux $f(u)=u^2/2$ in (1.1) on $\\mathbb{R}\\times\\mathbb{T}^2$, choose a large initial perturbation $u_0-\\tilde u \\in L^1\\cap L^\\infty$, and compute numerically the shifted $L^2$ distance $\\|u(t,\\cdot+X(t),\\cdot)-\\tilde u(\\cdot-\\sigma t)\\|_{L^2}$ over time; observing a decay rate slower than $t^{-1/4}$ or a shift $X(t)$ that grows without bound would disprove Theorem 1.2.","tokens_in":29450,"feed_emoji":"📉","tokens_out":8155,"duration_ms":71042,"temperature":0.7,"pith_summary":"The paper proves that large, bounded perturbations of a planar viscous shock in a multi-dimensional scalar viscous conservation law—on $\\mathbb{R}\\times\\mathbb{T}^2$ with a strictly convex flux—contract under the flow to the shifted shock profile, and that such perturbations decay in $L^2$ at the rate $(1+t)^{-1/4}$ provided they also have finite $L^1$ mass. The contraction is measured by a weighted relative entropy and holds up to a time-dependent shift that solves a feedback ODE. This is, the authors claim, the first quantitative convergence result to a planar shock for large (not small) perturbations, extending the one-dimensional theory of [8] to the multi-dimensional setting. If correct, it reduces a stability question for arbitrarily large deviations to a single nonlinear Poincaré inequality and the construction of a suitable entropy.","feed_headline":"Large shock perturbations decay at rate $t^{-1/4}$","feed_subtitle":"A weighted relative entropy and a feedback shift make multi-dimensional viscous shocks stable to big L2 perturbations.","key_machinery":"The argument uses the method of $a$-contraction with shifts: a weight function $a(\\xi)=1+\\lambda(u_--\\tilde u(\\xi))/\\varepsilon$ makes the entropy production positive, and a shift $X(t)$ solves the feedback ODE $\\dot X=\\Phi_\\varepsilon(Y)(2|B|+1)$ that enforces the contraction. The key new estimate is the multi-dimensional nonlinear Poincaré-type inequality (Proposition 2.1) on $[0,1]\\times\\mathbb{T}^2$: it bounds the cubic error $\\int W^3$ by the weighted dissipation $\\int z(1-z)|\\partial_z W|^2 + \\varepsilon^{-3/2}\\int |\\partial_{x'}W|^2$ for all $W$ with $\\|W\\|_{L^\\infty}\\le 1/\\varepsilon$ and $\\int W^2\\le M$. This inequality, together with a truncation of the large values of $|\\eta'(u)-\\eta'(\\tilde u)|$, yields the contraction and then the decay through a Gagliardo-Nirenberg interpolation.","core_discovery":"The central result is Theorem 1.2: for any initial datum $u_0$ with $u_0 - \\tilde u \\in L^1\\cap L^\\infty(\\Omega)$, the solution $u(t,\\cdot)$ of (1.1) satisfies $\\|u(t,\\cdot+X(t),\\cdot)-\\tilde u(\\cdot-\\sigma t)\\|_{L^2(\\Omega)} \\le C/(1+t^{1/4})$ for all $t>0$, with a shift $X(t)$ that is absolutely continuous and uniformly bounded in time. Theorem 1.1 provides the underlying contraction: the weighted relative entropy $\\int a(\\xi)\\,\\eta(u^X|\\tilde u)\\,d\\xi dx'$ is non-increasing in time once the shift is chosen by the ODE (3.2). Together they give the first quantitative large-perturbation decay estimate for a planar viscous shock in multi-dimensions.","pith_inferences":["The same $a$-contraction machinery may yield decay for planar shocks in physical systems with a single convex entropy (for example, Navier-Stokes type systems), provided a multi-dimensional Poincaré inequality analogous to Proposition 2.1 is available.","The $t^{-1/4}$ rate appears to be an artifact of the Gagliardo-Nirenberg interpolation with $L^1$; with stronger integrability or a better lower bound on the diffusion, the method might yield a $t^{-1/2}$ rate.","A numerical experiment with the Burgers flux and a large rectangular bump perturbation could directly test whether the weighted relative entropy is monotone and whether the shift remains bounded, probing the sharpness of Proposition 2.1.","The periodic transverse directions are used to obtain explicit Poincaré constants; on a fully unbounded domain $\\mathbb{R}^3$ the cubic term would require a different control, so the proof does not automatically transfer outside the periodic setting."],"forward_implications":["The contraction inequality (1.5) gives a global-in-time $L^2$ bound for the shifted perturbation, so planar viscous shocks are stable to arbitrarily large $L^2$ perturbations in $\\mathbb{R}\\times\\mathbb{T}^2$.","If the initial perturbation is also in $L^1$, the shifted solution converges to the shock profile in $L^2$ with the explicit rate $t^{-1/4}$, while the shift $X(t)$ remains uniformly bounded.","The same decay rate holds in $\\mathbb{R}\\times\\mathbb{T}^{n-1}$ for every $n\\ge2$, as noted in Remark 1.3.","The uniform bound on the shift follows from the $L^1$ contraction of the scalar equation, so the decay statement is about the shape of the perturbation rather than about a shift drifting to infinity."],"supporting_citations":[{"why":"Supplies the one-dimensional contraction framework, the entropy construction (A1)-(A2), the relative entropy identity (Lemma 2.2), and the shift ODE whose multi-dimensional extension is proved here.","marker":"[8]"},{"why":"Provides the one-dimensional nonlinear Poincaré inequality and the algebraic Lemma 2.5 that Proposition 2.1 generalizes to $[0,1]\\times\\mathbb{T}^2$.","marker":"[12]"},{"why":"Gives the Poincaré-type inequality without constraints (Lemma 2.6) used to control the $L^2$ deviation of $W$ from its average in the proof of Proposition 2.1.","marker":"[18]"},{"why":"Supplies the $L^1$ contraction property (Lemma B.1) that yields the uniform bound on the shift $X(t)$.","marker":"[2]"},{"why":"Provides the Gagliardo-Nirenberg interpolation inequality on $\\mathbb{R}\\times\\mathbb{T}^2$ used in Appendix B to convert the diffusion estimate into the $t^{-1/4}$ $L^2$ decay.","marker":"[7]"},{"why":"Shows the analogous decay proof for the multi-dimensional Burgers equation, which Appendix B follows to prove Theorem 1.2.","marker":"[9]"}],"fun_headline_variants":["Multi-D shock decay: large perturbations at t^{-1/4}","First quantitative decay for planar shock perturbations","Weighted entropy tames large shock perturbations in multi-D","Planar shocks stabilize big L2 perturbations with decay","Large perturbation decay proven for multi-D viscous shocks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof leans on a multi-dimensional nonlinear Poincaré inequality (Proposition 2.1) that holds only for deviations $W$ with $\\|W\\|_{L^\\infty}\\le 1/\\varepsilon$ and under smallness constraints on the shock strength $\\varepsilon$ and the truncation parameter; if that inequality fails for the required class of perturbations, the contraction estimate and the decay theorem collapse.","fun_headline_variants_meta":{"raw":{"variants":["Multi-D shock decay: large perturbations at t^{-1/4}","First quantitative decay for planar shock perturbations","Weighted entropy tames large shock perturbations in multi-D","Planar shocks stabilize big L2 perturbations with decay","Large perturbation decay proven for multi-D viscous shocks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000743,"raw_usage":{"total_tokens":3283,"prompt_tokens":879,"completion_tokens":2404,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":2327}},"tokens_in":495,"tokens_out":2404,"duration_ms":18942,"temperature":1.0,"reasoning_tokens":2327,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:37:16.195090+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the Burgers flux $f(u)=u^2/2$ in (1.1) on $\\mathbb{R}\\times\\mathbb{T}^2$, choose a large initial perturbation $u_0-\\tilde u \\in L^1\\cap L^\\infty$, and compute numerically the shifted $L^2$ distance $\\|u(t,\\cdot+X(t),\\cdot)-\\tilde u(\\cdot-\\sigma t)\\|_{L^2}$ over time; observing a decay rate slower than $t^{-1/4}$ or a shift $X(t)$ that grows without bound would disprove Theorem 1.2.","supporting_citations":[{"cited_title":"L2-type contraction for shocks of scalar viscous conservatio n laws with strictly convex flux","cited_arxiv_id":null,"evidence_quote":"Supplies the one-dimensional contraction framework, the entropy construction (A1)-(A2), the relative entropy identity (Lemma 2.2), and the shift ODE whose multi-dimensional extension is proved here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the one-dimensional nonlinear Poincaré inequality and the algebraic Lemma 2.5 that Proposition 2.1 generalizes to $[0,1]\\times\\mathbb{T}^2$."},{"cited_title":"Hyperbolic conservation laws in continuum physics , vol","cited_arxiv_id":null,"evidence_quote":"Supplies the $L^1$ contraction property (Lemma B.1) that yields the uniform bound on the shift $X(t)$."},{"cited_title":"Stability of planar rarefaction waves for scalar viscous co nservation law under periodic perturbations","cited_arxiv_id":null,"evidence_quote":"Provides the Gagliardo-Nirenberg interpolation inequality on $\\mathbb{R}\\times\\mathbb{T}^2$ used in Appendix B to convert the diffusion estimate into the $t^{-1/4}$ $L^2$ decay."},{"cited_title":"$L^2$ decay for large perturbations of viscous shocks for multi-D Burgers equation","cited_arxiv_id":"2403.08445","evidence_quote":"Shows the analogous decay proof for the multi-dimensional Burgers equation, which Appendix B follows to prove Theorem 1.2."}],"review_version":1}