{"id":"b2c5e49d-2f1e-40e6-b96c-bfe38e1f1fa8","arxiv_id":"2608.06173","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For small viscosity and small wave strengths, the authors construct global smooth Navier-Stokes solutions that converge with explicit rates to the entropy solution of the Euler equations through two interacting same-family shocks.","lead":"This paper proves a vanishing viscosity limit for the compressible Navier-Stokes equations when two shock waves from the same characteristic family collide, producing an outgoing shock and a rarefaction wave. A generalist might read it because it establishes a rigorous convergence rate in a regime where the collision time grows as the inverse of the wave strength, a previously open case.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's quantifiers are unsupported: the proof requires ε ≤ Cδ but the statement lets ε and δ vary independently, and for δ ≪ ε the approximate collision time lies before the initial time.","rationale":"The reader's weakest-assumption analysis identifies exactly the load-bearing concern: the pre-collision interval [-t0, -Aε/δ²] requires ε ≲ δ, but Theorem 1.1 quantifies ε and δ independently. The stress-test pass confirms this by direct calculation: for δ = ε² the approximate collision time falls before the initial time, so the constructed initial data (2.13) does not exist as the starting point of Theorem 3.1. This is not a cosmetic or typographical issue; the entire anti-derivative closure on [−t0, −Aε/δ²] and the smallness of the exponential terms in Proposition 3.1 depend on the interval having positive length with δ/ε large. The concern is repairable by reformulating the theorem with the condition ε ≤ Cδ or by making ε0 depend on δ, so the appropriate verdict remains CONDITIONAL. Other potential issues noted in the full text, such as the special preparation of initial data, are disclosed in the proof and are not central to the validity of the argument. The core method is credible and the gap is a quantifier/stating issue rather than a fundamental mathematical obstruction, so no change to the reader's conditional verdict is needed.","tokens_in":50780,"tokens_out":8907,"duration_ms":83441,"concrete_test":"Set δ = ε². Then t0 = 1/(s_l - s_r) ∼ c/δ = cε^{-2} while Aε/δ² = Aε^{-3}. For ε < c/A, one has Aε^{-3} > cε^{-2}, so -Aε/δ² < -t0 and the pre-collision interval [-t0, -Aε/δ²] is empty; the initial data (2.13) in Theorem 3.1 is undefined. If the authors add the hypothesis ε ≤ Cδ, verify that the entire proof in Sections 3–6 goes through unchanged on the admissible set, and recompute the constants ε0 and δ0 so that ε0 ≤ Cδ0 and the interval condition holds for all admissible pairs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof is built on the approximate collision time τ = -Aε/δ² (Section 2.4, eq. (2.13); Theorem 3.1), and it requires the pre-collision interval [-t0, -Aε/δ²] to be nonempty. Since t0 = 1/(s_l - s_r) ∼ 1/δ, the required inequality is Aε/δ² < t0 ∼ C/δ, i.e. ε ≲ δ. No such relation appears in Theorem 1.1, which states the result for any ε ∈ (0, ε0) and δ_l ∼ δ_r ≤ δ0 with ε0, δ0 independent. For δ ≪ ε the interval is empty; for example, δ = ε² gives t0 ∼ ε^{-2} while Aε/δ² = Aε^{-3}, so -Aε/δ² < -t0 and the initial data (2.13) cannot serve as a starting point for the anti-derivative estimates. The same condition enters the smallness of the exponential bounds in Proposition 3.1, e.g. e^{-cAδ/ε} in (3.9) is small only when δ/ε is large. Thus the theorem as stated is not supported by the proof. This is a statement-level gap, likely repairable by adding the condition ε ≤ Cδ or by making ε0 depend on δ, but it is load-bearing because every pre-collision estimate depends on the interval having positive length.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the vanishing viscosity limit for 1D compressible Navier-Stokes equations (1.1) in the regime of two interacting shocks from the same characteristic family. The entropy solution of the Euler system consists of two incoming 2-shocks before a collision time t0, and a composite outgoing 2-shock plus a 1-rarefaction wave after the collision. The authors construct a smooth approximate wave pattern using viscous shock profiles and a smooth rarefaction wave, and they introduce an approximate collision time t0 - Aε/δ² to split the analysis. Before this time, an anti-derivative method yields H²-type energy estimates; after this time, a weighted relative entropy method with time-dependent shifts yields H¹-type estimates. Theorem 1.1 claims global smooth Navier-Stokes solutions for all ε∈(0,ε0) and δ_l∼δ_r≤δ0, converging to the Euler entropy solution in L^p with explicit rates C(δ_l+δ_r)^{1/2} ε^{1/p} before t0 and additional time-dependent terms after t0.","tokens_in":50985,"tokens_out":17795,"duration_ms":160165,"significance":"If the main theorem and its proof are correct, this is a substantial advance in the vanishing viscosity problem for interacting shocks: same-family shock interaction is genuinely more singular than different-family interaction because the collision time grows like 1/δ, and the post-collision wave pattern is a shock-rarefaction composite. The proof is entirely analytic and constructive: the approximate collision time, the shift ODE, and the weight function are chosen with explicit constants rather than fitted to the solution, and the claimed convergence rates are explicit and falsifiable. The reliance on prior work is for standard viscous shock profiles and rarefaction wave properties, not for the interaction estimates. The overall architecture of the proof is coherent and the techniques are potentially adaptable to related problems, provided the parameter-relation gaps identified below are resolved.","major_comments":[{"comment":"The theorem's quantifier order is not supported by the proof. The proof requires the approximate collision time -Aε/δ² to lie strictly inside the pre-collision interval [-t0, 0] (equivalently t0 - Aε/δ² > 0). Since t0 = 1/(s_l - s_r) ∼ C/δ, this forces Aε/δ² < C/δ, i.e. ε ≤ Cδ for some constant C. Theorem 1.1 states the result for any ε∈(0,ε0) and δ_l∼δ_r≤δ0 with ε0 and δ0 independent, which permits δ≪ε. For example, if δ=ε², then t0∼ε^{-2} while Aε/δ² = Aε^{-3}, so -Aε/δ² < -t0 and the interval [-t0, -Aε/δ²] is empty; the initial data (2.13) cannot serve as a starting point for the anti-derivative estimates, and Theorem 3.1 is vacuous or inapplicable. The same condition is needed for the exponential factors such as e^{-cAδε} in (3.8) to be small. This is a load-bearing gap, though it is likely repairable by adding the condition ε ≤ Cδ to Theorem 1.1 and Theorems 3.1-3.2, or by making ε0 depend on δ.","section":"Theorem 1.1, Section 1.3, Section 2.4, Theorem 3.1"},{"comment":"The estimate (3.8), ∥(ϕ,ψ)(τ)∥² + ε∫∥ψ_y∥² ≤ Ce^{-cAδε}, does not imply the smallness used in (4.40). In the proof of Theorem 1.1, the bound ∥v(τ,·)-V(τ,·)∥² ≤ C(δ_l+δ_r)ε is obtained by bounding ∥ϕ∥² by C(δ_l+δ_r)ε, but (3.8) only gives ∥ϕ∥² ≤ Ce^{-cAδε}. For fixed δ as ε→0 (the regime of the theorem), e^{-cAδε} → 1, so this does not tend to zero and cannot be absorbed into the O(δ ε) terms in (4.40). This is not a mere presentation issue: the claimed convergence rate as ε→0 relies on the perturbation L² norm being small, and the stated exponential bound does not provide that smallness for the parameter range of the theorem. The authors should either correct the exponent in (3.8) to a quantity that is actually small in the relevant regime (e.g., a polynomial in δε) or revise the proof of (4.40) to use a different estimate.","section":"Theorem 3.1, Eq. (3.8) and Section 4, Eq. (4.40)"}],"minor_comments":[{"comment":"The same symbol (v_*, u_*) is used both for the pre-collision intermediate state in (1.6) and for the post-collision intermediate state in (1.12), despite the text saying these are different states. This causes an apparent contradiction: δ_l := v_* - v_- is the first-order left shock strength, while later δ_1 := v_* - v_- is claimed to be third order. Please use distinct notation for the two intermediate states and clarify the definition of δ_1.","section":"Section 1.1"},{"comment":"The transition interval [t0 - Aε/δ², t0] is repeatedly described as having 'width of order O(ε)'. Its actual width is Aε/δ², which is larger than O(ε) when δ is small; for example, if δ=√ε the width is of order one. The estimates in the paper do not appear to rely on this width being O(ε), but the statement is inaccurate and should be corrected to 'width O(ε/δ²)'.","section":"Sections 1.3 and 4"},{"comment":"There are several typos and formatting errors, e.g., 'T reatment' in the table of contents, 'Y ield' in the proof of Lemma 6.6, and inconsistent spacing around equations. These do not affect the mathematics but should be cleaned up.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a genuinely difficult problem and the overall strategy is plausible, but the statement-level quantifier gap and the inconsistency between (3.8) and (4.40) are load-bearing. The quantifier issue is easy to fix by adding ε ≤ Cδ to the theorems, but the exponential bound problem may require re-deriving the L² perturbation estimate to obtain a polynomially small bound in δε; if that is not possible, the claimed convergence rate for fixed δ as ε→0 is not justified. I recommend major revision rather than rejection, since the core ideas and the a priori framework are promising and the defects appear repairable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a real step forward: same-family interacting shocks for the p-system in the vanishing viscosity limit, with collision time ~1/δ and a post-collision shock+rarefaction, was open. The paper deserves a serious referee. Second, the theorem as stated is not fully supported by the proof, because the proof needs ε ≤ Cδ but the statement lets ε and δ vary independently.\n\nWhat is actually new: the approximate collision time -Aε/δ² and the two-phase argument (anti-derivative before, weighted relative entropy after). The rates in Theorem 1.1 are explicit and credible, and the estimates in Sections 5–6 are detailed and honestly presented. They also disclose the prepared initial data (2.13), though the abstract overstates by calling it the vanishing viscosity limit without emphasizing that the initial data are chosen as the approximate profile.\n\nSoft spots, in proportion. The quantifier issue is load-bearing. The pre-collision interval [-t0, -Aε/δ²] has length about C/δ - Aε/δ². The proof requires this to be positive and of order 1/δ, so Aε/δ² ≲ 1/δ, i.e. ε ≲ δ/A. Theorem 1.1 states the result for any ε∈(0,ε0) and δ≤δ0 with no relation. For δ much smaller than ε, the interval is empty and the anti-derivative argument cannot start. The same ordering makes the exponential e^{-cAδ/ε} in (3.9) small; otherwise it is not. This is not a minor typo: every pre-collision estimate depends on that interval. It is likely repairable by adding ε ≤ Cδ (or by making ε0 depend on δ), but as written the theorem overclaims.\n\nSecond, some stated bounds look off: (3.9) and the surrounding text have powers of δ and ε that are not all consistent with the claimed smallness; the claim that the transition interval has width O(ε) is only true for fixed δ, since the width is Aε/δ², not uniformly O(ε). These are probably fixable by clarifying the parameter regime, but they affect verifiability.\n\nWho is this for: specialists in vanishing viscosity and stability of viscous shock/rarefaction waves. The techniques are substantial and likely transferable to non-isentropic and multi-dimensional cases. I would send it to a serious referee, with a clear request to check the parameter ordering and tighten the statement. I wouldn't cite it in its current form; I'd wait for the corrected version.","headline":"Genuinely new configuration and a serious proof idea, but the theorem as stated outruns the argument: the proof needs an ε≲δ ordering that the statement never imposes.","tokens_in":51611,"tokens_out":4151,"would_cite":false,"duration_ms":43917,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["76N10","35Q35","35Q30","35Q31","76N06"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two shocks from the same family collide; the vanishing viscosity limit holds through the collision with explicit $L^p$ rates.","keywords":["Vanishing viscosity limit","compressible Navier-Stokes equations","interacting shocks","same family shocks","rarefaction waves","anti-derivative method","relative entropy","a-contraction method"],"falsifier":"Take $\\delta=\\varepsilon^2$ with a fixed large constant $A$. Then $-A\\varepsilon/\\delta^2=-A/\\varepsilon$ lies before the initial time $-t_0$ for small $\\varepsilon$, so the proof's pre-collision interval $[-t_0,-A\\varepsilon/\\delta^2]$ is empty. Checking whether the claimed global smooth solutions and convergence rates still exist in this $\\delta\\ll\\varepsilon$ regime, or finding a counterexample, would settle whether Theorem 1.1 needs the extra condition $\\varepsilon\\lesssim\\delta$.","tokens_in":50468,"feed_emoji":"🌊","tokens_out":9177,"duration_ms":97607,"temperature":0.7,"pith_summary":"The paper establishes that the vanishing viscosity limit can be rigorously justified for the one-dimensional compressible Navier-Stokes equations when the underlying Euler solution contains two shocks from the same characteristic family that collide. This case is more singular than collisions of shocks from different families because the collision time grows like the inverse wave strength and the outgoing wave pattern mixes a shock with a rarefaction wave. The authors prove that, for suitably small wave strengths and viscosity, there are global smooth Navier-Stokes solutions converging to the entropy solution of the Euler equations in every $L^p$ space with $p\\in[2,+\\infty)$, at rates written explicitly in terms of the wave strength and viscosity. If correct, this supplies a quantitative vanishing-viscosity theorem for an overtaking shock interaction, including the collision point itself.","feed_headline":"Vanishing viscosity proven for two colliding same-family shocks","feed_subtitle":"Smooth viscous solutions converge to the Euler shock-rarefaction pattern, even through the collision point, with explicit rates.","key_machinery":"Before the approximate collision time $\\tau=-A\\varepsilon/\\delta^2$, the argument uses anti-derivative variables $(\\Phi,\\Psi)$ for the perturbation around the sum of two shifted viscous shock profiles; the energy estimates close only up to this carefully chosen time, producing exponentially small bounds with a large constant $A$. At that time the solution is transferred to a different approximate pattern: a smooth approximate rarefaction wave superposed with a shifted viscous shock, where the shift $X(\\tau)$ is governed by a weighted relative entropy and $a$-contraction relation with weight $a=1+(\\lambda/\\delta_2)(v^s-v_*)$. The transition interval between the two patterns has width $O(\\varepsilon)$, and controlling the error there is what lets the proof pass through the collision.","core_discovery":"The central claim is Theorem 1.1: there exist positive constants $\\varepsilon_0$ and $\\delta_0$ such that for any $\\varepsilon\\in(0,\\varepsilon_0)$ and wave strengths $\\delta_l\\sim\\delta_r\\le\\delta_0$, the Cauchy problem for the compressible Navier-Stokes equations has a family of global smooth solutions $(v^\\varepsilon,u^\\varepsilon)$ converging as $\\varepsilon\\to0^+$ to the entropy solution $(V,U)$ of the Euler equations. The convergence holds in $L^p(\\mathbb{R})$ for every $p\\in[2,+\\infty)$ with the explicit bounds $\\|(v^\\varepsilon-V,u^\\varepsilon-U)(t,\\cdot)\\|_{L^p}\\le C(\\delta_l+\\delta_r)^{1/2}\\varepsilon^{1/p}$ for $0\\le t\\le t_0$, plus $C(\\delta_l+\\delta_r)(t-t_0)^{1/(2p)}\\varepsilon^{1/(2p)}$ for $t\\ge t_0$. The proof treats the pre-collision phase, the collision point, and the post-collision shock-rarefaction composite as one connected picture, using an approximate collision time to close uniform energy estimates before merging into a shifted composite wave after the collision.","pith_inferences":["The proof's stopping time condition suggests that Theorem 1.1 as stated may hide an implicit ordering between the wave strength and the viscosity: when $\\delta$ is much smaller than $\\varepsilon$, the interval on which the initial data are constructed is empty, so a separate argument would be needed to cover that regime.","A natural extension is to non-isentropic compressible Navier-Stokes equations with two interacting same-family shocks, where the outgoing wave pattern would also include a contact discontinuity; the paper explicitly leaves this as future work.","The shift $X(\\tau)$ is designed to track how the rarefaction wave pushes the outgoing shock; a numerical test of the viscous $p$-system near the collision could check whether the predicted shift matches the actual shock location.","The approximate collision time scales as $\\varepsilon/\\delta^2$, which matches the expected viscous interaction width divided by the relative shock speed, so the same device may transfer to scalar conservation laws with curved flux near shock coalescence."],"forward_implications":["For any fixed time before the collision, the $L^p$ error between the viscous solution and the Euler entropy solution is $O((\\delta_l+\\delta_r)^{1/2}\\varepsilon^{1/p})$, so the long $1/\\delta$ pre-collision time scale does not ruin the convergence rate.","After the collision the error bound contains the additional factor $(\\delta_l+\\delta_r)(t-t_0)^{1/(2p)}\\varepsilon^{1/(2p)}$, giving a polynomial-in-$\\varepsilon$ rate for every fixed $p\\ge2$ even as time advances.","The approximate collision time creates a transition layer of width $O(\\varepsilon)$ near the true collision, and the paper shows the approximate composite wave matches the exact entropy solution there with the desired rate.","The global-in-time uniform estimates simultaneously imply convergence in every $L^p$ space with $p\\in[2,+\\infty)$, rather than only in $L^2$.","The same framework applies in Eulerian coordinates, as noted in the paper, so the Lagrangian proof is not an obstruction to the physically standard formulation."],"supporting_citations":[{"why":"Supplies the anti-derivative method for interacting shocks from different characteristic families that this paper adapts to the same-family case.","marker":"[14]"},{"why":"Provides the weighted relative entropy and a-contraction framework for composite shock-rarefaction waves used after the collision.","marker":"[21]"},{"why":"Gives the smooth approximate rarefaction wave constructed from the Burgers equation, used in the post-collision composite profile.","marker":"[31]"},{"why":"Provides local existence and viscous-limit estimates for piecewise smooth solutions via anti-derivative techniques, cited for the pre-collision setup.","marker":"[7]"},{"why":"Establishes the single-shock vanishing viscosity limit that this paper extends to two interacting shocks.","marker":"[8]"},{"why":"Gives existence and exponential decay estimates for viscous shock profiles, used throughout the wave-interaction estimates.","marker":"[22]"},{"why":"Provides the Poincaré-type inequality behind the a-contraction estimates controlling the shifted shock.","marker":"[19]"}],"fun_headline_variants":["Explicit rates for vanishing viscosity in same-family shock collisions","Navier-Stokes to Euler: global smooth solutions past shock collision","Proof: viscosity limit survives same-family shock collision","Two same-family shocks: vanishing viscosity yields shock-rarefaction","From Navier-Stokes to Euler: shock collision with explicit rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof requires the wave strength to be at least a fixed multiple of the viscosity, because the pre-collision analysis stops at the time $-A\\varepsilon/\\delta^2$ and this time must lie before the actual collision; when $\\delta$ is much smaller than $\\varepsilon$, that interval is empty and the constructed initial data cannot start the argument.","fun_headline_variants_meta":{"raw":{"variants":["Explicit rates for vanishing viscosity in same-family shock collisions","Navier-Stokes to Euler: global smooth solutions past shock collision","Proof: viscosity limit survives same-family shock collision","Two same-family shocks: vanishing viscosity yields shock-rarefaction","From Navier-Stokes to Euler: shock collision with explicit rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000741,"raw_usage":{"total_tokens":3391,"prompt_tokens":1113,"completion_tokens":2278,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":729,"completion_tokens_details":{"reasoning_tokens":2194}},"tokens_in":729,"tokens_out":2278,"duration_ms":17996,"temperature":1.0,"reasoning_tokens":2194,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:22:00.602541+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $\\delta=\\varepsilon^2$ with a fixed large constant $A$. Then $-A\\varepsilon/\\delta^2=-A/\\varepsilon$ lies before the initial time $-t_0$ for small $\\varepsilon$, so the proof's pre-collision interval $[-t_0,-A\\varepsilon/\\delta^2]$ is empty. Checking whether the claimed global smooth solutions and convergence rates still exist in this $\\delta\\ll\\varepsilon$ regime, or finding a counterexample, would settle whether Theorem 1.1 needs the extra condition $\\varepsilon\\lesssim\\delta$.","supporting_citations":[{"cited_title":"and Yang, T.:Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks.Sci","cited_arxiv_id":null,"evidence_quote":"Supplies the anti-derivative method for interacting shocks from different characteristic families that this paper adapts to the same-family case."},{"cited_title":"and Wang, Y.:Time-asymptotic stability of composite waves of viscous shock and rarefaction for barotropic Navier-Stokes equations.Adv","cited_arxiv_id":null,"evidence_quote":"Provides the weighted relative entropy and a-contraction framework for composite shock-rarefaction waves used after the collision."},{"cited_title":"Pure Appl","cited_arxiv_id":null,"evidence_quote":"Gives the smooth approximate rarefaction wave constructed from the Burgers equation, used in the post-collision composite profile."},{"cited_title":"Rational Mech","cited_arxiv_id":null,"evidence_quote":"Provides local existence and viscous-limit estimates for piecewise smooth solutions via anti-derivative techniques, cited for the pre-collision setup."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the single-shock vanishing viscosity limit that this paper extends to two interacting shocks."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives existence and exponential decay estimates for viscous shock profiles, used throughout the wave-interaction estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Poincaré-type inequality behind the a-contraction estimates controlling the shifted shock."}],"review_version":1}