{"id":"3306e7a6-d597-4ca7-b197-2ba5ccf5155a","arxiv_id":"2506.17156","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Viscous solutions of 1D conservation laws converge to the inviscid preshock at sharp rate nu^{1/4} in L∞, with Hölder thresholds 1/3 for the shocking and 2/3 for the nonshocking component, and a universal viscous Burgers profile near shock formation.","lead":"This paper proves exact rates for how viscous solutions of one-dimensional conservation laws approach the frictionless solution as a shock forms: in the strongest norm the difference shrinks like the quarter root of viscosity, with precise smoothness thresholds and a universal shock profile. It is the first sharp result of this kind for systems with physical, partially degenerate viscosity.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main results are conditional on the unproved nondegenerate expansion (H6), deferred to companion [7]; until [7] is available, Corollaries 1.4–1.6 are not independently established.","rationale":"The paper is a serious matched-asymptotics analysis, and I found no internal contradiction in the main estimates: the ν^{1/4} rate and the Hölder thresholds are consistent with the inverse-cubic scaling and the anisotropic inner region, and the lower-bound arguments are plausible. The most load-bearing premise, however, is (H6): it fixes the leading-order structure ψ_0 = 𝔲 e1, the homogeneity hierarchy, the inner coordinates, the viscous Burgers profile, and hence all sharp rates. The manuscript explicitly delegates the proof of (H6) to an in-preparation companion [7] and only conjectures genericity. This is not a flaw in the internal logic, but it means the central claims are conditional on an external unverified input. The reader's CONDITIONAL verdict is therefore appropriate, and I see no reason to change it. A full check of [7]—especially the complete derivative-error estimate in (H6)—is the decisive test.","tokens_in":74428,"tokens_out":17446,"duration_ms":193296,"concrete_test":"Independently verify (H6): complete and referee companion [7], checking that the full derivative-error estimate O(𝔡^{L-2m-3n+2}) holds for all m,n on the claimed open set of C∞ near-simple-wave data satisfying (H3), with no hidden additional hypotheses. If the companion appears and the estimate checks out, the present paper's conclusions stand; if [7] proves only a leading-order cubic expansion or requires extra nondegeneracy, then the sharp rates and universal profile in Corollaries 1.4–1.6 should be regarded as conditional on an unverified hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.2 states (H6) as an assumption on the inviscid solution: near the origin ψ^(0) admits the expansion ψ^(0) = Σ_{ℓ=0}^L ψ_ℓ + O(𝔡^{L-2m-3n+2}), with ψ_0 = 𝔲 e1 and ψ_ℓ homogeneous polynomials. The paper then says 'In [7], we show that (H6′) is satisfied for an open set of initial data' and 'We leave this question to future investigation.' Every central result—Theorem 1.3 and Corollaries 1.4–1.6—invokes (H1)–(H6), so all displayed rates (ν^{1/4}, the Hölder thresholds 1/3 and 2/3, and the universal viscous Burgers profile) inherit the truth of (H6). The present manuscript proves none of (H6); it only uses it. If the first singularity is degenerate, occurs on an interval, or involves multiple characteristics, the inverse-cubic scaling that produces the ν^{1/4} rate and the 1/3/2/3 thresholds is not the operative one. The paper itself acknowledges this limitation in the introduction. Thus the central claim is exactly as secure as the unpublished companion [7], and the current text does not by itself establish applicability to Navier–Stokes.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the vanishing viscosity limit for one-dimensional systems of viscous conservation laws (1.1), up to the time at which the inviscid solution forms a first, nondegenerate shock. Under hypotheses (H1)--(H6), the authors construct a matched asymptotic expansion: an outer expansion in integer powers of the viscosity away from the preshock, a rescaled inner expansion near the preshock, and a doubly indexed ``grid'' expansion that mediates the matching between them. The main results are Theorem 1.3 (the true solution is approximated to arbitrary order in Sobolev norms by the approximate solution), Corollary 1.4 (sharp L∞ convergence rate of order ν^{1/4}), Corollary 1.5 (Hölder convergence thresholds 1/3 for the shocking component and 2/3 for the nonshocking components), and Corollary 1.6 (after the blow-up (1.5), the rescaled solution converges locally uniformly to a universal viscous Burgers profile). The proof is carried out through a long sequence of quantitative estimates, Propositions 3.2, 5.10, 6.1, 7.1, 7.11, and 8.3 being the main milestones, and is closed via Serre's local well-posedness framework for degenerate parabolic systems.","tokens_in":74709,"tokens_out":13317,"duration_ms":125506,"significance":"If the hypotheses are satisfied, this is a substantial advance: it gives the first sharp strong-norm rates for the vanishing viscosity limit up to shock formation for systems with degenerate physical viscosity, including in-principle application to compressible Navier--Stokes type systems. The paper's methodology is largely parameter-free: the ν^{1/4} rate, the Hölder thresholds 1/3 and 2/3, and the universal profile all come from anisotropic scaling and matched asymptotics rather than fitted constants. The construction of the leading inner term relies on the published, parameter-free analysis of viscous Burgers in [22]. The explicit estimates for outer, inner, and grid terms are extensive and internally coherent, and the decoupling between shocking and nonshocking characteristics is a genuine technical contribution.","major_comments":[{"comment":"The central hypothesis (H6), namely the nondegenerate inverse-cubic formation expansion of the inviscid solution, is assumed in Theorem 1.3 and Corollaries 1.4--1.6, but it is not proved in this manuscript. The text states that (H6') is shown in the companion paper [7], which is listed as ``In preparation,'' and that the genericity question is left to future investigation. Consequently, the displayed rates ν^{1/4}, the thresholds 1/3 and 2/3, and the universality of the Burgers profile are all conditional on an unpublished verification. Since the abstract and introduction claim that the results apply to the compressible Navier--Stokes equations, the manuscript should either include a proof of (H6) for a nontrivial class of data covering Navier--Stokes, or make the conditional character of the main theorems and of the applicability claims explicit in the abstract and introduction.","section":"Section 2.2, (H6)"},{"comment":"Corollary 1.5 asserts convergence of the nonshocking component ω^(ν) to ω^(0) in L∞_t C^β_x if and only if β < 2/3. The proof in Section 8.1 establishes convergence for β < 2/3, but the sharpness statement is not proved there: the failure at the endpoint is only asserted for the shocking component σ at α = 1/3 (via (8.11) and the following comparison). The argument for the 2/3 threshold for ω is deferred to Remark 8.1, which gives a heuristic scaling argument and a toy model but not a complete rigorous lower bound for ‖ω^(ν)−ω^(0)‖_{C^{2/3}}. The statement should be strengthened by a proof of the failure at β = 2/3 for the actual nonshocking components, or weakened to a one-sided convergence statement.","section":"Corollary 1.5 and Section 8.1"},{"comment":"Theorem 1.3 states that for any s,p ≥ 0 there exists K ∈ N such that ‖ψ^(ν)−ψ_app^K‖_{H^s} ≤ Cν^p, but the approximation constructed in Section 8 depends on two truncation parameters K and L (the outer and inner orders), and Theorem 8.7 estimates the error by ν^{min{(1−2β)K, Lβ/3}−Λ_n}. The statement of Theorem 1.3 suppresses the parameter L and the fact that both K and L must be chosen large depending on s and p. This is not a mathematical obstruction, but the statement should be aligned with the actual construction (e.g., by indexing the approximate solution by the pair (K,L)).","section":"Theorem 1.3 versus Theorem 8.7"}],"minor_comments":[{"comment":"The left side of (4.13) is displayed as [ν^{1/4}∂_T + A^⊥_⊥(0)]∂_X Ω_in_ℓ; from (4.9) and from the grid equation (4.16) the intended operator appears to be ν^{1/4}∂_T + A^⊥_⊥(0)∂_X applied to Ω_in_ℓ. Please clarify the notation.","section":"Proposition 4.1, Eq. (4.13)"},{"comment":"The proof says ``We say a polynomial in (t, 𝔲, log 𝔡) has polyhomogeneity,'' while the class Q_{h;d} was announced as polynomials in (t, 𝔲, 𝔪, log 𝔡). Since 𝔪 = 𝔡^{−2}, this is likely a harmless omission, but the definition should be stated consistently.","section":"Section 5.1, proof of Proposition 5.1"},{"comment":"The statement that l_{I'}(ψ^(ν))∂_xψ^(ν) converges in L∞_t C^α_x if and only if α < −1/3 invokes negative Hölder spaces without definition; this should be either defined or rephrased in terms of the difference quotients used in the preceding discussion.","section":"Remark 8.1"},{"comment":"There are occasional typos and repeated words, e.g., ``the the inviscid equation'' near the beginning of Section 3.1 and ``We again emphasize again'' in Section 4.1. A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main theorems are conditional on hypothesis (H6), whose verification is delegated to the unpublished companion [7]. This is a structural issue rather than a defect in the arguments conditional on (H6): the manuscript is honest about the dependency, but the advertised applicability to Navier--Stokes is not yet independently established. The second major comment concerns an ``if and only if'' statement whose sharpness for the nonshocking component is not proved in the main text. I would encourage the editor to consider whether the companion paper should be required to be available or at least summarized in an appendix before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the substance is real: this is the first serious attempt at sharp strong-norm vanishing viscosity rates for systems with degenerate physical viscosity up to preshock time, and the ν^{1/4} L∞ rate, the 1/3 vs 2/3 Hölder asymmetry, and the universal Burgers profile all come out of scaling and matched asymptotics, not fitted parameters. Second, the load-bearing hypothesis (H6)—the inverse-cubic nondegenerate preshock expansion—is not proved here. It is delegated to companion paper [7], listed as in preparation. The text says so plainly, but the consequence is that Corollaries 1.4–1.6 are conditional theorems until [7] appears.\n\nWhat is genuinely new: the scalar Burgers case was handled by two of the authors in [22] (published, parameter-free). This paper extends that to genuinely nonlinear systems via the criminal inner ansatz and the doubly-indexed grid expansion that decouples shocking from nonshocking characteristics. The 2/3 Hölder threshold for nonshocking components is new, and Remark 8.1 works hard to show it is intrinsic rather than an artifact of frozen coefficients. The matching machinery, especially the treatment of the anisotropic diffusive zone and the shadows cast along transverse characteristics, is careful, and the paper is honest about its own limitations, including the open global well-posedness for Navier–Stokes–Fourier.\n\nSoft spots, in proportion. The main one is (H6): until [7] exists, the advertised rates cannot be independently verified from this text. That is not a scandal, but it is a fact about the current public record. Second, several estimate classes are sketched rather than shown—derivative bounds in Proposition 5.10, parts of the bootstrap in Proposition 5.5 (\"we omit these similar calculations\"), pieces of Lemma 5.8. A referee will have real work to do. Third, no formal or computational verification; for a proof this long and intricate, that is a genuine risk factor, not a flaw in itself. The stress-test note lands, though I would soften it: the paper flags exactly this dependency in its introduction, and the published [22] gives some credibility to the machinery.\n\nBottom line: a conditional but substantial contribution, honestly framed. It deserves a serious referee. I would send it, with instructions that the referee verify the deferred estimates and confirm the status of [7].","headline":"The sharp-rates machinery is real and probably right, but the main theorems inherit their key hypothesis from an unpublished companion, so the public record is conditional.","tokens_in":75245,"tokens_out":3664,"would_cite":true,"duration_ms":38814,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35L65","35L67","35B25","35Q35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that small viscosity replaces a forming inviscid shock with a sharp, universal viscous profile: the $L^\\infty$ difference is of exact order $\\nu^{1/4}$, the shocking component converges in H\\\"older spaces exactly below…","keywords":["vanishing viscosity","shock formation","conservation laws","matched asymptotic expansion","viscous Burgers","Hölder regularity","nondegenerate shock","Navier–Stokes equations"],"falsifier":"Measure $\\|\\psi^{(\\nu)}-\\psi^{(0)}\\|_{L^\\infty}$ for a viscous system satisfying (H1)\\textendash{}(H5) whose first singularity is degenerate, occurs on an interval, or involves multiple characteristics. If the difference scales with an exponent other than $1/4$, or if the rescaled profile $\\Psi^{(\\nu)}$ fails to converge to the unique viscous Burgers solution, the central claim is false. In the paper's own regime, the two-sided bound $C^{-1}\\nu^{1/4}\\le\\|\\psi^{(\\nu)}-\\psi^{(0)}\\|_{L^\\infty}\\le C\\nu^{1/4}$ can be checked directly.","tokens_in":74228,"feed_emoji":"💥","tokens_out":8727,"duration_ms":81346,"temperature":0.7,"pith_summary":"This paper asks what happens when a small viscosity $\\nu$ is added to a 1D conservation law just before its inviscid solution forms a shock. It proves that the viscous solution exists up to the shock time and converges to the inviscid solution in $L^\\infty$ at the sharp two-sided rate $\\nu^{1/4}$. It also identifies the precise shape of the viscous regularization: after rescaling time by $\\nu^{1/2}$, space by $\\nu^{3/4}$, and amplitude by $\\nu^{1/4}$, the solution approaches a unique profile that solves the scalar viscous Burgers equation. A reader should care because this is the first sharp strong-norm description of vanishing viscosity for systems up to shock formation, and it covers physically degenerate diffusion such as compressible Navier\\textendash{}Stokes.","feed_headline":"Sharp vanishing-viscosity rate found at shock formation","feed_subtitle":"Viscous solutions stay within a precise power of the inviscid preshock, then settle into a universal Burgers profile.","key_machinery":"The machinery is the nondegenerate inverse-cubic preshock. The profile $\\mathfrak{u}$ is defined implicitly by $x=a|t|\\mathfrak{u}+b\\mathfrak{u}^3$, and the cubic distance $\\mathfrak{d}=(|t|+3a^{-1}b\\mathfrak{u}^2)^{1/2}$ measures proximity to the singularity in the natural anisotropic scaling. Hypothesis (H6) asserts that the inviscid solution has a full asymptotic expansion in polynomials in $(t,\\mathfrak{u},\\mathfrak{m})$, where $\\mathfrak{m}=a\\partial_x\\mathfrak{u}$, with leading term $\\mathfrak{u}e_1$. Under the rescaling $T=\\nu^{-1/2}t$, $X=\\nu^{-3/4}x$, $\\Psi=\\nu^{-1/4}\\psi$, the leading inner term satisfies the scalar viscous Burgers equation, and the paper matches the outer expansion to this inner expansion through a doubly-indexed grid of correctors whose horizontal sums give the outer terms and whose vertical sums give the inner terms.","core_discovery":"Under hypotheses (H1)\\textendash{}(H6), the paper's central claim is that the viscous solution $\\psi^{(\\nu)}$ of the system exists on $[t_0,0]\\times\\mathbb{R}$ and is approximated to arbitrary Sobolev order by a matched asymptotic expansion built from simple building blocks. The sharp consequences are that $C^{-1}\\nu^{1/4}\\le\\|\\psi^{(\\nu)}-\\psi^{(0)}\\|_{L^\\infty}\\le C\\nu^{1/4}$; that the shocking component converges in $L^\\infty_t C^\\alpha_x$ if and only if $\\alpha<1/3$, while the nonshocking components converge for $\\beta<2/3$; and that after the blow-up $T=\\nu^{-1/2}t$, $X=\\nu^{-3/4}x$, $\\Psi=\\nu^{-1/4}\\psi$, the blown-up solution tends locally uniformly to $U e_1$, where $U$ is the unique viscous Burgers solution matching the inverse cubic at infinity. The $2/3$ threshold for nonshocking components is not an artifact of the coordinate choice: it is forced by the off-diagonal diffusive term $\\nu B^\\perp_1\\,\\partial_x^2\\sigma$ whenever the diffusion is not diagonal in the eigenbasis.","pith_inferences":["Editorial inference: if the nondegenerate formation expansion (H6) is indeed generic, as the authors expect, then the $\\nu^{1/4}$ rate and universal Burgers profile should be the default observation in numerical studies of shock formation, while degenerate or multi-characteristic shocks should show different exponents.","Editorial inference: the dimensions give a testable signature: the difference $\\psi^{(\\nu)}-\\psi^{(0)}$ maintains order $\\nu^{1/4}$ over a spatial scale $\\nu^{3/4}$, and this ratio is exactly why $C^{1/3}$ convergence fails; checking the support scale of the difference is a direct numerical check.","Editorial inference: the same anisotropic blow-up and grid-matching strategy should transfer to 1D reductions of higher-dimensional symmetric flows and to repeated nonshocking eigenvalues; a repeated shocking eigenvalue would replace scalar viscous Burgers by a vector-valued analogue."],"forward_implications":["Existence of smooth viscous solutions up to the inviscid shock time is obtained for a class of degenerate viscous systems, including the compressible Navier\\textendash{}Stokes\\textendash{}Fourier equations, where large-data global existence was previously unavailable.","The vanishing viscosity limit holds in $L^\\infty$ with a sharp two-sided rate $\\asymp\\nu^{1/4}$, so the convergence cannot be improved to any smaller exponent.","H\\\"older convergence has sharp thresholds: the shocking component converges exactly for H\\\"older exponents below $1/3$, and nonshocking components below $2/3$; the difference is forced by the off-diagonal viscous cross-term.","Near the first singularity, the rescaled solution converges to a unique viscous Burgers profile determined only by two cubic coefficients and the effective diffusion coefficient, so the small-scale structure of a nascent shock is universal.","The outer and inner expansions match to arbitrary order, so quantities built from the solution near shock formation can be computed from explicit building blocks: inviscid hyperbolic terms, viscous Burgers terms, and linear transport or advection-diffusion terms."],"supporting_citations":[{"why":"Companion paper establishing the nondegenerate formation expansion (H6) on an open set of initial data; the present paper's hypotheses inherit this verification burden.","marker":"[7]"},{"why":"Constructs and studies the unique viscous Burgers solution that matches the inverse cubic at infinity; this solution is the leading inner term and the universal profile in Corollary 1.6.","marker":"[22]"},{"why":"Provides the local well-posedness and energy estimates, in entropic variables, used to close the difference between the approximate and true solutions.","marker":"[68]"},{"why":"Supplies the structural conditions on degenerate diffusion, constant nullity and entropy dissipation, that underlie hypothesis (H2).","marker":"[69]"},{"why":"Pioneered matched asymptotic expansions for viscous limits about shocks; the present outer, inner, and grid construction is modeled on this strategy.","marker":"[38]"},{"why":"Foundational Kawashima\\textendash{}Shizuta conditions for local well-posedness of hyperbolic\\textendash{}parabolic systems, which motivate the assumptions on the diffusion matrix.","marker":"[46]"}],"fun_headline_variants":["Shock formation: sharp ν^{1/4} vanishing-viscosity rate","Vanishing viscosity: universal Burgers profile at shock blow-up","Viscous shocks: exact convergence rates and universal tail","Sharp ν^{1/4} convergence for viscous shock formation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire sharp-rate conclusion rests on the nondegenerate formation expansion (H6): the inviscid solution must approach shock formation through a single characteristic with an inverse-cubic preshock that admits the full asymptotic expansion in $\\mathfrak{u}$, and if that structure fails, the $\\nu^{1/4}$ rate and universal Burgers profile are not expected to hold.","fun_headline_variants_meta":{"raw":{"variants":["Shock formation: sharp ν^{1/4} vanishing-viscosity rate","Vanishing viscosity: universal Burgers profile at shock blow-up","Viscous shocks: exact convergence rates and universal tail","Sharp ν^{1/4} convergence for viscous shock formation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000253,"raw_usage":{"total_tokens":1560,"prompt_tokens":939,"completion_tokens":621,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":547}},"tokens_in":555,"tokens_out":621,"duration_ms":6646,"temperature":1.0,"reasoning_tokens":547,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:10:53.986809+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure $\\|\\psi^{(\\nu)}-\\psi^{(0)}\\|_{L^\\infty}$ for a viscous system satisfying (H1)\\textendash{}(H5) whose first singularity is degenerate, occurs on an interval, or involves multiple characteristics. If the difference scales with an exponent other than $1/4$, or if the rescaled profile $\\Psi^{(\\nu)}$ fails to converge to the unique viscous Burgers solution, the central claim is false. In the paper's own regime, the two-sided bound $C^{-1}\\nu^{1/4}\\le\\|\\psi^{(\\nu)}-\\psi^{(0)}\\|_{L^\\infty}\\le C\\nu^{1/4}$ can be checked directly.","supporting_citations":[{"cited_title":"Shock formation in 1D conservation laws I: Inviscid structure","cited_arxiv_id":null,"evidence_quote":"Companion paper establishing the nondegenerate formation expansion (H6) on an open set of initial data; the present paper's hypotheses inherit this verification burden."},{"cited_title":"Local existence for viscous system of conservation laws:𝐻𝑠-data with𝑠 > 1+𝑑/2","cited_arxiv_id":null,"evidence_quote":"Provides the local well-posedness and energy estimates, in entropic variables, used to close the difference between the approximate and true solutions."},{"cited_title":"The structure of dissipative viscous system of conservation laws","cited_arxiv_id":null,"evidence_quote":"Supplies the structural conditions on degenerate diffusion, constant nullity and entropy dissipation, that underlie hypothesis (H2)."},{"cited_title":"Viscous limits for piecewise smooth solutions to systems of conservation laws","cited_arxiv_id":null,"evidence_quote":"Pioneered matched asymptotic expansions for viscous limits about shocks; the present outer, inner, and grid construction is modeled on this strategy."},{"cited_title":"Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics","cited_arxiv_id":null,"evidence_quote":"Foundational Kawashima\\textendash{}Shizuta conditions for local well-posedness of hyperbolic\\textendash{}parabolic systems, which motivate the assumptions on the diffusion matrix."}],"review_version":1}