{"id":"aa254f64-0252-42dd-88e1-9d4defe35b23","arxiv_id":"2412.21040","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every integer n>=1, smooth 1D Euler data can form a pre-shock cusp with Holder exponent 1/(2n+1), and the set of such data is a codimension-(2n-2) Banach manifold in W^{2n+2,infinity}.","lead":"This paper proves that one-dimensional gas dynamics can form shock precursors with a whole ladder of possible sharpness: for any integer n, smooth initial data can produce a first singularity that is a cusp with Holder exponent 1/(2n+1). A reader should care because it shows that stable and increasingly unstable singularity shapes are not isolated examples but an infinite hierarchy with precisely quantified finite-codimension stability.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Omitted stability estimates in §6.4 and sketch in §7 leave the codimension-(2n-2) manifold claim unproven","rationale":"The reader correctly identifies the a priori bounds in §5.4 and the inequality (4.8) as quantitative premises for the existence of the singularity and the cusp expansion. However, the most load-bearing gap in the written proof is the omission of the stability estimates in §6.4 and the sketchiness of §7. These sections are directly used to prove that the function fn has the Lipschitz and derivative properties needed for the implicit function theorem, which in turn establishes the finite-codimension manifold structure—a central claim of the paper. The reader's verdict is CONDITIONAL, and this concern reinforces that conditional status: the result may be true, but the presented proof is incomplete at a crucial juncture. I partially agree with the reader's weakest_assumption because while the §5 bounds are indeed fundamental, the §6.4/§7 gap is a more concrete and unambiguously load-bearing omission. Given the explicit acknowledgements of omitted proofs, the correct verdict remains CONDITIONAL: the paper should not be rejected, but it cannot be accepted without completing these arguments.","tokens_in":542,"tokens_out":3841,"duration_ms":136978,"concrete_test":"Re-derive Proposition 6.6 by writing out the induction on β_x for ∂β ˚Kλ using identity (6.1g) and the estimates from §5.4. Specifically, verify the step from β_x = m to β_x = m+1 for all β_t ≤ 2n. If the inequality (|β|+1)^2 ||∂β ˚Kλ||/(|β|! M C_x^{β_x} C_t^{β_t}) ≤ Bλ C_t B_z ε cannot be shown, the stability estimates fail and the implicit function theorem argument in Proposition 8.10 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central novelty is the finite-codimension stability statement. To prove it, the paper must show that the function fn (defined in (8.19)) has a surjective derivative with respect to the λ coordinates, and that the solution map depends Lipschitz-continuously on the initial data. These facts rely on the stability estimates of §6.4 and §7. However, §6.4 states that the proofs are omitted ('For this reason, they will be omitted.'), and §7 provides only a sketch. Without Proposition 6.6 and the analogous bounds in §7, the bounds (8.28c) and the Lipschitz continuity of T* and ~ηx are not established. Consequently, the invertibility of D_λ f_n is unverified, so the existence of the graph λ* in Proposition 8.10 is not rigorously proven. This is not a mere technicality: the codimension-2 (and higher) cases are the new content beyond the stable n=1 case, and they depend exactly on these omitted estimates.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies finite-time gradient catastrophe formation for the 1D compressible Euler equations with non-constant entropy. It introduces differentiated Riemann variables composed with fast-acoustic characteristics and proves a priori estimates in Lagrangian coordinates. The main result, stated as Theorem 9.1 and abbreviated as Theorem 1.1, asserts that for each integer n≥1 there is a codimension-(2n−2) Banach submanifold M_n of (W^{2n+2,∞}(T))^3 such that solutions from M_n form a first gradient singularity at a unique point (y*,T*), with T*=2/(1+α)+O(ε), the velocity and sound speed having a cusp of the form b0+b1(y−y*)^{1/(2n+1)}+O(|y−y*|^{2/(2n+1)}), with b1=−(2n+1)^{1/(2n+1)}[1+O(ε)], and with z and k remaining C^{1,1/(2n+1)} up to T*. The proof combines Lagrangian energy estimates, stability estimates with respect to perturbations of the initial data, an implicit-function argument to construct the finite-codimension manifold, and a Puiseux-inversion step to extract the cusp profile.","tokens_in":83561,"tokens_out":7338,"duration_ms":70282,"significance":"If the proof is completed, this is a significant contribution: it would give the full hierarchy of C^{0,1/(2n+1)} cusp singularities for smooth 1D Euler data, recovering the stable n=1 case and the unstable n=2 case of Buckmaster–Iyer under weaker regularity, with a characteristic method that avoids self-similar profiles. The explicit leading-order coefficient b1, the n-independent range of validity, and the Puiseux inversion are concrete and non-circular strengths. However, the finite-codimension part of the claim depends on stability estimates in Section 6.4 whose proofs are explicitly omitted and on a solution-continuity argument in Section 7 that is only sketched; these are load-bearing, not cosmetic.","major_comments":[{"comment":"The paragraph at the start of §6.4 states that the proofs of Proposition 6.6 and Corollaries 6.7–6.8 “will be omitted” because they are analogous to §5.4. These λ-stability bounds are load-bearing: Lemma 8.9 and the proof of Proposition 8.10 use (8.28c), which is Corollary 6.8, to show that D_λ f_n = Id + small error. Without a complete proof of these estimates, the existence of the graph λ* and hence the codimension-(2n−2) manifold M_n for n≥2 is not rigorously established. The authors should either write out the full induction or give a detailed reduction to Proposition 5.5, specifying the modified weights, constants M and L_n, and all changed terms, rather than leaving the verification to the reader.","section":"§6.4"},{"comment":"Section 7 is presented as a sketch. Proposition 7.1 and the culminating bounds (7.13a)–(7.13b) are stated after a calculation that suppresses the terms coming from (7.3a)–(7.3b), including differences of ∂_t~K, ∂_x~K, ∂_t~Z and ∂_x~Z multiplied by (η_x−~η_x) and (Σ−~Σ). These bounds are what make T* and ~η_x Lipschitz in the initial data in Proposition 8.8, which in turn gives the Lipschitz dependence of (˚x,˚T) and f_n used in §8.4. A sketch is insufficient at this point in the argument; the energy estimates for the difference system should be checked term by term with the same precision as in §5.3, or the absent estimates should be provided explicitly.","section":"§7"},{"comment":"The computation of D_λ f_n and the contraction estimate for f_n treat ∂_{λ_j} T* and ∂_{λ_j}(˚x,˚T) as genuine derivatives. The paper only proves that λ↦T* is Lipschitz (Proposition 8.8) and then uses Rademacher's theorem to obtain almost-everywhere derivative bounds (8.27a)–(8.27b). It is not explained why the chain-rule calculation for ∂_{λ_j} f_n^i is valid at points where T* is not differentiable, nor how the almost-everywhere derivative control implies the uniform bound ‖Id − D_λ f_n‖ ≤ 1/9 needed in Lemma C.1. Since Proposition 8.10 is the step that constructs the manifold, this differentiability-to-contraction issue must be addressed explicitly.","section":"§8.4, Proposition 8.10"}],"minor_comments":[{"comment":"The initial condition for the fast-acoustic characteristics is written as η(x,0)=0, but the subsequent use of η_x(x,0)=1 throughout the paper shows that this should be η(x,0)=x.","section":"Eq. (3.3)"},{"comment":"There are typographical errors such as “adiabtic exponent” and “discontintuous”; these should be corrected.","section":"§1.1"},{"comment":"The sentence “neither vacuum formation nor finite-time implosion is not possible” contains a double negative and should be rewritten.","section":"§3.3"},{"comment":"The notation w0 is overloaded: the fixed profile defined in (8.1) and the general initial-data component w0 are both denoted w0, which can be confusing in the manifold construction.","section":"§8.1"},{"comment":"Theorem 1.1 uses the notation O_γ(ε) for the blowup time, while Theorem 9.1 uses O_α(ε); since α=(γ−1)/2 this is consistent, but the dependence on n should be stated uniformly in the introductory theorem.","section":"§1.3"}],"recommendation":"major_revision","confidential_remarks":"The claimed result is important and the overall strategy is credible, but the omitted proofs in §6.4 and the sketch in §7 are exactly the parts that establish the finite-codimension stability statement. I do not see circularity or fitted parameters: the cusp exponent and leading coefficient are derived from the characteristic inversion, not imposed. I would recommend requesting a revision in which the omitted stability estimates and the difference estimates are fully written out; if the authors can provide them, the paper likely merits publication in a strong analysis journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, the construction is genuinely new: for each n≥1 it exhibits smooth 1D Euler data whose first singularity is a C^{0,1/(2n+1)} cusp, covering all exponents in the odd-denominator hierarchy without self-similar analysis. The differentiated Riemann variables in Lagrangian coordinates are the right tool, the Puiseux inversion in Section 9 is clean, and the recovery of the n=1 stable cusp and the n=2 Buckmaster–Iyer result with milder assumptions is real progress. Second, the headline codimension-(2n−2) stability claim is not actually proven in the written text. Section 6.4 states the estimates are omitted, Section 7 is explicitly a sketch, and Proposition 8.10 plus the manifold conclusion depend on exactly those missing bounds. This is not a manufactured gap; the paper tells you the proofs are absent. Without Proposition 6.6 and the full difference estimates from §7, the surjectivity of D_λ f_n and the Lipschitz dependence of T* and η_x on the data are unverified, so the graph λ* in Proposition 8.10 is not rigorously established. The cusp expansion in Section 9 is conditional on being on that manifold, so the main theorem as stated is incomplete. I want to be fair: the omitted estimates are advertised as analogues of the detailed §5.4 estimates, and the line of argument is credible. A determined author could likely fill them. But “completely analogous” is not a proof for a claim of this precision, and the codimension statement is the selling point of the abstract. For a reader, the paper is worth studying for the framework and the existence side, but I would not cite the codimension theorem in its current form. A serious referee should engage, but ask for the missing estimates to be written out before accepting the stability claim. This deserves peer review, not desk rejection, because the core idea is strong and the gaps look repairable.","headline":"Infinite cusp hierarchy for 1D Euler is a strong and plausible result, but the codimension-stability proof relies on omitted estimates, so the flagship claim is not established as written.","tokens_in":84117,"tokens_out":1851,"would_cite":false,"duration_ms":22441,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35L67","35Q31","35L65","35B44"],"pacs":[],"model":"deepseek-v4-flash","headline":"Smooth one-dimensional Euler flows can form a first singularity shaped as a cusp with exactly $C^{0,1/(2n+1)}$ Hölder regularity for any integer $n \\geq 1$, and the initial data doing so form a codimension-$(2n-2)$ manifold.","keywords":["1D Euler equations","gradient catastrophe","pre-shock","Hölder cusp singularity","shock formation","Riemann variables","finite-codimension stability","characteristic methods"],"falsifier":"For the explicit data $u_0 = \\sigma_0 = \\tfrac12 w_0$, $S_0 = 0$ with $w'_0(y) = -1 + y^{2n}$ (periodicized, positive, zero mean), the solution reduces to Burgers dynamics, and the theorem predicts a first singularity at $T_* = 2/(1+\\alpha)$ with log-log slope $-2n/(2n+1)$ for $|\\partial_y u|$ versus $|y-y_*|$ and leading coefficient $(2n+1)^{1/(2n+1)}$; a numerical simulation for $n = 2, 3, 4$ measuring the local exponent at the first time $\\min_x \\eta_x = 0$, or a simulation showing $\\eta_x$ vanishes at two distinct points before any gradient blowup, would settle the central claim directly.","tokens_in":83171,"feed_emoji":"📉","tokens_out":14161,"duration_ms":126662,"temperature":0.7,"pith_summary":"The paper proves that the one-dimensional Euler equations of gas dynamics admit an infinite hierarchy of finite-time gradient catastrophes, all reached from smooth, compressive, non-vacuous initial data. For each integer $n \\geq 1$ there is a codimension-$(2n-2)$ Banach manifold of initial data in $W^{2n+2,\\infty}$ such that the solution develops its first gradient singularity at a single point $(y_*,T_*)$ with $T_* = 2/(1+\\alpha) + O(\\varepsilon)$, and at that instant the velocity and sound speed form a cusp $u(y,T_*) = b_0 + b_1 (y-y_*)^{1/(2n+1)} + O(|y-y_*|^{2/(2n+1)})$ with explicit leading coefficient $b_1 = -(2n+1)^{1/(2n+1)}(1+O(\\varepsilon))$; equivalently, the gradient has precisely $C^{0,1/(2n+1)}$ Hölder regularity. The hierarchy matters because the pre-shock cusp is the transition state from smooth flow to shock: a theory of shock formation must account for which cusps can form and how many tuning conditions they impose, and here $n=1$ is the fully stable cubic-root cusp while each higher $n$ demands $2n-2$ additional constraints on the data. The paper also records that the formal $n \\to \\infty$ limit of the cusp profiles is the classical Riemann data, but the radius of validity collapses, so a direct bridge to the Riemann problem remains open.","feed_headline":"Infinite hierarchy of cusp singularities for 1D Euler","feed_subtitle":"Every integer n gives a finite-time cusp of exact Hölder power 1/(2n+1); the higher cusps cost 2n−2 constraints.","key_machinery":"The machinery is the study of differentiated Riemann variables pulled back along the fast-acoustic characteristic flow $\\eta$, defined by $\\partial_t \\eta = \\lambda_3(\\eta,t)$ with $\\lambda_3 = u + \\alpha\\sigma$. In these Lagrangian coordinates the product $\\eta_x \\mathring{W}$ stays as smooth as the initial data even though $\\partial_y w$ blows up, and the system for $(\\Sigma, \\eta_x, \\eta_x\\mathring{W}, \\mathring{Z}, \\mathring{K})$ has polynomial right-hand sides. The key inequality $\\eta_x \\mathring{W} \\leq -\\tfrac12 + 4\\eta_x$, proved by bootstrap, makes the weighted $L^q$ energies for $\\mathring{Z}$ and $\\mathring{K}$ contractive, and after sending $q \\to \\infty$ yields uniform $W^{2n+2,\\infty}$ bounds up to the blowup time. The first singularity is then characterized exactly as the first time $\\eta_x$ vanishes at a single point $x_*$; imposing the vanishing of the first $2n-1$ $x$-derivatives of $\\eta_x$ there gives $2n$ equations in two unknowns, which the implicit function theorem solves as a codimension-$(2n-2)$ Lipschitz graph in initial-data space. Finally, polynomial inversion (an analytic Puiseux-Newton expansion) produces the inverse of the map $x \\mapsto y = \\eta(x,T_*)$ as $(y-y_*)^{1/(2n+1)}$ plus controllable error, converting the smooth expansion of $w \\circ \\eta$ into the cusp formulas.","core_discovery":"The central claim, Theorem 9.1, is that for every integer $n \\geq 1$ there exists a codimension-$(2n-2)$ Banach submanifold $M_n$ of $(W^{2n+2,\\infty}(\\mathbb{T}))^3$, containing the reference data $(u_0,\\sigma_0,S_0) = \\tfrac12(w_0,w_0,0)$ with $w'_0(y) = -1 + y^{2n}$ near the distinguished point, such that every solution from data in $M_n \\cap B_\\varepsilon$ forms its first gradient singularity for $u$ and $\\sigma$ at a unique point $(y_*,T_*)$, with $T_* = 2/(1+\\alpha) + O(\\varepsilon)$ and $y_* = \\tfrac12 + O(\\varepsilon)$ on the torus. At that instant the velocity and sound speed obey the cusp expansion $u(y,T_*) = b_0 + b_1 (y-y_*)^{1/(2n+1)} + O(|y-y_*|^{2/(2n+1)})$, and $\\sigma$ similarly, with $b_0 = \\tfrac52 + O(\\varepsilon)$ and $b_1 = -(2n+1)^{1/(2n+1)}[1+O(\\varepsilon)]$, while the entropy $S$ remains $C^{1,1/(2n+1)}$ uniformly; away from $y_*$ the solution stays $W^{2n+2,\\infty}$ smooth. The gradient blows up like $(y-y_*)^{2n/(2n+1)}$, which is exactly the statement that the pre-shock has $C^{0,1/(2n+1)}$ Hölder regularity. For $n=1$, $M_1$ is open (codimension zero) and the result recovers the established stable $C^{0,1/3}$ pre-shock; for $n=2$ it recovers the previously known codimension-2 unstable $C^{0,1/5}$ cusp, and for all $n \\geq 3$ it gives new singularity profiles.","pith_inferences":["Because the proof never invokes self-similar profiles and uses only the transport structure of the dominant characteristic family, the same flatness-to-cusp dictionary should transfer to other one-dimensional hyperbolic conservation systems with a genuinely nonlinear fast wave family, with the cusp exponent $1/(2n+1)$ fixed by the order of flatness of the initial compression alone.","The codimension-$(2n-2)$ statement implies a selection rule for generic data: small random perturbations of the $n$-th reference data for $n \\geq 2$ should almost surely fall onto the $n=1$ stratum and produce a $C^{0,1/3}$ cusp, so the higher cusps are observable only with deliberately tuned data; this is testable numerically by adding small generic noise and measuring the local Hölder exponent a","The degenerate $n \\to \\infty$ limit suggests a double-scaling regime in which the flatness order $n$ and the perturbation amplitude $\\varepsilon$ are linked (for instance $\\varepsilon \\sim n^{-c}$), possibly giving a quantitative route toward the Riemann-problem data at a controlled rate; the paper leaves this connection open.","The entropy $S$ remains strictly smoother than the velocity and sound speed through the singularity, so the cusp is a property of the acoustic family alone; this separated-regularity structure could serve as a marker for detecting pre-shock formation in numerical or experimental settings."],"forward_implications":["The $n=1$ case reproduces the fully stable $C^{0,1/3}$ pre-shock from an open set of $W^{4,\\infty}$ data, so the classical picture of one-dimensional shock formation is contained as the first element of the hierarchy.","For every $n \\geq 2$ there are smooth initial data producing a first singularity with Hölder exponent $1/(2n+1)$, so infinitely many distinct cusp types are dynamically reachable, not merely the generic cubic-root cusp.","Each higher cusp is exactly finitely stable: the set of its initial data has codimension $2n-2$ in $W^{2n+2,\\infty}$, so $2n-2$ independent tuning conditions are necessary and sufficient to avoid collapsing to the generic cusp.","The explicit profile (leading coefficient $-(2n+1)^{1/(2n+1)}$, blowup time $2/(1+\\alpha)$ independent of $n$) gives quantitative, checkable predictions for what a numerical observation of the pre-shock should see.","With $L$ additional derivatives of the initial data the cusp expansion extends to a truncated Puiseux series of order $L$; the paper flags that the $n \\to \\infty$ limit degenerates, so connecting the hierarchy to the classical Riemann problem remains an open problem."],"supporting_citations":[{"why":"Supplies the characteristic-coordinate framework and the differentiated Riemann variables on which the entire proof is built.","marker":"[42]"},{"why":"The prior self-similar-analysis construction of the stable C^{0,1/3} pre-shock whose n=1 case this paper recovers by characteristics.","marker":"[9]"},{"why":"The only prior construction of an unstable (codimension-2) C^{0,1/5} cusp, which the present n=2 case recovers under milder regularity assumptions.","marker":"[8]"},{"why":"The original construction of the C^{0,1/3} pre-shock in the p-system, serving as the base case of the hierarchy.","marker":"[30]"},{"why":"Provides the Burgers-equation Puiseux expansion for initial slopes with flat minima, the scalar model whose exact computations seed the cusp formulas.","marker":"[23]"},{"why":"Establishes the analogous codimension-(2n-2) sets giving C^{0,1/(2n+1)} cusps for dispersive and dissipative perturbations of Burgers, the closest existing scalar statement of the finite-codimension phenomenon.","marker":"[40]"},{"why":"Used to rule out vacuum formation and implosion in one dimension, ensuring the first singularity is a gradient blowup characterized by the Eulerian blowup criterion.","marker":"[14]"},{"why":"The authors' earlier characteristics-based treatment of the C^{0,1/3} cusp in azimuthal symmetry, whose polynomial-inversion appendix is generalized here.","marker":"[39]"}],"fun_headline_variants":["Infinite cusp hierarchy for 1D Euler: each n gives Hölder 1/(2n+1)","Every integer n: a finite-time cusp of exact Hölder power 1/(2n+1) in 1D Euler","1D Euler cusps: infinite hierarchy, codimension-2n-2 stability for all n","Every n>0: a sharper cusp singularity in 1D Euler, exact Hölder exponent"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction rests on a quantitative bootstrap: the differentiated Riemann variables, pulled back to the fast-acoustic coordinates, must stay uniformly bounded up to order $2n+1$ with the specific inequality $\\eta_x \\mathring{W} \\leq -\\tfrac12 + 4\\eta_x$ making the energy estimates contractive, and if those bounds failed, the blowup time could not be identified with the first zero of $\\eta_x$ and the cusp expansion would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Infinite cusp hierarchy for 1D Euler: each n gives Hölder 1/(2n+1)","Every integer n: a finite-time cusp of exact Hölder power 1/(2n+1) in 1D Euler","1D Euler cusps: infinite hierarchy, codimension-2n-2 stability for all n","Every n>0: a sharper cusp singularity in 1D Euler, exact Hölder exponent"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00092,"raw_usage":{"total_tokens":4044,"prompt_tokens":1143,"completion_tokens":2901,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":759,"completion_tokens_details":{"reasoning_tokens":2787}},"tokens_in":759,"tokens_out":2901,"duration_ms":20729,"temperature":1.0,"reasoning_tokens":2787,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:03:47.463959+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the explicit data $u_0 = \\sigma_0 = \\tfrac12 w_0$, $S_0 = 0$ with $w'_0(y) = -1 + y^{2n}$ (periodicized, positive, zero mean), the solution reduces to Burgers dynamics, and the theorem predicts a first singularity at $T_* = 2/(1+\\alpha)$ with log-log slope $-2n/(2n+1)$ for $|\\partial_y u|$ versus $|y-y_*|$ and leading coefficient $(2n+1)^{1/(2n+1)}$; a numerical simulation for $n = 2, 3, 4$ measuring the local exponent at the first time $\\min_x \\eta_x = 0$, or a simulation showing $\\eta_x$ vanishes at two distinct points before any gradient blowup, would settle the central claim directly.","supporting_citations":[{"cited_title":"The geometry of maximal d evelopment and shock formation for the Euler equations in mu ltiple space dimen- sions","cited_arxiv_id":null,"evidence_quote":"Supplies the characteristic-coordinate framework and the differentiated Riemann variables on which the entire proof is built."},{"cited_title":"For mation of shocks for 2D isentropic compressible Euler","cited_arxiv_id":null,"evidence_quote":"The prior self-similar-analysis construction of the stable C^{0,1/3} pre-shock whose n=1 case this paper recovers by characteristics."},{"cited_title":"Formation of unstab le shocks for 2D insentropic compressible Euler","cited_arxiv_id":null,"evidence_quote":"The only prior construction of an unstable (codimension-2) C^{0,1/5} cusp, which the present n=2 case recovers under milder regularity assumptions."},{"cited_title":"Description de la formation d’un choc dans le p-système","cited_arxiv_id":null,"evidence_quote":"The original construction of the C^{0,1/3} pre-shock in the p-system, serving as the base case of the hierarchy."},{"cited_title":"Singularity formation for burgers equation with transvers e viscosity","cited_arxiv_id":null,"evidence_quote":"Provides the Burgers-equation Puiseux expansion for initial slopes with flat minima, the scalar model whose exact computations seed the cusp formulas."},{"cited_title":"Gradient blow-u p for dispersive and dissipative perturbations of the Burge rs equation","cited_arxiv_id":null,"evidence_quote":"Establishes the analogous codimension-(2n-2) sets giving C^{0,1/(2n+1)} cusps for dispersive and dissipative perturbations of Burgers, the closest existing scalar statement of the finite-codimension phenomenon."},{"cited_title":"Shock forma tion in the compressible euler equations and related system s","cited_arxiv_id":null,"evidence_quote":"Used to rule out vacuum formation and implosion in one dimension, ensuring the first singularity is a gradient blowup characterized by the Eulerian blowup criterion."},{"cited_title":"A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy","cited_arxiv_id":"2302.01289","evidence_quote":"The authors' earlier characteristics-based treatment of the C^{0,1/3} cusp in azimuthal symmetry, whose polynomial-inversion appendix is generalized here."}],"review_version":1}