10⁹ into 109. The quotes are reproduced exactly as the extraction left them and have NOT been repaired — the meaning of such a code point belongs to the one PDF it came from, and guessing it would produce a clean, confident, unverifiable quote.x 10 9 may be x 10⁹, chi 2 5 may be chi-squared with five degrees of freedom, and 1 2 may be one half. Read any expression here against the published paper before relying on it.phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | ill-posedness of the Prandtl boundary layer equation— “This theorem expresses strong linear ill-posedness of the Prandtl equation in the Sobolev framework.” | |
| claim-type | existence— “We prove here that it is linearly ill-posed in Sobolev type spaces.” | |
| method — evidence | — “The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solutions with non-degenerate critical points.” | |
| mechanism | THE EXPECTED ANSWER IS STATED BEFORE IT IS OVERTURNED: on the strength of the inviscid result the authors say it seems reasonable to bet on well-posedness, and then show the equation is linearly ill-posed in that very setting— “On the basis of the inviscid result, it seems reasonable to bet for well-posedness of the Prandtl equation (1.3) in Sobolev type spaces. The aim of this paper is to show that it is actually linearly ill-posed in this framework.” | mechanisms[0] |
| mechanism | the destabilisation is attributed to TWO ingredients together - viscosity, and critical points in the base velocity profile - and the authors immediately state what that conjunction buys them: the result does not contradict the positive results for the inviscid case or for monotonic data— “As we shall see later on, the reason for illposedness is a strong destabilization mechanism due to two ingredients: viscosity, and critical points in the base velocity profile. In particular, it does not contradict the positive results obtained in the inviscid case and for monotonic data.” | mechanisms[1] |
| mechanism | THE PDE QUESTION IS REDUCED TO A CONDITION ON AN ORDINARY DIFFERENTIAL EQUATION: an asymptotic analysis in the high tangential frequency limit turns ill-posedness for the linearised system into a spectral condition for a reduced ODE— “It relies on an asymptotic analysis of (1.6), in the high tangential frequency limit. Thanks to this analysis, we show that ill-posedness for the PDE (1.6) comes down to a “spectral condition” for a reduced ODE, namely:” | mechanisms[2] |
| mechanism | the instability is located at a SCALE rather than at a place: it happens at high tangential frequencies with a typical time that shrinks with them, and at that timescale the time dependence of the base velocity does not matter - which lets the authors replace the real equation with a simpler frozen one— “As we shall see, it takes place at high tangential frequencies, say O(1/ε), and has a typical time O(√ε). At this timescale, the time dependence of the base velocity (us(t, y), 0) will not play an important role. Thus, to understand the instability mechanism, we can consider the simpler equation” | mechanisms[3] |
| mechanism | A FREE PARAMETER IS ADDED TO THE ANSATZ PRECISELY BECAUSE THE SYSTEM IS OVERDETERMINED - the jump and boundary conditions supply too many constraints - and the paper says that this is what justifies introducing it— “Let us point out that this system is a priori overdetermined, as jump and boundary conditions provide too many constraints. This justifies the introduction of the parameter τ in the Ansatz (2.5).” | mechanisms[4] |
| mechanism | the corrector's ROLE IS STATED AS CANCELLATION: the inviscid solution is not smooth at the critical point, and the shear layer term, which concentrates there, exists to cancel the discontinuities the regular part leaves behind— “However, it has a jump at y = a, together with its second derivative. The role of the shear layer vsl ε , which concentrates near y = a, is to cancel these discontinuities.” | mechanisms[5] |
| mechanism | the sign of a second derivative would split the argument in two, and a change of variables collapses the split: if the condition holds for one sign then a transformed pair satisfies the other, so BOTH CASES yield a solution with the property the instability needs— “If U ′′s (a) < 0, it is exactly the system in (SC). If on the contrary U ′′s (a) > 0, and if (τ, W ) satisfies the system in (SC), then (τ := τ , W := W ) satisfies the above system. In both − m τ < 0. cases, back to the original system (2.6), condition (SC) gives a solution (τ, V ) with” | mechanisms[6] |
| mechanism | WELL-POSEDNESS IN THE ANALYTIC SETTING IS PROVED IN THE SAME PAPER, as a foil: the Cauchy problem for the linearised equation is locally well-posed there, which is what makes the Sobolev result a statement about the function space rather than about the equation— “In short, the Cauchy problem for (1.6) is locally well-posed in the analytic setting.” | mechanisms[7] |
| mechanism | the analytical result is accompanied by numerical computation whose stated purpose is to show the mechanism is EFFECTIVE rather than merely possible— “We end up the paper with numerical computations, which emphasize that our instability mechanism is effective.” | mechanisms[8] |
| extends | prior well-posedness results for the Prandtl equation (analytic or monotone data)— “This problem is known to be well-posed for analytic data [13, 10], or for data with monotonicity properties [11, 15].” | claim_relations[0] |
| builds-on | reference [8] (analysis of the inviscid version of the Prandtl equation)— “Interestingly, the strong instability is due to vicosity, which is coherent with well-posedness results obtained for the inviscid version of the equation [8].” | claim_relations[1] |
| builds-on | reference [8] (study of the inviscid spectral problem and its evolution equation)— “This spectral problem, as well as the corresponding evolution equation, have been studied exhaustively in [8].” | claim_relations[2] |
| sense | ill-posedness - LINEAR, and only that: the theorem is about the linearization around solutions with non-degenerate critical points, so the bare claim 'the Prandtl equation is ill-posed' drops the word that bounds the result— “This theorem expresses strong linear ill-posedness of the Prandtl equation in the Sobolev framework. It is a consequence of an instability process, that holds at high tangential frequencies” | senses[0] |
| sense | the expected answer - STATED AND THEN OVERTURNED: the paper records that the inviscid result made well-posedness the reasonable bet, and then proves the opposite in this framework— “On the basis of the inviscid result, it seems reasonable to bet for well-posedness of the Prandtl equation (1.3) in Sobolev type spaces. The aim of this paper is to show that it is actually linearly ill-posed in this framework” | senses[1] |
| sense | the source of instability - VISCOSITY, which is the term usually read as stabilising: the paper names viscosity and critical points in the base profile as the two ingredients— “As we shall see later on, the reason for illposedness is a strong destabilization mechanism due to two ingredients: viscosity, and critical points in the base velocity profile” | senses[2] |
| sense | Inviscid-limit difficulty - no uniform vorticity bound as viscosity vanishes, corresponding physically to a boundary layer— “One challenging open problem of fluid dynamics is to understand the inviscid limit of the Navier-Stokes equations ∂tuν + uν uν + pν ∇ − · ∇ ν∆uν =… [checked against the archived source, not this rendering] | senses[3] |
| sense | Prandtl asymptotics - formally natural but not uniformly valid because boundary-layer separation can occur | senses[4] |
| sense | Tangential regularity obstacle - one derivative is lost without horizontal diffusion to balance it Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[5] |
| sense | Analytic setting - sufficient for local boundary-layer justification but filters instability mechanisms Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[6] |
| sense | Inviscid Cauchy problem - characteristic solution with finite tangential derivative loss and weak well-posedness Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[7] |
| sense | Viscous destabilization - viscosity and non-degenerate critical points jointly underlie the Sobolev failure Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[8] |
| sense | Strong linear ill-posedness - high-frequency perturbations grow on a square-root-frequency timescale Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[9] |
| sense | Spectral reduction - PDE ill-posedness is reduced to a condition on a lower-dimensional ODE Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[10] |
| sense | Spectral-condition problem - existence of a heteroclinic orbit for the reduced ordinary differential equation Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[11] |
| sense | Evans-function computation - shooting with Runge-Kutta integration followed by Newton-Raphson root finding Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[12] |
| sense | Direct numerical test - finite differences on a stretched grid and Crank-Nicholson time stepping from random data Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[13] |
| sense | Numerical confirmation - eigenvalue and eigenmode comparisons agree with theory and show the mechanism dominates linear dynamics Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | senses[14] |
| edge | viscosity —causes→ ill-posedness Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[0] |
| edge | critical point —causes→ ill-posedness Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concept_edges[1] |
| evidence status note | concept — THE TERM IS NOT IN THE QUOTE - the quote's 'this equation' is the inviscid version of (1.3), named in the sentence just before it ('the inviscid version of (1.3) is analyzed'). That sentence is not quoted because a displaced fragment of the equation breaks it. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[26] |
| evidence status note | concept — THE TERM IS NOT IN THE QUOTE - text extraction moved part of the quote's first sentence out of it: the words 'yields a singular perturbation problem' now stand just after the quote. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[40] |
| for which use | limitation — the asymptotic result's reach: THE RELEVANCE OF THIS ASYMPTOTICS IN Lp, OR ITS RELEVANCE IN THE ABSENCE OF RAYLEIGH INSTABILITIES, ARE STILL OPEN ISSUES Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[0] |
| for which use | limitation — the ill-posedness result, which is confined to its setting: IT DOES NOT CONTRADICT THE POSITIVE RESULTS OBTAINED IN THE INVISCID CASE AND FOR MONOTONIC DATA Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[1] |
| for which use | limitation — the stronger part of the theorem: SO FAR, WE HAVE NOT MANAGED TO IMPROVE THE APPROXIMATION OF THE PREVIOUS PARAGRAPH FOR general base profiles Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[2] |
| for which use | limitation — the boundary-layer asymptotics as a description of the flow: the paper states its validity is not clear and that physicists including Prandtl himself emphasised it may not hold uniformly in space and time Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[3] |
| for which use | limitation — experimental agreement with the boundary-layer description: it is stated to fit many experiments upstream from the separation zone, which does not extend to the separated region Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[4] |
| for which use | limitation — the mechanism as described in section 2: it is derived for a base velocity whose time variation is neglected, and accounting for the original system requires modifications Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[5] |
| evidence status note | mechanism — 2026-08-16: A FRAGMENT IS DISPLACED INSIDE THIS QUOTE - damage shape 5, a word-order scramble. The staged text reads "satisfies the above system. In both - m tau < 0. cases, back to the original system (2.6), condition (SC) gives a solution (tau, V ) with". The fragment "Im tau < 0" belongs at the END of the sentence, completing "gives a solution (tau, V) with Im tau < 0"; the extraction has lifted it out and dropped it between "In both" and "cases". Read as: "In both cases, back to the original system (2.6), condition (SC) gives a solution (tau, V) with Im tau < 0." The quote is an exact substring and the file is out of order here. No guard detects this shape; found by reading. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | mechanisms[6] |
| relation | Evans function → spectral condition — In this last section, we present numerical illustrations of the instability process. 5.1 Numerical test of (SC) To check (SC) numerically, it is more convenient to reformulate it with an Evans function Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[6] |
| replication | untested | |
| no type for this | vorticity field → inviscid limit — Lack of uniform vorticity bounds is the main mathematical difficulty in taking the inviscid limit. Difficulty-for is absent from the governed relation types. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[0] |
| no type for this | parabolic scaling → normal direction — The boundary-layer asymptotics applies a parabolic scaling in the normal direction. Applied-in-direction is absent from the governed relation types. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[1] |
| no type for this | loss of one derivative → standard energy estimates — The unbalanced loss of one tangential derivative prevents standard energy estimates from applying. Prevents is absent from the governed relation types. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[2] |
| no type for this | Rayleigh instability → formal asymptotics — Rayleigh instability was used in prior work to show that the boundary-layer asymptotics does not always hold in a Sobolev space. Invalidates-in-a-setting is absent. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[3] |
| no type for this | finite loss of x-derivatives → (weakly) well-posed — Only finite tangential derivative loss is the stated basis for calling the inviscid Cauchy problem weakly well-posed. Supports-status is absent from the governed types. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[4] |
| no type for this | spectral condition → ill-posedness — Ill-posedness of the PDE is reduced to a spectral condition for a reduced ODE. Reduces-to is absent from the governed relation types. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[5] |
| no type for this | Evans function → spectral condition — The spectral condition is reformulated with an Evans function for numerical checking. Reformulates is absent from the governed relation types. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[6] |
| no type for this | shooting method → Newton-Raphson procedure — Backward shooting supplies the function on which Newton-Raphson locates a zero. Sequential-input-to is absent from the governed relation types. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[7] |
| no type for this | finite differences → Crank-Nicholson scheme — The direct simulation combines finite-difference spatial discretization with Crank-Nicholson time stepping. Combined-with is absent from the governed relation types. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | unhoused_relations[8] |
| limitation | the relevance of the asymptotics in Lp and in the absence of Rayleigh instabilities remains open Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[0] |
| limitation | The negative result is confined to the viscous Sobolev setting and leaves the known positive (inviscid and monotonic-data) results intact. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[1] |
| limitation | The refined approximation behind the stronger part of the theorem could not be obtained for general base profiles, forcing a restriction to special quadratic-near-the-critical-point profiles. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[2] |
| limitation | The originator of the model is cited as doubting that it holds uniformly, and the paper states outright that the validity of the asymptotics is not clear. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[3] |
| limitation | The empirical support for the description is bounded to one side of the separation zone, and the paper says which side. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[4] |
| limitation | The instability mechanism is derived with the time variation of the base velocity dropped, and carrying it back to the real system is stated to require modification. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | limitations[5] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | The Prandtl equation is linearly ill-posed in Sobolev-type spaces, shown by constructing unstable quasimodes at high tangential frequencies around solutions with non-degenerate critical points. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[0] |
| finding | Strikingly, the strong instability is caused by VISCOSITY — consistent with well-posedness of the inviscid version of the equation. Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.findings[1] |
| definition | Prandtl equation Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[0] |
| evidence status note | definition — QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. | content_tags.definitions[0] |
| definition | ill-posedness Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[1] |
| definition | inviscid limit Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | content_tags.definitions[2] |
| key-terms | Prandtl, boundary layer, ill-posed, instability, Sobolev, viscous, equation, Navier-Stokes, analytic |
| concept | Prandtl equation Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[0] |
| concept | ill-posedness Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[2] |
| concept | viscosity Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[6] |
| concept | boundary layer separation Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[7] |
| concept | critical point Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[8] |
| concept | inviscid limit — not defined in this paper's own words; vanishing-viscosity limit of the Navier-Stokes system posed as the motivating fluid-dynamics problem Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[9] |
| concept | no-slip boundary condition — not defined in this paper's own words; boundary constraint in the motivating Navier-Stokes problem Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[10] |
| concept | vorticity field — not defined in this paper's own words; field lacking uniform bounds as viscosity tends to zero Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[11] |
| concept | boundary layer phenomenon — not defined in this paper's own words; near-boundary behavior corresponding to failure of uniform vorticity control Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[12] |
| concept | parabolic scaling — not defined in this paper's own words; normal-coordinate scaling used to represent the boundary layer Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[13] |
| concept | normal direction — not defined in this paper's own words; direction across the boundary in which the parabolic rescaling is applied Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[14] |
| concept | Prandtl equations — not defined in this paper's own words; boundary-layer equations formally obtained from the scaled Navier-Stokes ansatz Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[15] |
| concept | formal asymptotics — not defined in this paper's own words; natural boundary-layer expansion whose uniform validity is uncertain Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[16] |
| concept | Prandtl model — not defined in this paper's own words; boundary-layer description whose relevance and limitations motivate the paper Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[17] |
| concept | functional spaces — not defined in this paper's own words; regularity setting on which both well-posedness and asymptotic justification depend Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[18] |
| concept | tangential variable — not defined in this paper's own words; boundary-parallel coordinate in which regularity is difficult to control Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[19] |
| concept | x derivatives — not defined in this paper's own words; tangential derivatives for which the equation lacks control Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[20] |
| concept | loss of one derivative — not defined in this paper's own words; regularity deficit introduced when recovering the normal velocity from divergence Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[21] |
| concept | horizontal diffusion term — not defined in this paper's own words; missing stabilizing term that would balance tangential derivative loss Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[22] |
| concept | standard energy estimates — not defined in this paper's own words; standard estimates rendered inapplicable by unbalanced derivative loss Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[23] |
| concept | analytic framework — not defined in this paper's own words; high-regularity setting that filters out small-scale instability mechanisms Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[24] |
| concept | Rayleigh instability — not defined in this paper's own words; inviscid-flow mechanism previously used to invalidate the boundary-layer expansion in a Sobolev setting Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[25] |
| concept | inviscid version — not defined in this paper's own words; zero-diffusion Prandtl system admitting characteristic solutions and only finite derivative loss Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[26] |
| concept | finite loss of x-derivatives — not defined in this paper's own words; bounded tangential regularity loss compatible with weak well-posedness Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[27] |
| concept | (weakly) well-posed — not defined in this paper's own words; status of the inviscid Cauchy problem under finite derivative loss Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[28] |
| concept | strong destabilization mechanism — not defined in this paper's own words; viscosity-and-critical-point mechanism producing Sobolev ill-posedness Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[29] |
| concept | base velocity profile — not defined in this paper's own words; velocity profile whose critical points participate in destabilization Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[30] |
| concept | monotonic data — not defined in this paper's own words; data regime whose positive results are not contradicted by the critical-point theorem Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[31] |
| concept | shear velocity profile — not defined in this paper's own words; heat-equation solution about which the Prandtl system is linearized Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[32] |
| concept | linearization — not defined in this paper's own words; linear perturbation system constructed around the shear solution Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[33] |
| concept | analytic regularity — not defined in this paper's own words; exponential Fourier decay in the tangential coordinate Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[34] |
| concept | high tangential frequencies — not defined in this paper's own words; tangential regime in which perturbations exhibit rapid growth Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[35] |
| concept | unstable quasimodes — not defined in this paper's own words; high-frequency approximate modes built around non-degenerate critical points Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[36] |
| concept | spectral condition — not defined in this paper's own words; reduced ordinary-differential condition to which PDE ill-posedness is reduced Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[37] |
| concept | reduced ODE — not defined in this paper's own words; ordinary differential equation carrying the spectral condition Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[38] |
| concept | instability process — not defined in this paper's own words; high-frequency process responsible for strong linear ill-posedness Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[39] |
| concept | singular perturbation problem — not defined in this paper's own words; normal-coordinate problem arising in the high tangential-frequency limit Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[40] |
| concept | inviscid eigenelements — not defined in this paper's own words; zero-viscosity eigenpairs corrected when viscosity is restored Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[41] |
| concept | strong instability — not defined in this paper's own words; growth created by the square-root correction to the eigenvalue Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[42] |
| concept | heteroclinic orbits — not defined in this paper's own words; ODE trajectories whose existence establishes the spectral condition Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[43] |
| concept | holomorphic solution — not defined in this paper's own words; complex-domain ODE solution supplied by analytic-coefficient theory Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[44] |
| concept | complex change of variable — not defined in this paper's own words; rotation and rescaling used to obtain the needed decaying solution Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[45] |
| concept | formal analysis — not defined in this paper's own words; section-two construction from which the rigorous ill-posedness theorem is derived Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[46] |
| concept | numerical illustrations — not defined in this paper's own words; computations used to test effectiveness of the instability process Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[47] |
| concept | Evans function — not defined in this paper's own words; numerically convenient reformulation of the spectral condition Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[48] |
| concept | shooting method — not defined in this paper's own words; backward-integration approach for locating the complex spectral parameter Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[49] |
| concept | Runge-Kutta scheme — not defined in this paper's own words; numerical integrator used in the shooting calculation Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[50] |
| concept | Newton-Raphson procedure — not defined in this paper's own words; root-finding procedure applied after backward integration Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[51] |
| concept | direct simulations — not defined in this paper's own words; time-dependent numerical test of the proposed mechanism Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[52] |
| concept | finite differences — not defined in this paper's own words; spatial discretization used for direct simulation Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[53] |
| concept | stretched grid — not defined in this paper's own words; nonuniform spatial grid supporting the finite-difference discretization Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[54] |
| concept | Crank-Nicholson scheme — not defined in this paper's own words; time-stepping method used for the simulation Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[55] |
| concept | initial random data — not defined in this paper's own words; random gridpoint values used to initialize the numerical evolution Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[56] |
| concept | excellent agreement — not defined in this paper's own words; reported match confirming effectiveness and dominance of the mechanism Quotation withheld — this paper is not ours to reproduce, and the page has already quoted as much of it as the rights layer allows. The tag, its field and its confidence stand; the sentence is at the publisher. | concepts[57] |
When we have changed our mind. Some entries in the table begin with a word in capitals and the reason for it. EVIDENCE CORRECTED — the finding still stands, but we had quoted the wrong sentence from the paper, and the quotation has been replaced with the right one. RETRACTED — the entry itself was wrong; it stays on the page, marked, with the reason, rather than being deleted. WITHDRAWN AND REPLACED — the entry was wrong, and a corrected version of it appears in the entries below. Each mark carries its own reason, in brackets, because the reasons differ from one entry to the next. An entry we have WITHDRAWN is removed from this page and kept in the record, with its reason. A line reading Audit and a date is a later re-reading of the entry against its whole sentence: DOUBTFUL means it was kept, and the reason says what is loose about it. Quotation marks inside the tagger's sentence. Where a tag is the tagger's own sentence, a phrase in quotation marks inside it is borrowed from the paper — the tagger's wording around it is not. The paper's own sentence is the evidence quote for that row; on this page it is withheld with the article, and the article is at its publisher. Nothing inside quotation marks anywhere on this page is ours: if it is quoted, it is the paper's. Whose words a concept carries. A concept entry records a term the paper itself uses, so the term, the sense given for it, and its evidence are the paper's own words, quoted where the paper fixes them. This is the opposite of a sense entry in the tag table, which is the tagger's sentence about how the paper uses a term. Where a concept entry could not be grounded in the paper's words, the entry says so on its own row.
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