On the Ill-Posedness of the Prandtl Equation

David Gérard-Varet, Emmanuel Dormy · Journal of the American Mathematical Society / arXiv · 2010 · mathematics · 10.1090/S0894-0347-09-00652-3 · record gerardvaret2010-prandtl-illposedness
tag layer · index · concepts · authors · עברית

Where this record's text came from. faceted 2026-08 by reading the locally held text (pdftext); every evidence string verified as an exact substring of that text before merging. Claim type is coded existence — the theorem asserts that strong linear ill-posedness (and the unstable quasimodes realising it) exists — but the paper also states an explicit mechanism ('Quotation withheld.'), so mechanism would be a defensible second reading. method is mathematical-proof, though the paper adds a numerical component ('A numerical study of this instability is also provided.'). The second limitation quote is deliberately truncated mid-sentence: in the PDF extraction the sentence 'So far, we have not managed to improve the approximation ... for general us. This explains the technical restriction µ ∈ [0, 1/2) of part i).' is interleaved with inline-math fragments, so only the leading clause is quotable character-for-character. Displayed equations throughout the extraction are heavily scrambled; all quoted material is from running prose. 2026-08-14, 2026-08-16 and 2026-08-28 schema-0.3 passes: every quote verified as an exact substring of the staged text and anchored at merge time; concept terms verified present; every edge endpoint an existing node.

7 quotes on this page carry a character that renders as nothing. 35 such characters in all: a private-use glyph. The PDF this text came from used characters that survive extraction as code points no ordinary font draws, so the quote reads as though they were never there. That can change what a sentence says: a soft hyphen left by a line break puts a space inside a word, so document reads as docu ment; a flattened superscript turns 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.

6 quotes on this page may have lost a superscript, a subscript or a fraction slash. Nothing in them is corrupt — every character survived. What did not survive is POSITION: whether a digit sat above the line, below it, or between two others. So 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.

How to read this page. The article itself is not reproduced here — this paper is not cleared for redistribution, so what follows is the tag layer alone. The evidence sentences are withheld with the article. The table at the foot lists every tag the record carries; each row names the facet it belongs to, and the whose words column says whether the tag records the paper's own wording or the tagger's sentence about the paper. Every tag links to each other paper in the corpus carrying it. The machine-readable edition of the same thing is the TEI file beside this one, where the tags live in standOff, and the article is at its publisher via the link below. 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.

Other fields define these differently the same word, defined another way in another discipline
viscosity 2 fields

Tags on this paper

Every tag the record carries, each linking to every other paper in the corpus that carries it. A tag appears here whether or not its words occur in the article — the spans below are only the ones a sentence in this paper licenses. Of the 58 concept tags on this paper, 54 are grounded in a quoted sentence. 49 carry the tagger's gloss of what the term means in this paper, which is not something the article can license.

Prandtl equation boundary layer ill-posedness Cauchy problem Sobolev spaces Navier-Stokes equations viscosity boundary layer separation critical point inviscid limit no-slip boundary condition vorticity field boundary layer phenomenon parabolic scaling normal direction Prandtl equations formal asymptotics Prandtl model functional spaces tangential variable x derivatives loss of one derivative horizontal diffusion term standard energy estimates analytic framework Rayleigh instability inviscid version finite loss of x-derivatives (weakly) well-posed strong destabilization mechanism base velocity profile monotonic data shear velocity profile linearization analytic regularity high tangential frequencies unstable quasimodes spectral condition reduced ODE instability process singular perturbation problem inviscid eigenelements strong instability heteroclinic orbits holomorphic solution complex change of variable formal analysis numerical illustrations Evans function shooting method Runge-Kutta scheme Newton-Raphson procedure direct simulations finite differences stretched grid Crank-Nicholson scheme initial random data excellent agreement 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 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 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 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 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 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 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 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 the analytical result is accompanied by numerical computation whose stated purpose is to show the mechanism is EFFECTIVE rather than merely possible Prandtl ill-posed instability Sobolev viscous equation Navier-Stokes analytic mathematical-proof

Legend

relation to other work	`claim_relations`	how this paper stands to another: agrees, contradicts, extends, qualifies
claim type	`claim_type`	the shape of the claim: existence, mechanism, method, critique, prediction, null
concept edge	`concept_edges`	an asserted relation between two concepts, with the sentence asserting it
concept	`concepts`	an idea the paper works with, quoted where the paper uses it
content tag	`content_tags`	a finding, definition, quantity, statistic or variable named in the paper
limitation	`limitations`	what the paper says its evidence cannot support
mechanism	`mechanisms`	how the reported effect is said to come about
method	`method`	how the paper produced its evidence
phenomenon	`phenomenon`	the effect or regularity under study
sense	`senses`	a distinct use of a term, quoted where the paper fixes it
unrecordable relation	`unhoused_relations`	a relation the paper asserts that this vocabulary cannot express, with the reason

The paper

The full text is not reproduced here. This article is not cleared for redistribution, so the page carries the tag layer only. Its evidence sentences are withheld; the grounding status remains with each tag, and the article itself is at https://doi.org/10.1090/S0894-0347-09-00652-3. The reading view exists for the papers whose licence permits it.

Every tag on this page

claim_relations 3 · claim_type 1 · concept_edges 2 · concepts 54 · content_tags 5 · limitations 6 · mechanisms 9 · method 1 · phenomenon 1 · senses 15 · unhoused_relations 9

Relations on this page carry a quotation for the claim. Where a relation was separately evidenced — and so tested against the vocabulary for its type — the row says so; none of 2 relations here carry that mark. The rest are the tagger’s reading of the sentence quoted beside them.

This record holds no controls. A layer is empty when the tagger recorded nothing in it — which is not the same as the paper having nothing to record.

facet	tag	field	conf.	whose words
claim_type	existence Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_type`		the paper's words
phenomenon	ill-posedness of the Prandtl boundary layer equation Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`phenomenon`		the paper's words
method	mathematical-proof Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: mathematical-proof	`method`		the paper's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[0]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[1]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[2]`	0.85	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[3]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[4]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[5]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: 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. also in: 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	`mechanisms[6]`	0.95	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: 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	`mechanisms[7]`	0.85	the tagger's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: the analytical result is accompanied by numerical computation whose stated purpose is to show the mechanism is EFFECTIVE rather than merely possible	`mechanisms[8]`	0.95	the tagger's words
limitations	the relevance of the asymptotics in Lp and in the absence of Rayleigh instabilities remains open Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: 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	`limitations[0]`		the tagger's words
limitations	The negative result is confined to the viscous Sobolev setting and leaves the known positive (inviscid and monotonic-data) results intact. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: 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	`limitations[1]`		the tagger's words
limitations	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 record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: the stronger part of the theorem: Quotation withheld general base profiles	`limitations[2]`		the tagger's words
limitations	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 record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: 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	`limitations[3]`	1.0	the tagger's words
limitations	The empirical support for the description is bounded to one side of the separation zone, and the paper says which side. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: 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	`limitations[4]`	1.0	the tagger's words
limitations	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 record is not cleared for redistribution. The sentence is in the paper at the link above. what this limits: 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	`limitations[5]`	1.0	the tagger's words
concepts	Prandtl equation Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Prandtl equation	`concepts[0]`		the paper's words
concepts	ill-posedness Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: ill-posedness	`concepts[2]`		the paper's words
concepts	viscosity Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: viscosity	`concepts[6]`		the paper's words
concepts	boundary layer separation Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: boundary layer separation	`concepts[7]`		the paper's words
concepts	critical point Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: critical point	`concepts[8]`		the paper's words
concepts	inviscid limit Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: inviscid limit	`concepts[9]`	0.95	the paper's words
concepts	no-slip boundary condition Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: no-slip boundary condition	`concepts[10]`	0.95	the paper's words
concepts	vorticity field Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: vorticity field	`concepts[11]`	0.95	the paper's words
concepts	boundary layer phenomenon Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: boundary layer phenomenon	`concepts[12]`	0.95	the paper's words
concepts	parabolic scaling Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: parabolic scaling	`concepts[13]`	0.95	the paper's words
concepts	normal direction Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: normal direction	`concepts[14]`	0.95	the paper's words
concepts	Prandtl equations Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Prandtl equations	`concepts[15]`	0.95	the paper's words
concepts	formal asymptotics Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: formal asymptotics	`concepts[16]`	0.95	the paper's words
concepts	Prandtl model Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Prandtl model	`concepts[17]`	0.95	the paper's words
concepts	functional spaces Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: functional spaces	`concepts[18]`	0.95	the paper's words
concepts	tangential variable Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: tangential variable	`concepts[19]`	0.95	the paper's words
concepts	x derivatives Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: x derivatives	`concepts[20]`	0.95	the paper's words
concepts	loss of one derivative Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: loss of one derivative	`concepts[21]`	0.95	the paper's words
concepts	horizontal diffusion term Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: horizontal diffusion term	`concepts[22]`	0.95	the paper's words
concepts	standard energy estimates Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: standard energy estimates	`concepts[23]`	0.95	the paper's words
concepts	analytic framework Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: analytic framework	`concepts[24]`	0.95	the paper's words
concepts	Rayleigh instability Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Rayleigh instability	`concepts[25]`	0.95	the paper's words
concepts	inviscid version Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: 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. also in: inviscid version	`concepts[26]`	0.95	the paper's words
concepts	finite loss of x-derivatives Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: finite loss of x-derivatives	`concepts[27]`	0.95	the paper's words
concepts	(weakly) well-posed Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: (weakly) well-posed	`concepts[28]`	0.95	the paper's words
concepts	strong destabilization mechanism Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: strong destabilization mechanism	`concepts[29]`	0.95	the paper's words
concepts	base velocity profile Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: base velocity profile	`concepts[30]`	0.95	the paper's words
concepts	monotonic data Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: monotonic data	`concepts[31]`	0.95	the paper's words
concepts	shear velocity profile Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: shear velocity profile	`concepts[32]`	0.95	the paper's words
concepts	linearization Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: linearization	`concepts[33]`	0.95	the paper's words
concepts	analytic regularity Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: analytic regularity	`concepts[34]`	0.95	the paper's words
concepts	high tangential frequencies Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: high tangential frequencies	`concepts[35]`	0.95	the paper's words
concepts	unstable quasimodes Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: unstable quasimodes	`concepts[36]`	0.95	the paper's words
concepts	spectral condition Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: spectral condition	`concepts[37]`	0.95	the paper's words
concepts	reduced ODE Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: reduced ODE	`concepts[38]`	0.95	the paper's words
concepts	instability process Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: instability process	`concepts[39]`	0.95	the paper's words
concepts	singular perturbation problem Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: 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. also in: singular perturbation problem	`concepts[40]`	0.95	the paper's words
concepts	inviscid eigenelements Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: inviscid eigenelements	`concepts[41]`	0.95	the paper's words
concepts	strong instability Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: strong instability	`concepts[42]`	0.95	the paper's words
concepts	heteroclinic orbits Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: heteroclinic orbits	`concepts[43]`	0.95	the paper's words
concepts	holomorphic solution Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: holomorphic solution	`concepts[44]`	0.95	the paper's words
concepts	complex change of variable Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: complex change of variable	`concepts[45]`	0.95	the paper's words
concepts	formal analysis Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: formal analysis	`concepts[46]`	0.95	the paper's words
concepts	numerical illustrations Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: numerical illustrations	`concepts[47]`	0.95	the paper's words
concepts	Evans function Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Evans function	`concepts[48]`	0.95	the paper's words
concepts	shooting method Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: shooting method	`concepts[49]`	0.95	the paper's words
concepts	Runge-Kutta scheme Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Runge-Kutta scheme	`concepts[50]`	0.95	the paper's words
concepts	Newton-Raphson procedure Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Newton-Raphson procedure	`concepts[51]`	0.95	the paper's words
concepts	direct simulations Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: direct simulations	`concepts[52]`	0.95	the paper's words
concepts	finite differences Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: finite differences	`concepts[53]`	0.95	the paper's words
concepts	stretched grid Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: stretched grid	`concepts[54]`	0.95	the paper's words
concepts	Crank-Nicholson scheme Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Crank-Nicholson scheme	`concepts[55]`	0.95	the paper's words
concepts	initial random data Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: initial random data	`concepts[56]`	0.95	the paper's words
concepts	excellent agreement Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: excellent agreement	`concepts[57]`	0.95	the paper's words
unhoused_relations	vorticity field → inviscid limit Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. Lack of uniform vorticity bounds is the main mathematical difficulty in taking the inviscid limit. Difficulty-for is absent from the governed relation types.	`unhoused_relations[0]`	0.95	the tagger's words
unhoused_relations	parabolic scaling → normal direction Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. The boundary-layer asymptotics applies a parabolic scaling in the normal direction. Applied-in-direction is absent from the governed relation types.	`unhoused_relations[1]`	0.95	the tagger's words
unhoused_relations	loss of one derivative → standard energy estimates Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. The unbalanced loss of one tangential derivative prevents standard energy estimates from applying. Prevents is absent from the governed relation types.	`unhoused_relations[2]`	0.95	the tagger's words
unhoused_relations	Rayleigh instability → formal asymptotics Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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.	`unhoused_relations[3]`	0.95	the tagger's words
unhoused_relations	finite loss of x-derivatives → (weakly) well-posed Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. 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.	`unhoused_relations[4]`	0.95	the tagger's words
unhoused_relations	spectral condition → ill-posedness Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. Ill-posedness of the PDE is reduced to a spectral condition for a reduced ODE. Reduces-to is absent from the governed relation types.	`unhoused_relations[5]`	0.95	the tagger's words
unhoused_relations	Evans function → spectral condition Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. The spectral condition is reformulated with an Evans function for numerical checking. Reformulates is absent from the governed relation types. — the relation stated: Quotation withheld 5.1 Quotation withheld	`unhoused_relations[6]`	0.95	the tagger's words
unhoused_relations	shooting method → Newton-Raphson procedure Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. Backward shooting supplies the function on which Newton-Raphson locates a zero. Sequential-input-to is absent from the governed relation types.	`unhoused_relations[7]`	0.95	the tagger's words
unhoused_relations	finite differences → Crank-Nicholson scheme Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. The direct simulation combines finite-difference spatial discretization with Crank-Nicholson time stepping. Combined-with is absent from the governed relation types.	`unhoused_relations[8]`	0.95	the tagger's words
senses	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 Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[0]`	1.0	the tagger's words
senses	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 Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[1]`	1.0	the tagger's words
senses	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 Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[2]`	1.0	the tagger's words
senses	Inviscid-limit difficulty - no uniform vorticity bound as viscosity vanishes, corresponding physically to a boundary layer Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[3]`	0.95	the tagger's words
senses	Prandtl asymptotics - formally natural but not uniformly valid because boundary-layer separation can occur Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[4]`	0.95	the tagger's words
senses	Tangential regularity obstacle - one derivative is lost without horizontal diffusion to balance it Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[5]`	0.95	the tagger's words
senses	Analytic setting - sufficient for local boundary-layer justification but filters instability mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[6]`	0.95	the tagger's words
senses	Inviscid Cauchy problem - characteristic solution with finite tangential derivative loss and weak well-posedness Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[7]`	0.95	the tagger's words
senses	Viscous destabilization - viscosity and non-degenerate critical points jointly underlie the Sobolev failure Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[8]`	0.95	the tagger's words
senses	Strong linear ill-posedness - high-frequency perturbations grow on a square-root-frequency timescale Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[9]`	0.95	the tagger's words
senses	Spectral reduction - PDE ill-posedness is reduced to a condition on a lower-dimensional ODE Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[10]`	0.95	the tagger's words
senses	Spectral-condition problem - existence of a heteroclinic orbit for the reduced ordinary differential equation Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[11]`	0.95	the tagger's words
senses	Evans-function computation - shooting with Runge-Kutta integration followed by Newton-Raphson root finding Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[12]`	0.95	the tagger's words
senses	Direct numerical test - finite differences on a stretched grid and Crank-Nicholson time stepping from random data Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[13]`	0.95	the tagger's words
senses	Numerical confirmation - eigenvalue and eigenmode comparisons agree with theory and show the mechanism dominates linear dynamics Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`senses[14]`	0.95	the tagger's words
claim_relations	extends → prior well-posedness results for the Prandtl equation (analytic or monotone data) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[0]`		the tagger's words
claim_relations	builds-on → reference [8] (analysis of the inviscid version of the Prandtl equation) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[1]`		the tagger's words
claim_relations	builds-on → reference [8] (study of the inviscid spectral problem and its evolution equation) Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[2]`		the tagger's words
concept_edges	viscosity —causes→ ill-posedness Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 1 also in: viscosity · ill-posedness	`concept_edges[0]`		the tagger's words
concept_edges	critical point —causes→ ill-posedness Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. strength of the relation: 1 also in: critical point · ill-posedness	`concept_edges[1]`		the tagger's words
content_tags	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 record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[0]`		the tagger's words
content_tags	Strikingly, the strong instability is caused by VISCOSITY — consistent with well-posedness of the inviscid version of the equation. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[1]`		the tagger's words
content_tags	Prandtl equation Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: 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. also in: Prandtl equation	`content_tags.definitions[0]`		the tagger's words
content_tags	ill-posedness Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: ill-posedness	`content_tags.definitions[1]`		the tagger's words
content_tags	inviscid limit Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: inviscid limit	`content_tags.definitions[2]`		the tagger's words

Interpretive choices

Tags that could not be placed in the text:

none — every tag was located in the text

What was deliberately not tagged, and why, is recorded per paper by the tagger. Where a paper has no such record, this page says so rather than implying that nothing was declined.