On the Ill-Posedness of the Prandtl Equation

On the Ill-Posedness of the Prandtl Equation David Gérard-Varet Emmanuel Dormy

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On the Ill-Posedness of the Prandtl Equation 2010 Journal of the American Mathematical Society / arXiv 10.1090/S0894-0347-09-00652-3 0904.0434 gerardvaret2010-prandtl-illposedness gerardvaret2010-prandtl-illposedness.txt, the project's staged copy 82dc7379578e2e1ffc781bf74af4854cfdfb3fe6 arxiv-open mathematics

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them existence Quotation withheld because this record is not cleared for redistribution. ill-posedness of the Prandtl boundary layer equation Quotation withheld because this record is not cleared for redistribution. mathematical-proof mathematical-proof Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms 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 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. Quotation withheld because this record is not cleared for redistribution. mechanisms 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 Quotation withheld because this record is not cleared for redistribution. mechanisms the analytical result is accompanied by numerical computation whose stated purpose is to show the mechanism is EFFECTIVE rather than merely possible Quotation withheld because this record is not cleared for redistribution. the relevance of the asymptotics in Lp and in the absence of Rayleigh instabilities remains open 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 Quotation withheld because this record is not cleared for redistribution. The negative result is confined to the viscous Sobolev setting and leaves the known positive (inviscid and monotonic-data) results intact. 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 Quotation withheld because this record is not cleared for redistribution. 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. what this limits: the stronger part of the theorem: Quotation withheld general base profiles Quotation withheld because this record is not cleared for redistribution. 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. 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 Quotation withheld because this record is not cleared for redistribution. The empirical support for the description is bounded to one side of the separation zone, and the paper says which side. 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 Quotation withheld because this record is not cleared for redistribution. 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. 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 Quotation withheld because this record is not cleared for redistribution. Prandtl equation Prandtl equation Quotation withheld because this record is not cleared for redistribution. ill-posedness ill-posedness Quotation withheld because this record is not cleared for redistribution. viscosity viscosity Quotation withheld because this record is not cleared for redistribution. boundary layer separation boundary layer separation Quotation withheld because this record is not cleared for redistribution. critical point critical point Quotation withheld because this record is not cleared for redistribution. inviscid limit inviscid limit Quotation withheld because this record is not cleared for redistribution. no-slip boundary condition no-slip boundary condition Quotation withheld because this record is not cleared for redistribution. vorticity field vorticity field Quotation withheld because this record is not cleared for redistribution. boundary layer phenomenon boundary layer phenomenon Quotation withheld because this record is not cleared for redistribution. parabolic scaling parabolic scaling Quotation withheld because this record is not cleared for redistribution. normal direction normal direction Quotation withheld because this record is not cleared for redistribution. Prandtl equations Prandtl equations Quotation withheld because this record is not cleared for redistribution. formal asymptotics formal asymptotics Quotation withheld because this record is not cleared for redistribution. Prandtl model Prandtl model Quotation withheld because this record is not cleared for redistribution. functional spaces functional spaces Quotation withheld because this record is not cleared for redistribution. tangential variable tangential variable Quotation withheld because this record is not cleared for redistribution. x derivatives x derivatives Quotation withheld because this record is not cleared for redistribution. loss of one derivative loss of one derivative Quotation withheld because this record is not cleared for redistribution. horizontal diffusion term horizontal diffusion term Quotation withheld because this record is not cleared for redistribution. standard energy estimates standard energy estimates Quotation withheld because this record is not cleared for redistribution. analytic framework analytic framework Quotation withheld because this record is not cleared for redistribution. Rayleigh instability Rayleigh instability Quotation withheld because this record is not cleared for redistribution. inviscid version inviscid version 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. Quotation withheld because this record is not cleared for redistribution. finite loss of x-derivatives finite loss of x-derivatives Quotation withheld because this record is not cleared for redistribution. (weakly) well-posed (weakly) well-posed Quotation withheld because this record is not cleared for redistribution. strong destabilization mechanism strong destabilization mechanism Quotation withheld because this record is not cleared for redistribution. base velocity profile base velocity profile Quotation withheld because this record is not cleared for redistribution. monotonic data monotonic data Quotation withheld because this record is not cleared for redistribution. shear velocity profile shear velocity profile Quotation withheld because this record is not cleared for redistribution. linearization linearization Quotation withheld because this record is not cleared for redistribution. analytic regularity analytic regularity Quotation withheld because this record is not cleared for redistribution. high tangential frequencies high tangential frequencies Quotation withheld because this record is not cleared for redistribution. unstable quasimodes unstable quasimodes Quotation withheld because this record is not cleared for redistribution. spectral condition spectral condition Quotation withheld because this record is not cleared for redistribution. reduced ODE reduced ODE Quotation withheld because this record is not cleared for redistribution. instability process instability process Quotation withheld because this record is not cleared for redistribution. singular perturbation problem singular perturbation problem 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. Quotation withheld because this record is not cleared for redistribution. inviscid eigenelements inviscid eigenelements Quotation withheld because this record is not cleared for redistribution. strong instability strong instability Quotation withheld because this record is not cleared for redistribution. heteroclinic orbits heteroclinic orbits Quotation withheld because this record is not cleared for redistribution. holomorphic solution holomorphic solution Quotation withheld because this record is not cleared for redistribution. complex change of variable complex change of variable Quotation withheld because this record is not cleared for redistribution. formal analysis formal analysis Quotation withheld because this record is not cleared for redistribution. numerical illustrations numerical illustrations Quotation withheld because this record is not cleared for redistribution. Evans function Evans function Quotation withheld because this record is not cleared for redistribution. shooting method shooting method Quotation withheld because this record is not cleared for redistribution. Runge-Kutta scheme Runge-Kutta scheme Quotation withheld because this record is not cleared for redistribution. Newton-Raphson procedure Newton-Raphson procedure Quotation withheld because this record is not cleared for redistribution. direct simulations direct simulations Quotation withheld because this record is not cleared for redistribution. finite differences finite differences Quotation withheld because this record is not cleared for redistribution. stretched grid stretched grid Quotation withheld because this record is not cleared for redistribution. Crank-Nicholson scheme Crank-Nicholson scheme Quotation withheld because this record is not cleared for redistribution. initial random data initial random data Quotation withheld because this record is not cleared for redistribution. excellent agreement excellent agreement Quotation withheld because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. Evans function → spectral condition 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 Quotation withheld because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. Inviscid-limit difficulty - no uniform vorticity bound as viscosity vanishes, corresponding physically to a boundary layer Quotation withheld because this record is not cleared for redistribution. Prandtl asymptotics - formally natural but not uniformly valid because boundary-layer separation can occur Quotation withheld because this record is not cleared for redistribution. Tangential regularity obstacle - one derivative is lost without horizontal diffusion to balance it Quotation withheld because this record is not cleared for redistribution. Analytic setting - sufficient for local boundary-layer justification but filters instability mechanisms Quotation withheld because this record is not cleared for redistribution. Inviscid Cauchy problem - characteristic solution with finite tangential derivative loss and weak well-posedness Quotation withheld because this record is not cleared for redistribution. Viscous destabilization - viscosity and non-degenerate critical points jointly underlie the Sobolev failure Quotation withheld because this record is not cleared for redistribution. Strong linear ill-posedness - high-frequency perturbations grow on a square-root-frequency timescale Quotation withheld because this record is not cleared for redistribution. Spectral reduction - PDE ill-posedness is reduced to a condition on a lower-dimensional ODE Quotation withheld because this record is not cleared for redistribution. Spectral-condition problem - existence of a heteroclinic orbit for the reduced ordinary differential equation Quotation withheld because this record is not cleared for redistribution. Evans-function computation - shooting with Runge-Kutta integration followed by Newton-Raphson root finding Quotation withheld because this record is not cleared for redistribution. Direct numerical test - finite differences on a stretched grid and Crank-Nicholson time stepping from random data Quotation withheld because this record is not cleared for redistribution. Numerical confirmation - eigenvalue and eigenmode comparisons agree with theory and show the mechanism dominates linear dynamics Quotation withheld because this record is not cleared for redistribution. extends → prior well-posedness results for the Prandtl equation (analytic or monotone data) Quotation withheld because this record is not cleared for redistribution. builds-on → reference [8] (analysis of the inviscid version of the Prandtl equation) Quotation withheld because this record is not cleared for redistribution. builds-on → reference [8] (study of the inviscid spectral problem and its evolution equation) Quotation withheld because this record is not cleared for redistribution. viscosity —causes→ ill-posedness viscosity ill-posedness strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. critical point —causes→ ill-posedness critical point ill-posedness strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. 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 because this record is not cleared for redistribution. Strikingly, the strong instability is caused by VISCOSITY — consistent with well-posedness of the inviscid version of the equation. Quotation withheld because this record is not cleared for redistribution. Prandtl equation Prandtl equation 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. Quotation withheld because this record is not cleared for redistribution. ill-posedness ill-posedness Quotation withheld because this record is not cleared for redistribution. inviscid limit inviscid limit Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://doi.org/10.1090/S0894-0347-09-00652-3.