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SUMMARY:Linear instabilities of the Prandtl equations via the harmonic osc
 illator - Francesco  De Anna (Julius-Maximilians-Universität Würzburg)
DTSTART:20250819T130000Z
DTEND:20250819T134500Z
UID:TALK234724@talks.cam.ac.uk
DESCRIPTION:Boundary layers are regions near a solid boundary where fluid 
 flows exhibit strong gradients\, often giving rise to instabilities\, turb
 ulence\, and wave phenomena such as Tollmien&ndash\;Schlichting modes. The
  Prandtl equations are among the most widely studied models\, intended to 
 represent the leading-order behavior of viscous boundary layers in the van
 ishing-viscosity limit. In a seminal result\, G&eacute\;rard-Varet and Dor
 my proved the ill-posedness of these equations in Sobolev spaces around ce
 rtain non-monotonic shear flows\, by implicitly constructing unstable quas
 i-eigenmodes whose growth in time is governed by a Gevrey-class 2 disper
 sion relation (a regularity level intermediate between Sobolev and analyti
 c classes).\nIn collaboration with J. Kortum (University of W&uuml\;rzburg
 )\, we show that these unstable quasi-eigenmodes can in fact be constructe
 d explicitly\, using hypergeometric functions of the first kind and approp
 riate eigenfunctions of the harmonic oscillator. This explicit formulation
  helps clarify the nature of the instability mechanism and offers further 
 insight into the conditions required for ill-posedness in more regular fun
 ction spaces\, such as the Gevrey classes.
LOCATION:Seminar Room 1\, Newton Institute
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