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DTSTART:20190331T010000
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DTSTART:20191027T010000
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DTSTART;TZID=Europe/Paris:20190514T093000
DTEND;TZID=Europe/Paris:20190514T120000
DTSTAMP:20260621T162906
CREATED:20190514T073000Z
LAST-MODIFIED:20211025T103550Z
UID:8493-1557826200-1557835200@www.math.ens.psl.eu
SUMMARY:Nonuniqueness for the Navier–Stokes equations and model equations / Singularities in Fluid Mechanics
DESCRIPTION:JG: In this talk\, I will discuss fundamental properties of the solutions to the incompressible Navier–Stokes equations in three dimensions. After reviewing the classical local well-posedness results\, I will explain how numerical simulations suggest local ill-posedness at the borderline of the known results. I will discuss a plausible scenario of non-uniqueness from smooth initial data through finite-time blow-up. Finally\, I will describe how this scenario is actually happening in a model equation sharing the same fundamental properties as the Navier–Stokes equations.KM: Singularities of the Navier-Stokes equations occur when some derivative of the velocity field is infinite at any point of a field of flow (or\, in an evolving flow\, becomes infinite at any point within a finite time). Such singularities can be mathematical but unphysical (as e.g. in two-dimensional flow near a sharp corner) in which case they can be ‘resolved’ by improving the physical model considered; or they can be physical but non-mathematical (as e.g. in the case of cusp singularities at a fluid/fluid interface) in which case resolution of the singularity may involve incorporation of additional physical effects; these examples will be briefly reviewed. The `finite-time singularity problem’ for the Navier-Stokes equations will then be discussed and a new analytical approach will be presented; here it will be shown that there is indeed a singularity of the ‘physical but non-mathematical’ type\, in that\, at suf- ficiently high Reynolds number\, vorticity can be amplified by an arbitrarily large factor within a finite time. In this case\, the singularity is resolved by three-dimensional vortex re- connection in a manner that admits analytical description.
URL:https://www.math.ens.psl.eu/evenement/nonuniqueness-for-the-navier-stokes-equations-and-model-equations-singularities-in-fluid-mechanics/
LOCATION:ENS Salle W
CATEGORIES:Séminaire Analyse non linéaire et EDP
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