BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Département de mathématiques et applications - ECPv6.2.2//NONSGML v1.0//EN
CALSCALE:GREGORIAN
METHOD:PUBLISH
X-WR-CALNAME:Département de mathématiques et applications
X-ORIGINAL-URL:https://www.math.ens.psl.eu
X-WR-CALDESC:évènements pour Département de mathématiques et applications
REFRESH-INTERVAL;VALUE=DURATION:PT1H
X-Robots-Tag:noindex
X-PUBLISHED-TTL:PT1H
BEGIN:VTIMEZONE
TZID:Europe/Paris
BEGIN:DAYLIGHT
TZOFFSETFROM:+0100
TZOFFSETTO:+0200
TZNAME:CEST
DTSTART:20240331T010000
END:DAYLIGHT
BEGIN:STANDARD
TZOFFSETFROM:+0200
TZOFFSETTO:+0100
TZNAME:CET
DTSTART:20241027T010000
END:STANDARD
END:VTIMEZONE
BEGIN:VEVENT
DTSTART;TZID=Europe/Paris:20240402T093000
DTEND;TZID=Europe/Paris:20240402T123000
DTSTAMP:20260426T181828
CREATED:20230830T101315Z
LAST-MODIFIED:20240325T080508Z
UID:16729-1712050200-1712061000@www.math.ens.psl.eu
SUMMARY:A critical drift-diffusion equation: intermittent behavior
DESCRIPTION:This talk is about a simple but rich model problem at the cross section of stochastic homogenization and singular stochastic PDE: We consider a drift-diffusion process with a time-independent and divergence-free random drift that is of white-noise character. As already realized in the physics literature\, the critical case of two space dimensions is most interesting: The elliptic generator requires a small-scale cut-off for well- posedness\, and one expects marginally super-diffusive behavior on large scales. I will explain the criticality of the two-dimensional case in the introductory course by scaling arguments. \nIn the presence of an (artificial) large-scale cut-off at scale L\, as a consequence of standard stochastic homogenization theory and its notion of a corrector\, there exist harmonic coordinates with a stationary gradient F ; the merit of these coordinates being that under their lens\, the drift-diffusion process turns into a martingale. I will revisit (qualitative) stochastic homogenization theory\nin the introductory course. \nIt has recently been established that the second moments diverge as 𝔼 |F|² ~ √ln L as L ↑ ∞. We show that in this limit\, |F|²/𝔼 |F|² is not equi-integrable\, while |det F |/𝔼 |F|² converges to zero (in probability). This suggests that any limit\, if it exists\, will not admit a simple characterization. \nWe establish this asymptotic behavior by characterizing a proxy F̃  introduced in previous work as the solution of an Itô SDE w. r. t. the variable L\, and which implements the concept of a scale-by-scale homogenization. For this proxy\, we establish 𝔼|F̃ |⁴ ≫ (𝔼|F̃ |²)² and 𝔼(det F̃ )² ≲ 1. In view of the former property\, we assimilate this phenomenon to intermittency. I will try to elucidate the idea of scale-by-scale homogenization in the introductory course. \nThis is joint work with G. Chatzigeorgiou\, P. Morfe\, L. Wang\, and with C. Wagner.
URL:https://www.math.ens.psl.eu/evenement/felix-otto/
LOCATION:ENS – salle W\, 45 rue d'Ulm\, Paris\, 75005\, France
CATEGORIES:Séminaire Analyse non linéaire et EDP
END:VEVENT
BEGIN:VEVENT
DTSTART;TZID=Europe/Paris:20240430T093000
DTEND;TZID=Europe/Paris:20240430T123000
DTSTAMP:20260426T181828
CREATED:20230830T101424Z
LAST-MODIFIED:20240426T060337Z
UID:16733-1714469400-1714480200@www.math.ens.psl.eu
SUMMARY:CANCELED - Camillo De Lellis - CANCELED
DESCRIPTION:Mini-course: Flows of nonsmooth vector fields \nConsider a vector field v on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindelöf) Theorem states that\, if the vector field is Lipschitz in space\, for every initial datum x there is a unique trajectory γ starting at x at time 0 and solving the ODE γ'(t) = v(t\, γ(t)). The theorem looses its validity as soon as v is slightly less regular. However\, if we bundle all trajectories into a global map allowing x to vary\, a celebrated theory started by DiPerna and Lions in the 80es shows that there is a unique such flow under very reasonable conditions and for much less regular vector fields. This has a lot of repercussions to several important partial differential equations where the idea of « following the trajectories of particles » plays a fundamental role. In this lecture I will review the fundamental ideas of the original theory\, an alternative approach due to Gianluca Crippa and myself\, and review a series of natural questions. Some of these questions will be answered in the second part. \nLecture: Transport equations and anomalous dissipation \nAfter mentioning the fundamental theorems of DiPerna-Lions and Ambrosio on flows of Sobolev vector fields we will explore a number of sharpness questions related to them (a more detailed and elementary overiew of the theorems and of the sharpness questions will already have been given in the previous lecture\, but anyway this seminar will be arranged so that it can be followed independently from the latter). Many of these questions have been at least partially answered in the last few years and I will first survey what is the overall understanding which we have gained on them. I will then focus on a particular one and highlight its link with a phenomenon which is poorly understood at the rigorous mathematical level: the occurrence of the so-called anomalous dissipation in incompressible fluid dynamics.
URL:https://www.math.ens.psl.eu/evenement/camillo-de-lellis/
LOCATION:Jussieu —  salle 15-16-309\, 4 Place Jussieu\, Paris\, 75005\, France
CATEGORIES:Séminaire Analyse non linéaire et EDP
END:VEVENT
END:VCALENDAR