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Steady states and dynamics of the aggregation-diffusion equation - lecture 2

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Authors : Yao, Yao (Author of the conference)
CIRM (Publisher )

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Abstract : The aggregation-diffusion equation is a nonlocal PDE that arises in the collective motion of cells. Mathematically, it is driven by two competing effects: local repulsion modelled by nonlinear diffusion, and long-range attraction modelled by nonlocal interaction. In this course, I will discuss several qualitative properties of its steady states and dynamical solutions. Using continuous Steiner symmetrization techniques, we show that all steady states are radially symmetric up to a translation. (joint with Carrillo, Hittmeir and Volzone). Once the symmetry is known, we further investigate whether steady states are unique within the radial class, and show that for a given mass, the uniqueness/non-uniqueness of steady states is determined by the power of the degenerate diffusion, with the critical power being m = 2. (joint with Delgadino and Yan). I'll also discuss some properties on the long-time behavior of aggregation-diffusion equation with linear diffusion (joint with Carrillo, Gomez-Castro and Zeng), and global-wellposedness if Keller-Segel equation when coupled with an active advection term (joint with Hu and Kiselev).

Keywords : Aggregation-diffusion equation; radial symmetry; continuous Steiner symmetrization

MSC Codes :
35B35 - Stability of solutions of PDE
35K55 - Nonlinear parabolic equations
76B03 - Existence, uniqueness, and regularity theory

    Information on the Video

    Film maker : Hennenfent, Guillaume
    Language : English
    Available date : 22/07/2024
    Conference Date : 24/06/2024
    Subseries : Research School
    arXiv category : Analysis of PDEs
    Mathematical Area(s) : PDE
    Format : MP4 (.mp4) - HD
    Video Time : 00:58:49
    Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2024-06-25_Yao_2.mp4

Information on the Event

Event Title : Research School - Jean Morlet Chair - Frontiers in Interacting Particle Systems, Aggregation-Diffusion Equations & Collective Behavior / Ecole - Chaire Jean Morlet - Frontières dans les équations de systèmes de particules en interaction. Equations d'agrégation-diffusion et comportement collectif
Event Organizers : Carrillo, José Antonio ; Nouri, Anne
Dates : 24/06/2024 - 28/06/2024
Event Year : 2024
Event URL : https://conferences.cirm-math.fr/3050.html

Citation Data

DOI : 10.24350/CIRM.V.20194603
Cite this video as: Yao, Yao (2024). Steady states and dynamics of the aggregation-diffusion equation - lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20194603
URI : http://dx.doi.org/10.24350/CIRM.V.20194603

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