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Nonlinear analysis and PDEs seminar

Time: Tuesday, May 29, 2018, 9h30 - 11h puis 11h30 - 12h30

Roman Shvydkoy (University of Illinois at Chicago (UIC) - Personal website)

In this talk we will give an overview of recent developments in the modeling of hydrodynamic systems with self-organization. These models appear in a broad range of applications, from biological sciences to theories of social behavior. The common challenge in modeling such systems is to include realistic local interactions between agents and at the same time to build an analytically tractable PDE. First agent-based systems of this type appeared in seminal works of Cucker and Smale [1, 2]. We will walk through several refinements of the original system (on micro-, meso- and macroscopic levels) and introduce a new anisotropic topological model which responds well to analysis yet exhibits the most fundamental feature of collective dynamics ?? emergence of consensus. These models present new challenges to the actively developing theory of fractional parabolic equations with rough drift. The talk is based on a recent series of joint works with E. Tadmor [3, 4, 5, 6]. References [1] Felipe Cucker and Steve Smale. Emergent behavior in flocks. IEEE Trans. Automat. Control, 52(5):852??862, 2007. [2] Felipe Cucker and Steve Smale. On the mathematics of emergence. Jpn. J. Math., 2(1):197??227, 2007. [3] Roman Shvydkoy and Eitan Tadmor. Eulerian dynamics with a commutator forcing. Transactions of Mathematics and Its Applications, 1(1):tnx001, 2017. [4] Roman Shvydkoy and Eitan Tadmor. Eulerian dynamics with a commutator forcing II: Flocking. Discrete Contin. Dyn. Syst., 37(11):5503??5520, 2017. [5] Roman Shvydkoy and Eitan Tadmor. Eulerian dynamics with a commutator forcing III: Fractional diffusion of order 0 < α < 1. to appear in Physica D. [6] Roman Shvydkoy and Eitan Tadmor. Topological models of collective behavior. preprint.

 

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