Séminaires : Dynamical systems and PDEs

Equipe(s) Responsable(s)SalleAdresse
Géométrie et Dynamique
Sergei Kuksin, Dmitrii Treschev
Zoom-seminar, voir http://www.mathnet.ru/php/conference.phtml?option_lang=eng&eventID=9&confid=1832 Zoom
Paris 7 Diderot & Steklov Mathematical Institute Online Seminar

Séances à suivre

Orateur(s)Titre Date DébutSalleAdresseDiffusion
+ Sergey Denisov (University of Wisconsin-Madison) , Singularity formation in the contour dynamics for 2d Euler equation on the plane 25/11/2020 16:00 Zoom-seminar, voir http://www.mathnet.ru/php/conference.phtml?option_lang=eng&eventID=9&confid=1832 Zoom
Abstract: We will study 2d Euler dynamics of centrally symmetric pair of patches on the plane. In the presence of exterior regular velocity, we will show that these patches can merge so fast that the distance between them allows double-exponential upper bound which is known to be sharp. The formation of the 90 degree corners on the interface and the self-similarity analysis of this process will be discussed. For a model equation, we will discuss existence of the curve of smooth stationary solutions that originates at singular stationary steady state.
+ Séances antérieures

Séances antérieures

Orateur(s)Titre Date DébutSalleAdresse
+ Sergey Denisov (University of Wisconsin-Madison) , Singularity formation in the contour dynamics for 2d Euler equation on the plane 25/11/2020 16:00
Abstract: We will study 2d Euler dynamics of centrally symmetric pair of patches on the plane. In the presence of exterior regular velocity, we will show that these patches can merge so fast that the distance between them allows double-exponential upper bound which is known to be sharp. The formation of the 90 degree corners on the interface and the self-similarity analysis of this process will be discussed. For a model equation, we will discuss existence of the curve of smooth stationary solutions that originates at singular stationary steady state.
+ S. L. Tabachnikov Flavors of bicycle mathematics 28/10/2020 16:00
This talk concerns a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment. Surprisingly, this simple model is quite rich and has connections with several areas of research, including completely integrable systems. Here is a sampler of problems that I hope to touch upon: 1) The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. This mapping is a Moebius transformation, a remarkable fact that has various geometrical and dynamical consequences. 2) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. These two front tracks are related by the bicycle (Backlund, Darboux) correspondence, which defines a discrete time dynamical system on the space of curves. This system is completely integrable and it is closely related with another, well studied, completely integrable dynamical system, the filament (a.k.a binormal, smoke ring, local induction) equation. 3) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is an open problem, intimately related with Ulam's problem in flotation theory (in dimension two): is the round ball the only body that floats in equilibrium in all positions? This problem is also related to the motion of a charge in a magnetic field of a special kind. It turns out that the known solutions are solitons of the planar version of the filament equation.
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