Partial Differential Equations
in Mathematical Physics

Conference in honour of Alexander Komech's 75th birthday

25 May 2021

Zoom meeting ID: 931-7514-2429

Invited speakers with slides of their talks

Alexei Ilyin (Keldysh Institute of Applied Mathematics, Moscow)
Two-sided dimension estimates of the attractor of the damped driven Euler—Bardina equations in two and three dimensions (PDF)
We prove the existence of the global attractor of the damped and driven regularized Euler—Bardina equations and give an estimate of its dimension. The system is studied in the whole space, in a bounded domain and on the torus in two and three dimensions. In the periodic case it is shown that the upper bounds of the dimension are sharp as the regularization parameter alpha tends to zero.
A joint work with S.V.Zelik
TIME: Tuesday, 25 May 2021, 16:00 Moscow (15h00 Paris, 9:00 AM Montreal)

Sergei Kuksin (Institut de Mathématiques de Jussieu, Paris)
The K41 theory of turbulence and its rigorous one-dimensional model (PDF)
I will present three main laws from the Kolmogorov theory of turbulence ("the K41 model"), discuss their versions for one-dimensional fluid and will show that the latter may be rigorously justified for the 1d fluid, described by the Burgers equation, via a qualitative analysis of the dynamical system which the equation defines in Sobolev spaces.
The talk is based on a MS of my joint book with Alex Boritchev.
TIME: Tuesday, 25 May 2021, 17:00 Moscow (16h00 Paris, 10:00 AM Montreal)

Alexander Shnirelman (Concordia University, Montreal)
Soliton asymptotics in hydrodynamics (PDF)
We consider the motion of the ideal incompressible fluid in a bounded 2-dimensional domain which is described by the Euler equations. One of the basic problems about it is the large time behavior of solutions. Physical experiments and accurate computer simulations show that any flow eventually assumes the shape of a few large and hierarchically organized vortices; in course of their motion, those vortices preserve their individuality, and do not mix with each other. Such behavior looks paradoxical; it is sometimes called "negative viscosity". This phenomenon is analogous to the soliton asymptotics of solutions of nonlinear dispersive equations. In both cases we observe the apparent entropy loss: the diversity of the outcomes (i.e. solutions after a very long time) is much smaller that that of the initial conditions. The talk is devoted to the analysis of this paradox which is in fact a manifestation of some basic feature of systems studied in the nonequilibrium statistical mechanics.
TIME: Tuesday, 25 May 2021, 18:15 Moscow (17h15 Paris, 11:15 AM Montreal)

Organizing committee: Andrew Comech, Elena Kopylova

Organized by Texas A&M University and the Institute for Information Transmission Problems of Russian Academy of Sciences