Titles and Abstracts 

Lorenzo Brandolese (Université de Lyon 1)

Forcing rapid dissipation of Navier-Stokes flows

Abstract: We show that, by acting on a finite number of parameters of a compactly supported control force, we can increase the energy dissipation rate of any small solution of the Navier--Stokes equations in $\R^n$. The magnitude of the control force is bounded by a negative Sobolev norm of the initial velocity. Its support can be chosen to be contained in an arbitrarily small region, in time or in space.

Diego Chamorro  (Université d'Evry)

Some Liouville-type theorems for the Navier-Stokes equations.

Abstract: Uniqueness of the $\dot{H}^1$ solutions of the stationary 3D Navier-Stokes equations remains a challenging open problem. If we assume some additional hypotheses on the solutions (stated in terms of Lebesgue or Morrey spaces, which will ensure a suitable decay at infinity), then it is possible to obtain that the trivial solution is unique. Of course, there is a gap between the information given by the functional framework of the $\dot{H}^1$ solutions and the information added by the hypotheses. We will see however, by studying the fractional stationary Navier-Stokes problem, that this gap can almost be filled in this fractional case.

Laurent Desvillettes (Université Paris Diderot)

Modeling populations structured by a quantitative trait and by space by PDE

Abstract: We show on various examples how PDE theory (and especially results relative to large time asymptotics) can help to understand the evolution of structured populations appearing in some problems of biology related to biological invasions, speciation, etc.

Pierre Gilles Lemarie Rieusset  (Université d'Evry)

Non-trivial steady solutions for the Navier-Stokes equations on the 2D torus.

Abstract: Recently, non-trivial steady solutions for the Navier-Stokes equations have been built on the 4D torus (through convex integration). We discuss the problem arising in dimension 2 and provide a construction for such solutions, with the peculiarity that they are no longer square integrable, so that we have to deal more cautiously with the quadratic nonlinearity.

Giuseppe Toscani (Università di Pavia) 

Modelling contagious viral dynamics: a kinetic approach based on mutual utility

(Joint work with Giulia Bertaglia and Lorenzo Pareschi)

Abstract: The temporal evolution of a contagious viral disease is modelled as the dynamic progression of different classes of population with individuals interacting pairwise. This interaction follows a binary mechanism typical of kinetic theory, wherein agents aim to improve their condition with respect to a mutual utility target. To this end, we introduce kinetic equations of Boltzmann-type to describe the time evolution of the probability distributions of the multi-agent system. The interactions between agents are defined using principles from price theory, specifically employing Cobb-Douglas utility functions for binary exchange and the Edgeworth box to depict the common exchange area where utility increases for both agents. Several numerical experiments  highlight the significance of this mechanism in driving the phenomenon toward endemicity.

Mattia Zanella (Università di Pavia)

Trends to equilibrium for many-agent systems in swarm manufacturing

Abstract: We study equilibration rates for nonlocal Fokker-Planck equations arising in swarm manufacturing. The PDEs of interest possess a time-dependent nonlocal diffusion coefficient and a nonlocal drift, modeling the interaction of a large system of agents. The emerging steady profile is characterized by a uniform spreading over a portion of the domain. The result follows by combining entropy methods for quantifying the decay of the solution towards its quasi-stationary distribution with the properties of the quasi-stationary profile.