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## Some problems in bio-mathematics and mathematical physics

• Date October 8-9, 2015
• Room GSSI Main Lecture Hall
• Speaker many speakers

SPEAKERS

• Carlo Boldrighini (University La Sapienza, Italy)
• Giorgio Fusco (University of L'Aquila, Italy)
• Stefan Luckhaus (University of Leipzig, Germany)
• Stefano Olla (University Paris-Dauphine, France)
• Mario Pulvirenti (University La Sapienza, Italy)
• Angela Stevens (Max Plank Institute for Mathematics, Germany)
• Livio Triolo (University of Roma "Tor Vergata", Italy)
• Maria Eulalia Vares (Instituto de Matematica, UFRJ, Brazil)

TENTATIVE PROGRAM

October 8:

• 10.00 Opening
• 10.15-11.15 Stefano Olla. Non-equilibrium macroscopic evolution of chain of oscillators with conservative noise.
• 12.00-13.00 Angela Stevens. Chemotaxis, hydrodynamic limit, and formal approximations of reinforced random walks.
• Lunch.
• 14.15-15.15 Mario Pulvirenti. On the size of chaos in the Boltzmann-Grad linit for hard speres.
• 15.15-16.15 Giorgo Fusco. Dissipative dynamics near a finite dimensional manifold.

October 9:

• 9.15-10.15 Stephan Luckhaus. Problems in microscopic plasticity.
• 11.00-12.00 Carlo Boldrighini. Blow-up for complex and real solution of the 3-d Navier-Stokes equations: theory and computer simulations.
• 12.15-13.15 Maria Eulalia Vares. Dynamic random walk on contact process environment.
• Lunch
• 15.45-16.15. Livio Triolo. Concluding Remarks.

ABSTRACTS

Carlo Boldrighini: Blow-up for complex and real solution of the 3-d Navier-Stokes equations: theory and computer simulations. (Joint work with S. Frigio and P. Maponi)
We present results of computer simulations of a class of complex solutions of the 3-d Navier-Stokes equations in $\R^{3}$ for which Li and Sinai have proved a finite-time blow-up. The simulations show that near the critical time the energy concentrates around one or two points of the physical space. We also give results on the behavior of some related real solutions.

Giorgio Fusco: Dissipative dynamics near a finite dimensional manifold
We consider an evolutionary equation $u_t=F(u)$, $u(0)=u_0 which admits a Lyapunov functional$J:H\rightarrow\R$,$H$an Hilbert space. We let$\mathcal{M}\subset H$be a finite dimensional embedded manifold and derive a sufficient condition on the structure of the graph$\mathcal{G}^J$of$J$in a tubular neighborhood$\mathcal{N}$of$\mathcal{M}$ensuring that u_0\in\mathcal{N}\quad\Rightarrow\quad u(t,u_0)\in\mathcal{N},\;\text{ for }\;t\in[0,T)\] where either$T=+\infty$or$u(T,u_0)$belongs to the lateral boundary of$\mathcal{N}\$.
As an application of the abstract result we review the phenomenon of {\it Slow Motion} for Allen-Cahn and Cahn-Hilliard equations in one space dimension.
This is joint work with P.Bates and G.Karaly.

Stephan Luckhaus: Problems in microscopic plasticity
Plastic deformation of metals are governed by the motion of dislocations.The basic problem is how to connect the theory of elastic fields with dislocations to atomistic theories on the one hand and the upscaling , that is the description of the collective behaviour of dislocations on the other hand. We try to present a few problems and conjectures.

Stefano Olla: Non-equilibrium macroscopic evolution of chain of oscillators with conservative noise.
In the non-equilibrium evolution of a one dimensional chain of anharmonic oscillators we expect two main space-time scales: a hyperbolic scale where the evolution is ballistic-mechanic, dominated by tension gradients and governed by Euler equations, and a super-diffusive scale where the evolution, at constant tension, depends on the gradients of the temperature and is governed by a fractional heat equation. This conjecture can be proven for a harmonic chain with random exchange of momentum between nearest neighbor particles. Non-acoustic chains (tensionless) will instead behave diffusively.

Mario Pulvirenti: On the size of chaos in the Boltzmann-Grad linit for hard speres.
I present a quantitative analysis of the low-density limit of a hard sphere system based on the study of a set of functions (correlation errors) measuring the deviations in time from the statistical independence of particles (propagation of chaos). In the context of the BBGKY hierarchy, a correlation error of order k measures the event where k tagged particles are connected by a chain of interactions preventing the factorization. Provided k is not too large and the time sufficiently small, such an error goes to zero with the hard spheres diameter to the power Ck, C>0. This requires a new analysis of many recollision events, and improves previous estimates.
This analysis is based on a joint work with S. Simonella.

Angela Stevens: Chemotaxis, hydrodynamic limit, and formal approximations of reinforced random walks (Joint work with S. Grosskinsky, D. Marahrens,J.J.L. Velazquez)
Chemotaxis is an important and common mechanism during structure formation of developmental cell systems. A PDE-system of cross-diffusion type, the so-called Keller-Segel model, provides a macroscopic description for this phenomenon.
In the first part of the talk the first equation of this system is derived as hydrodynamic limit from a stochastic interacting particle system on the lattice, where the attractive chemical signal is assumed to be stationary with a slowly varying mean.
In the second part of the talk the qualitative behavior of the full PDE-system is compared to the respective behavior of the formally related reinforced random walk model.

Maria Eulalia Vares: Dynamic random walk on contact process environment
In this talk I will discuss ideas and results obtained in collaboration with Thomas Mountford. We consider a random motion in the integers whose rates are determined by an underlying supercritical contact process in equilibrium. A CLT is proven, valid for all supercritical infection rates for the environment.
Reference: T. Mountford, M.E. Vares: Random walks generated by equilibrium contact processes.
Electron. J. Prob. 20(3), 17pp (2015).