L3

(Marta Lewicka)

Variational methods have been extensively used in the past decades to rigorously derive nonlinear models in the description of thin elastic films. In this context, natural growth or differential swelling-shrinking lead to models where an elastic body aims at reaching a space-dependent metric. We will describe the effect of such, generically incompatible, prestrain metrics on the singular limits' bidimensional models. We will discuss metrics that vary across the specimen in both the midplate and the thin (transversal) directions. We will also cover the case of the oscillatory prestrain, exhibit its relation to the non-oscillatory case via identifying the effective metrics, and discuss the role of the Riemann curvature tensor in the limiting models.

(Shankar Venkataramani)

Using the bidimensional models for pre-strained Elasticity, that Marta will discuss in her talk, I will discuss some contrasts between the mechanics of thin objects with non-negative curvature (plates, spherical shells, etc) and the mechanics of hyperbolic sheets, i.e. soft/thin objects with negative curvature. I will motivate the need for new "geometric" methods for discretizing the relevant equations, and present some of our preliminary work in this direction.

This is joint work with Toby Shearman and Ken Yamamoto.

The derivation of first-order nonlinear transport PDEs via interacting particles subject only to deterministic forces is crucial in the socio-biological sciences and in the real world applications (e.g. vehicular traffic, pedestrian movements), as it provides a rigorous justification to a "continuum" description in situations more naturally described by a discrete approach. This talk will collect recent results on the derivation of entropy solutions to scalar conservation laws (arising e.g. in traffic flow) as many particle limits of "follow-the-leader"-type ODEs, including extensions to the case with Dirichlet boundary conditions and to the Hughes model for pedestrian movements (the results involve S. Fagioli, M. D. Rosini, G. Russo). I will then describe a recent extension of this approach to nonlocal transport equations with a "nonlinear mobility" modelling prevention of overcrowding for high densities (in collaboration with S. Fagioli and E. Radici).