We consider a coagulation type system modeling submonolayer deposition. This study focuses on establishing asymptotic scaling (or similarity) profiles and rates of convergence of the solutions to those profiles. In an earlier study (Costin et al. in Commun. Inf. Syst., 13 (3013), pp. 183-200), other authors have established those scaling profiles by using asymptotic expansions methods. However, in order to aditionally obtain the above mentioned rates of convergence, higher order estimates are needed. Here we show how center manifolds estimates provide an alternative way of establishing those profiles while obtaining, at the same time, those required higher order estimates. A new interesting feature we prove is that, although the memory of the initial condition is completely lost in the limiting profile, the rate of convergence to this profile preserve information about the large cluster tail of the initial condition.

This is a joint work with Fernando Pestana da Costa and Rafael Sasportes.

We use a newly developed theory of forcing for surface homeomorphisms to obtain a Poincaré-Bendixson like result for orientation preserving homeomorphisms of the 2-sphere with zero topological entropy.

If $f$ is such a map and is not a pseudo-rotation, we show that for every $x$, there exists a power of $f$ such that the omega limit of $x$ must be either:

1. A cycle made of the union of unlinked fixed points and points heteroclinic to them.
2. A set rotating with irrational speed around a fixed point and possibly this fixed point.
3. An "infinitely renormalizable" set where the restriction of the dynamics is semi-conjugate to the odometer.

Joint work with P. Le Calvez.

In this work we consider an optimal design problem for two-component fractured media for which a macroscopic strain is prescribed. Within the framework of structured deformations, we derive an integral representation for the relaxed energy functional. We start from an energy functional accounting for bulk and surface contributions coming from both constituents of the material; the relaxed energy densities, obtained via a blow-up method, are determined by a delicate interplay between the optimization of sharp interfaces and the diffusion of microcracks. This model has the far-reaching perspective to incorporate elements of plasticity in optimal design of composite media.

This is a joint work with M. Morandotti (SISSA) and Elvira Zappalle (U. degli Studi di Salerno).

We consider stationary monotone mean-field games (MFGs) and study the existence of weak solutions. We introduce a regularized problem that preserves the monotonicity and prove the existence of solutions to the regularized problem. Next, using Minty's method, we establish the existence of solutions for the original MFGs. Finally, we examine the properties of these weak solutions in several examples.

We study the global asymptotic behaviour of solutions for some families of $n$-dimensional non-autonomous delay differential equations (DDEs), which encompass a large number of structured population models.

Some classes of monotone DDEs (with possible infinite delay) are first analysed: by using comparative results from the theory of cooperative systems [4], some criteria for persistence and permanence are given [2]. We then consider a family of non-autonomous DDEs obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ODEs. By exploiting the stability and the monotone character of the linear ODE, and by using comparison techniques with auxiliary monotone systems, we are able to establish sufficient conditions for both the extinction of all the populations and the permanence of the system [3].

In the case of DDEs with autonomous coefficients, sharper results are obtained, even in the case of reducible community matrices, improving or extending criteria in recent literature (see e.g. [1]).

References

1. J. Arino, L. Wang, G. S. K. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theor. Biol. 241 (2006), 109-119.
2. T. Faria, Persistence and permanence for a class of functional differential equations with infinite delay, J. Dyn. Diff. Equ.(2016). DOI 10.1007/s10884-015-9462-x.
3. T. Faria, R. Obaya, A. M. Sanz, Asymptotic behaviour for non-monotone delayed perturbations of monotone non-autonomous linear ODEs, submitted (2016). http://arxiv.org/abs/1607.05033.
4. H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 1995.

The notion of regularity was introduced by Lyapunov and plays an important role in the stability theory, particularly in the context of ergodic theory, as a consequence of Oseledets' multiplicative ergodic theorem. Somewhat surprisingly, we were able to establish the existence of a structure of Oseledets type for any nonregular dynamics. In particular, this allowed us to obtain lower and upper bounds for the Lyapunov exponents in terms of the growth rates of the singular values.

This is joint work with Claudia Valls.

The equations of motion for a system of point vortices on an oriented Riemannian surface of finite topological type is presented.

The equations are obtained from a Green's function on the surface. The uniqueness of the Green's function is established under hydrodynamic conditions at the surface's boundaries and ends. The hydrodynamic force on a point vortex is computed using a new weak formulation of Euler's equation adapted to the point vortex context. An analogy between the hydrodynamic force on a massive point vortex and the electromagnetic force on a massive electric charge are presented as well as the equations of motion for massive vortices. Any noncompact Riemann surface admits a unique Riemannian metric such that a single vortex in the surface does not move (Steady Vortex Metric). Some examples of surfaces with steady vortex metric isometrically embedded in $\mathbb{R}^3$ are presented.