6th IST-IME Meeting

September 5-9, 2016
Instituto Superior Técnico, Lisbon, Portugal

# Abstracts

Miguel Abreu, Instituto Superior Técnico, Universidade de Lisboa

## On the mean Euler characteristic of Gorenstein toric contact manifolds

In this talk I will prove that the mean Euler characteristic of a Gorenstein toric contact manifold is equal to half the normalized volume of the corresponding toric diagram. I will also give some immediate applications of this result. This is joint work with Leonardo Macarini.

Luis Barreira, Instituto Superior Técnico, Universidade de Lisboa

## Lyapunov regularity via singular values

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.

## Propagation of regularity for positive definite kernels and functions

We show that, for positive definite kernels, if specific forms of regularity (continuity, $\mathcal{S}_n$-differentiability or holomorphy) hold locally on the diagonal, then they must hold globally on the whole domain of positive-definiteness. This local-to-global propagation of regularity is a consequence of the algebraic structure induced by the non-negativity of the associated bilinear forms up to order $5$. Consequences of these results for topological groups and for positive definite and exponentially convex functions are explored.
Xavier Carvajal, IM, Universidade Federal do Rio de Janeiro

## Local well-posedness and dissipative limit of high dimensional KdV-type equations

Considered in this work is an $n$-dimensional dissipative version of the Korteweg-deVries equation. Our goal here is to investigate the well-posedness issue for the associated initial value problem in the anisotropic Sobolev spaces. We also study the limit behavior of this equation when the dissipative effects are reduced.

Alexandre Nolasco de Carvalho, ICMC, Universidade de São Paulo

## Structural stability of uniform attractors: topological and geometrical

We present a careful description of the relationship between pullback and uniform attractors, leading to a detailed description of the uniform attractor and providing the understanding of its dynamical structures. That description is used to show continuity (upper and lower semicontinuity) and structural stability (topological and geometrical) of uniform attractors, at least for a non-autonomous perturbation of a semigroup.

Radoslaw Czaja, University of Silesia at Katowice

## Attractors for Dynamical Systems with Impulses

In this talk I will formulate a suitable definition of a global attractor for impulsive dynamical systems, which model the evolution of a continuous process interrupted by abrupt changes of state.

An impulsive dynamical system (IDS) consists of a continuous semigroup $\{\pi(t) : t\geq 0\}$ on a metric space $X$, a nonempty closed subset $M$ of $X$ called an impulsive set, which is “transversal” to the flow of the semigroup, and a continuous function $I : M \to X$ called an impulsive function. Whenever a trajectory for the semigroup $\pi$ hits the set $M$ the impulsive function $I$ redirects it to a new state, defining an impulsive trajectory. Assuming that all impulsive trajectories exist for all times $t\geq 0$, we obtain a possibly discontinuous semigroup $\{\tilde{\pi}(t):t\geq 0\}$.

To describe the long-time behavior of $\tilde{\pi}($ we introduce the notion of a global attractor $\mathcal{A}\subset X$, which is precompact, $\mathcal{A} = \overline{\mathcal{A}}\setminus M$, $\tilde{\pi}$-invariant and $\tilde{\pi}$-attracting all bounded subsets of $X$. Such sets are more suitable for impulsive dynamical systems and better describe their dynamics than compact global attractors known for continuous semigroups.

I will present several properties for this class of precompact global attractors and a theorem on existence of such attractors. The theory has applications to chosen ordinary and partial differential equations with impulsive functions. This is a joint work with E. M. Bonotto, A. N. Carvalho and R. Collegari from the University of São Paulo in São Carlos and M. C. Bortolan from the Federal University of Santa Catarina, Brazil ([1, 2]).

### References

1. E. M. Bonotto, M. C. Bortolan, A. N. Carvalho, R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, Journal of Differential Equations, 259 (2015), 2602{2625, doi:10.1016/j.jde.2015.03.033.
2. E. M. Bonotto, M. C. Bortolan, R. Collegari, R. Czaja, Semicontinuity of attractors for impulsive dynamical systems, to appear in Journal of Differential Equations, doi:10.1016/j.jde.2016.06.024.
João Paulo Dias, Universidade de Lisboa

## On a quasilinear non-local Benney system

We consider the quasilinear non-local Benney system

$\begin{cases}i u_t + u_{xx} = |u|^2 u + buv \\ \displaystyle v_t + a \left( \int_{\mathbb{R}^+} v^2 \ dx \right) v_x = -b(|u|^2)_x , \quad (x,t) \in \mathbb{R}^+ \times [0,\infty[ . \end{cases}$

We study the existence and uniqueness of the local strong solutions to the initial boundary value problem, their possible blowup, the existence of global weak solutions and we exhibit bound-state solutions in some special cases.This talk is based on a submitted paper by J. P. Dias and F. Oliveira with the same title.

Tomasz Dlotko, University of Silesia at Katowice

## Fractional Navier-Stokes Equations

I will present shortly the results concerning fractional generalization of the Navier-Stokes problem reported in two recent papers [1, 2]. The abstracts of that two manuscripts are copied below.

The following reference list contains the most important references; an excellent paper [4], forming a base for the local in time solvability, by Yoshikazu Giga and Tetsuro Miyakawa, and recent publications [6, 5] by Jiahong Wu containing earlier studies of the fractional Navier-Stokes equation.

#### Abstract 1

We consider the Navier-Stokes equation (N-S) in dimensions two and three as limits of the fractional approximations. In 2-D the N-S problem is critical with respect to the standard $L^2$ a priori estimates and we consider its regular approximations with the fractional power operator $(-P\Delta)^{1+\alpha}$, $\alpha \gt 0$ small, where $P$ is the projector on the space of divergence-free functions. In 3-D different properties of the N-S problem with respect to the standard $L^2$ a priori estimate are obtained and the 3-D regular approximating problem involves fractional power operator $(-P\Delta)^s$ with $s\gt \frac{5}{4}$.

Using Dan Henry's semigroup approach and the Giga-Miyakawa estimates we construct regular solutions to such approximations. The solutions are global in time, unique, smooth and regularized through the equation in time. Solution to 2-D and 3-D N-S equations are obtained next as a limit of such regular solutions of the approximations. Moreover, since the nonlinearity of the N-S equation is of quadratic type, the solutions corresponding to small initial data and small $f$ are shown to be global in time and regular.

#### Abstract 2

We consider fractional Navier-Stokes equations in a bounded smooth domain $\Omega\in\mathbb{R}^N$, $N\ge 2$. Following the geometric theory of abstract parabolic problems we give the detailed analysis concerning existence, uniqueness, regularization and continuation properties of the solutions. Using these generalizations we construct next global solution of the original Navier-Stokes problem. Focusing finally on the 3-D model with zero external force we show that such solutions regularize after a certain time $T=T(\|u_0\|_{X_2})$.

### References

1. J. W. Cholewa, T. Dlotko, Fractional Navier-Stokes equations, submitted, January 2016.
2. T. Dlotko, Navier-Stokes equation and its fractional approximations, Appl. Math. Optim., DOI 10.1007/s00245-016-9368-y (2016).
3. T. Dlotko, M. B. Kania, C. Sun, Quasi-geostrophic equation in $\mathbb{R}^2$, J. Differential Equations 259 (2015), 531-561.
4. Y. Giga, T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), 267-281.
5. J. Wu, The quasi-geostrophic equation and its two regularizations, Comm. Partial Differ. Equ. 27 (2002), 1161-1181.
6. J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003), 284-312.

## Asymptotic behaviour for some classes of non-autonomous delay differential equations

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.
Rui Loja Fernandes, University of Illinois at Urbana-Champaign

## A classification of non-commutative integrable systems

A non-commutative integrable system (NCIS) on a symplectic manifold $(X^{2n}, \Omega)$ is given by a collection of functions $\{f_1, \dots, f_k\}$ where $(k\geq n)$, satisfying the following two assumptions:

1. Involutivity: the first $r=2n-k$ functions Poisson commute with all $k$ functions:

$\{f_i,f_j\}=0,\qquad (i=1,\dots,r; j=1,\dots,k).$
2. Independence: the functions are independent almost everywhere: $\operatorname{d} f_1\wedge \dots \wedge\operatorname{d}f_k\neq 0 \qquad \text{on a dense open set}.$

When $k=n$ one recovers the classical notion of a commutative integrable system (CIS). The same way a CIS is related to a Lagrangian fibration, NCIS are related to isotropic fibrations. In this lecture I will explore this relationship and some beautiful connections with Poisson geometry, integral affine geometry and symplectic groupoids, leasing to a classification of regular NCIS.

This talk is based on various ongoing collaborations with Marius Crainic (Utrecht), David Martinez Torres (PUC-Rio), Daniele Sepe (UFF-Rio), Camille Laurent-Gengoux (Metz) and Pol Vanhaeck (Poitiers).

Bernold Fiedler, Freie Universität Berlin

## Sturm global attractors: the three-dimensional balls

Sturm attractors $\mathcal{A}$ are global attractors of dissipative PDEs $u_t = u_{xx}+f(x,u,u_x),$ for $0 \lt x \lt 1$, , say with Neumann boundary conditions and hyperbolic equilibria $v$. The collection of unstable manifolds $c_v=W^u(v)$ forms a regular complex of cells $c_v \in \mathcal{C}$, in the sense of algebraic topology. Caveat: this is true for Sturm attractors, but totally wrong in general! We call $\mathcal{C}$ the Sturm complex.

We describe all $3$-ball Sturm complexes, i.e., all $\mathcal{C}$ which arise from the closure of a single $3$-cell $c_0$. We relate our description to the associated Fusco-Rocha meanders, and to the braid $x\mapsto(x,v(x),v_x(x))$ of equilibrium spaghetti. In particular we explain why, against all intuition, the top and bottom equilibria must be chosen as adjacent corners, in the octahedral $3$-cell complex, rather than antipodally opposite.

All results hold, equally, for the Jacobi systems studied by Waldyr M. Oliva. This is joint work with Carlos Rocha, and contains artwork by Anna Karnauhova.

Giorgio Fusco, Universitá degli Studi dell’Aquila

## On the existence of an heteroclinic connection between two global minimizers of the the Ginzburg-Landau functional

We assume that $W: \mathbb{R}^m\rightarrow \mathbb{R}$ is a nonnegative potential with exactly two nondegenerate zeros $\pm a \in \mathbb{R}^m$ and that there exist two (unique modulo translation) heteroclinic orbits $\overline{u}_- \ne \overline{u}_+$ connecting $-a$ to $a$, that is two minimizers of the Ginzburg-Landau functional $J(\varphi)= \int_ {\mathbb{R}} (W(\varphi)+\frac12|\varphi’|^2) \ ds$:

$J(\overline u_\pm) = \min_A J(\varphi) , \quad A\equiv \{\varphi \in W^{1,2}_{loc}(\mathbb{R};\mathbb{R}^m): \lim_{s\rightarrow\pm\infty} \varphi(s)=\pm a\} .$

M. Schatzman in her remarkable paper Asymmetric heteroclinic double layers considered the PDE $\Delta u=W_u(u) , \quad W_u=(D_{u_1}W,\dots, D_{u_m}W)^T$ and, under a nondegeneracy condition on $\overline u_\pm$, proved the existence of a solution $u^S: \mathbb{R}^2\rightarrow\mathbb{R}^m$ that connects $\overline u_-$ to $\overline u_+$ in the sense that

\begin{align*}\lim_{y\rightarrow\pm\infty} u^S(x,y) & = \pm a , \\\lim_{x\rightarrow\pm\infty} u^S(x,y) & = \overline u_\pm(y-\eta_\pm)\end{align*} for some $\eta_\pm \in \mathbb{R}$.

Assuming that $W$ is a $C^\infty$ function, we give an alternative elementary proof of this result. Our approach is based on the idea of regarding $u^S$ as an Heteroclinic map $\mathbb{R} \ni x \rightarrow u^S(x, \cdot) \in A$ that connects the minimizers $\overline u_\pm(\cdot-\eta_\pm)$ of the Effettive Potential $J(\varphi)$.

Antonio Galves, IME, Universidade de São Paulo and Neuromat

## Infinite systems of interacting chains with memory of variable length - a stochastic model for biological neural nets

We consider a new class of non Markovian processes with a countable number of interacting components. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. The system evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian, and the stochastic chains with memory of variable length which have finite state space.

These features make it suitable to describe the time evolution of biological neural systems. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time. This construction implies uniqueness of the stationary process.

Finally we consider the case where the interactions between components are given by a critical directed Erdös-Rényi-type random graph with a large but finite number of components. In this framework we obtain an explicit upper-bound for the correlation between successive inter-spike intervals which is compatible with previous empirical findings. This is a joint work with E. Löcherbach.

Ronaldo Garcia, IME, Universidade Federal de Goiás

## Partially Umbilic Singularities of Hypersurfaces of $\mathbb{R}^4$ and of Plane Fields of $\mathbb{R}^3$

In this talk will be established the geometric structure of the lines of principal curvature of a hypersurface immersed in $\mathbb{R}^4$ in a neighborhood of the set $\mathcal{S}$ of its principal curvature singularities, consisting of the points at which at least two principal curvatures are equal. Under generic conditions defined by appropriate transversality hypotheses it is proved that $\mathcal{S}$ is the union of regular smooth curves $\mathcal{S}_{12}$ and $\mathcal{S}_{23}$, consisting of partially umbilic points, where only two principal curvatures coincide. This curve is partitioned into regular arcs consisting of points of Darbouxian types $D_1$, $D_2$, $D_3$, with common boundary at isolated semi-Darbouxian transition points of types $D_{12}$ and $D_{23}$.

Also a similar result is obtained for principal lines associated to plane fields of $\mathbb{R}^3$.

Pedro Girão, Instituto Superior Técnico, Universidade de Lisboa

## On the occurrence of mass inflation for the Einstein-Maxwell-scalar field system with a cosmological constant and an exponential Price law

In this talk we will consider spherically symmetric characteristic initial data for the Einstein-Maxwell-scalar field system with a cosmological constant, with initial data on the outgoing initial null hypersurface satisfying a Price law, and we will study the extendibility of the corresponding maximal globally hyperbolic development. This is a joint work with João L. Costa, José Natário and Jorge D. Silva and this talk is the continuation of the one by Jorge Drumond Silva.

Diogo Gomes, King Abdullah University of Science and Technology

## Stationary mean-field games and variational inequalities

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.

Jair Koilller, Fundação Getúlio Vargas, Rio de Janeiro

## Simple variational splines on manifolds

Riemannian splines have been studied since the 1990's and in fact one of the leading groups is Fatima Leite's at Coimbra. A simple spline is a curve controlled by a force, whose normal component deviates it from the geodesic path, whereas the tangential component controls the speed. We revisit the theme with ideas from Geometric Mechanics.

Philippo Lappicy, Freie Universität Berlin

## Einstein Constraints: A Dynamical Approach

The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a single quasilinear parabolic equation on the sphere with various singularities, and nonlinearity being the prescribed scalar curvature of space. We focus on self-similar solutions of Schwarzschild type. Those describe, for example, the initial data of black holes. We give a detailed study of the axially symmetric solutions, since the domain is now one dimensional and nodal properties can be used to describe certain asymptotics of the rescaled self-similar solutions. In particular, we mention examples for certain prescribed scalar curvatures.

Jaume Llibre, Universitat Autònoma de Barcelona

## Centers: their integrability and relations with the divergence

We shall recall briefly how can be the local phase portraits of the equilibrium points of an analytic differential system in the plane, and we shall put our attention in the centers. First in the kind of integrability of the different types of centers, and after in the focus–center problem, i.e. how to distinguish a center from a focus. This is a difficult problem which is not completely solved. We shall provide some new results using the divergence of the differential system.

José Matias, Instituto Superior Técnico, Universidade de Lisboa

## Optimal Design of Fractured Media with Prescribed Macroscopic Strain

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).

Joan Solà Morales, Universitat Politècnica de Catalunya

## Convergence to steady-state and boundary layer profiles in a linear chromatography system

These are results of an ongoing joint work with J. Menacho. They are mostly contained in a paper of the same title appeared in SIAM J. Appl. Math. 75 (2015), no. 2, pp. 745-761. We study the hyperbolic system of equations of the so called Linear Transport Model in a True Moving Bed chromatography device with four ports. By using methods based on a suitable energy-functional we show that all solutions approach exponentially a unique steady-state solution. Then, with the use of Asymptotic Analysis techniques we calculate the limit profiles of these steady-state solutions when the mass transfer coefficient between the liquid and solid phases tends to infinity. Along this singular limit sharp boundary layers appear near some ports. We are able to obtain explicit and simple formulas for these limit profiles.

## Orbital Stability of Periodic Traveling-wave Solutions for a Dispersive Equation

In this talk we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding wave speed. Essentially, our method establishes that if the linearized operator has only one negative eigenvalue which is simple and zero is a simple eigenvalue the orbital stability is determined provided that a convenient condition about the average of the wave is satisfied. We use our approach to prove the orbital stability of periodic waves associated with the Kawahara equation.

José Natário, Instituto Superior Técnico, Universidade de Lisboa

## An elementary derivation of the Montgomery phase formula for the Euler top

We give an elementary derivation of the Montgomery phase formula for the motion of an Euler top, using only basic facts about the Euler equation and parallel transport on the 2-sphere (whose holonomy is seen to be responsible for the geometric phase). We also give an approximate geometric interpretation of the geometric phase for motions starting close to an unstable equilibrium point.

Sérgio Oliva, IME, Universidade de São Paulo

## Human mobility in epidemic models and nonlocal diffusions

Following Brockmann, where human mobility is introduced in a simple SIR model, we get a Reaction Diffusion equation with fractional power diffusion. The first interesting mathematical and epidemiological question is how to characterize the existence of positive equilibrium in these equations. We also present a correlation network between occurrences of reported cases of dengue between cities in the state of Rio de Janeiro, Brazil.

Henrique Oliveira, Instituto Superior Técnico, Universidade de Lisboa

## The sensitivity of evolutionary entropy to perturbations in the micro-local variables

A large class of biological and socioeconomic dynamical systems have a hierarchical structure with time and energy values ranging from the molecular level to the macroscopic level. A macroscopic variable which has come to play a role in understanding the relation between the different scales in the hierarchy is evolutionary entropy. This statistical parameter, a function of the interaction at the micro-level, describes the diversity of pathways of energy flows between the elements that compose the micro-level. Evolutionary entropy characterize the robustness or stability of the hierarchy, that is the rate at which macroscopic variables that describe the system return to their steady state condition after a random perturbation.

This talk is concerned with an analysis of the response or sensitivity of evolutionary entropy to perturbations in the microscopic parameters that describe the hierarchy.

The problem we address is an elaboration of studies which originally began in the study of demographic networks. We will appeal to the methods explored in those studies to derive a general expression for the response of entropy to changes in the microscopic variables that describe the interaction that defines the hierarchy.

Joint work with Lloyd Demetrius, Harvard University and Max Planck, Berlin.

## Well-posedness for multicomponent Schrödinger-gKdV systems and stability of solitary waves with prescribed mass

In this talk we discuss the well-posedness issues of the associated initial value problem, the existence of nontrivial solutions with prescribed $L^2$-norm, and the stability of associated solitary waves for two classes of coupled nonlinear dispersive equations. The first model describes the nonlinear interaction between two Schrödinger type short waves and a generalized Korteweg-de Vries type long wave and the second one describes the nonlinear interaction of two generalized Korteweg-de Vries type long waves with a common Schrödinger type short wave. The results here extend many of the previously obtained results for two-component coupled Schrödinger-Korteweg-de Vries systems.

This is a joint work with Adan J. Fernandes and Santosh Bhattarai.

## Scattering for a 3D Coupled Nonlinear Schrödinger System

We consider the three-dimensional cubic nonlinear Schrödinger system $\begin{cases} i \partial_t u + \triangle u + (|u|^2+\beta |v|^2)u = 0 , \\ i \partial_t v + \triangle v + (|v|^2+\beta |u|^2)v = 0 . \end{cases}$ Let $(P;Q)$ be any ground state solution of the above Schrödinger system. We show that for any initial data $(u_0,v_0)$ in $H^1(\mathbb{R}^3) \times H^1(\mathbb{R}^3)$ satisfying $M(u_0,v_0)A(u_0,v_0) < M(P,Q)A(P,Q)$ and $M(u_0,v_0)E(u_0,v_0) < M(P,Q)E(P,Q)$, where $M(u,v)$ and $E(u,v)$ are the mass and energy (invariant quantities) associated to the system, the corresponding solution is global in $H^1(\mathbb{R}^3) \times H^1(\mathbb{R}^3)$ and scatters. Our approach is in the same spirit of Duyckaerts-Holmer-Roudenko, where the authors considered the 3D cubic nonlinear Schrödinger equation. Joint work with L.G. Farah (UFMG).

Paulo Piccioni, IME, Universidade de São Paulo

## Infinitely many solutions to the Yamabe problem on noncompact manifolds

I will discuss the existence of infinitely many complete metrics with constant positive scalar curvature on prescribed conformal classes on certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, $S^m \times R^d$ and $S^m \times H^d$. As a consequence, one obtains infinitely many periodic solutions to the singular Yamabe problem on $S^m \setminus S^k$, for all $0 \leq k \lt (m−2)/2$. I will also show that all Bieberbach groups are periods of bifurcating branches of solutions to the Yamabe problem on $S^m\times R^d$. This is a joint work with R. Bettiol, UPenn.

João Teixeira Pinto, Instituto Superior Técnico, Universidade de Lisboa

## Center manifolds and rates of convergence to scaling profiles in a deposition model

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.

## Explosion of smoothness for conjugacies

Let $f$ and $g$ be piecewise $C^r$ maps of the interval, with $r > 1$ and non-flat at the discontinuity sets $C_f$ and $C_g$, respectively, and let $h$ be a topological conjugacy between $f$ and $g$. We note that the maps $f$ and $g$ can be discontinuous and/or have different lateral derivatives (zero, finite or infinite) at the non-flat discontinuity sets $C_f$ and $C_g$, respectively. Let $A$ be a cycle of intervals of $f$, whose $supp A$ is a chaotic topological attractor. We prove that, if $h$ is $C^1$ in a single point with non-zero derivative then the conjugacy is a $C^r$ diffeomorphism.

Clodoaldo Ragazzo, IME, Universidade de São Paulo

## Hydrodynamic Vortex on Surfaces

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.

Carlos Rocha, Instituto Superior Técnico, Universidade de Lisboa

## On a definition of Morse-Smale evolution processes

We consider a definition of Morse-Smale evolution process that extends the notion of Morse-Smale dynamical system to the nonautonomous framework. In particular we consider nonautonomous perturbations of autonomous systems. In this case our definition of Morse-Smale evolution process holds for perturbations of Morse-Smale autonomous systems with or without periodic orbits. We establish that small nonautonomous perturbations of autonomous Morse-Smale evolution processes derived from certain nonautonomous differential equations are Morse-Smale evolution processes. We apply our results to examples of scalar parabolic semilinear differential equations generating evolution processes and possessing periodic orbits.

This is a joint work with Radoslaw Czaja and Waldyr Oliva.

Pedro Salomão, IME, Universidade de São Paulo

## Systems of transversal sections for Reeb flows on $\mathbb{R}P^3 \# \mathbb{R}P^3$

Global surfaces of section for three-dimensional flows allows one to study dynamics via the first return map. When such global surfaces of section do not exist, it is still possible to consider the so called systems of transversal sections: a transversal foliation to the vector field in the complement of a finite set of periodic orbits. Such systems may provide interesting information about the underlying dynamics. In this talk, I will introduce the notion of systems of transversal sections for Reeb flows on the connected sums of real projective three spaces. In particular, I will present sufficient conditions so that a set formed by three periodic orbits admits a system of transversal section in its complement.

This result is motivated by the restricted three body problem for energies slightly above the first Lagrange value. This is joint work with N. de Paulo (UFSC) and U. Hryniewicz (UFRJ).

Jorge Drumond Silva, Instituto Superior Técnico, Universidade de Lisboa

## On black holes and the strong cosmic censorship in spherical symmetry

This talk serves as an introduction to the problem of the stability of the Cauchy horizon and the future extend​a​bility of the maximal globally hyperbolic development for the Einstein-Maxwell-scalar field system with a cosmological constant.

We will start by reviewing some basic notions of General Relativity, in particular the structure of black holes in spherical symmetry, leading to the formulation of the celebrated strong cosmic censorship conjecture, from a PDE perspective.

We will finish by describing the characteristic initial value problem for the study of the Einstein-Maxwell-scalar field system inside a black hole, with an exponential Price law over the event horizon, which is the object of recent joint work with João L. Costa, Pedro M. Girão and José Natário.

This will be a prelude to the talk by Pedro M. Girão, that will be focused on the last topic.

Fábio Tal, IME, Universidade de São Paulo

## Zero entropy homeomorphisms of the sphere

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.

Marco Antonio Teixeira, Universidade de Campinas

## The fold-fold singularity

First of all, mention that non-smooth systems are driven by applications and they play an intrinsic role in a wide range of technological areas and many fundamental difficulties arise, even for local phenomena. The aim of this paper is to survey a qualitative and geometric analysis on the dynamics of a 3D fold-fold singularity following various texts existing in the literature.

Gláucio Terra, IME, Universidade de São Paulo

## Vakonomic vs Nonholonomic Mechanics Revisited

The purpose of this talk is to present conditions under which vakonomic and nonholonomic mechanics for linearly constrained mechanical systems coincide. The main result states that, if the vakonomic vector field is tangent to the Whitney sum $\mathcal{D} \oplus (\mathcal{D}^0)^{(1)}$, where $\mathcal{D}$ is the constraint distribution and $(\mathcal{D}^0)^{(1)}$ is the first derived system of the annihilator $\mathcal{D}^0 \subset T^*M$ of $\mathcal{D}$ in the cotangent bundle of the configuration manifold $M$, then the the projection on $\mathcal{D}$ of the vakonomic trajectories with initial condition in this Whitney sum coincide with the nonholonomic trajectories.