8th IST-IME Meeting

September 5-9, 2022
Instituto Superior Técnico
Lisboa, Portugal

# Abstracts

Miguel Abreu
Instituto Superior Técnico, Universidade de Lisboa

## Periodic orbits of Reeb flows and the Ehrhart polynomial of toric diagrams

An embedding of the $2n+1$ dimensional sphere in the real $2n+2$ dimensional vector space gives rise to a $1$-parameter autonomous flow on the sphere, called the characteristic flow. If the image of the sphere by the embedding bounds a star shaped domain, the corresponding characteristic flow is a Reeb flow. A long standing and important conjecture, which is still very much open, states that any of these Reeb flows on the $2n+1$ dimensional sphere has at least $n+1$ geometrically distinct periodic orbits. In this talk I will present illustrative examples motivated by this conjecture and its possible generalization to toric contact manifolds. These are completely determined by certain rational convex polytopes, called toric diagrams, and I will describe in particular the role played by their Ehrhart polynomial. This is part of joint work with Leonardo Macarini and Miguel Moreira (arXiv:2202.00442).

Nicholas Alikakos
University of Athens

## Some Estimates for the Triple Junction Problem on the Plane

There is a well-established analogy between minimal surfaces and a class of solutions of the scalar two-phase Allen-Cahn equation, discovered by De Giorgi in the 70s, that has been thoroughly understood in the decade 2000-2010. The basic object in this case is the solution of a simple ODE that is analogous to a plane.

On the other hand for three or more equally preferred phases a vector order parameter is required, and the basic object is a PDE solution of the Allen-Cahn system that connects the three phases that is analogous to a singular minimal cone. Such solutions are fully understood in the presence of symmetry (general reflection point group), but not at all in the general case.

In this lecture I will begin by stating the triple junction problem on the plane. Then I will present some joint work with Zhiyuan Geng that is motivated by this problem, which hopefully will be useful for its solution.

José Arrieta

## Boundedness of solutions of nonautonomous logistic equations

We consider a reaction diffusion equation in a bounded domain with logistic reaction term of the type  $\lambda u-n(t,x)u^\rho$  where the function  $n(t,\cdot)$  vanishes in a time dependent subset $K(t)$. We are interested in understanding how the set $K(t)$ affects the asymptotic behavior of the solutions, specially we would like to decide when the solutions are bounded as times goes to infinity.

Luís Barreira
Instituto Superior Técnico, Universidade de Lisboa

## Spectra of delay equations

I will describe a classification of the spectra of nonautonomous delay equations. This is joint work with Claudia Valls.

Alexandre Carvalho
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos

## Inertial Manifolds, Saddle Point Property and Exponential Dichotomy: A unified theory

Inertial manifold theory, saddle point property and exponential dichotomy have been treated as different topics in the literature with different proofs.

As a common feature, they all have the purpose of ‘splitting’ the space to understand the dynamics. We present a unified proof for the inertial manifold theorem, which as a local consequence yields the saddle-point property with a fine structure of invariant manifolds and the roughness of exponential dichotomy.

Simão Correia
Instituto Superior Técnico, Universidade de Lisboa

## Stable perturbations of self-similar solutions for the modified KdV at the blow-up time

Self-similar solutions for the modified Korteweg-de Vries equation are of both physical and mathematical interest. They describe the asymptotic behavior of solutions for large times and blow-up at the initial time. Due to their scaling invariance, they present several critical features (time decay, spatial decay and regularity), which means that the existing theory is not applicable. In this talk, I will show that one can define the flow around these solutions. Moreover, given any smooth perturbation, one can build a solution to the mKdV which behaves, at the blow-up time, as the self-similar solution plus the perturbation. This is joint work with R. Côte and L. Vega.

Fernando Pestana da Costa

## A coagulation type model for silicosis

We introduce an infinite dimensional system of ordinary differential equations modelling silicosis, and discuss results on existence, uniqueness, basic properties of solutions, and, for a family of coefficients, the structure of equilibria and their linear stability properties, which allow us to understand the local bifurcation of equilibria.

The reported results are based on joint works with P. Antunes, M. Drmota, M. Grinfeld, J. Pinto and R. Sasportes.

University of Silesia in Katowice

## Finite-dimensional attractors for dissipative fourth order problems in $\mathbb{R}^{N}$

A Cauchy problem for a dissipative fourth order parabolic equation in $\mathbb{R}^N$ with a general potential is considered
\begin{align*} & u_{t}+\Delta^2u=g(x)+m(x)u+f_0(x,u),\ t>0,\ x\in\mathbb{R}^{N}, \\ & u(0,x)=u_0(x),\ x\in\mathbb{R}^{N}.\end{align*} Using the quasi-stability method by Chueshov and Lasiecka an estimate from above of the fractal dimension of a global attractor is derived. It is also shown that the global attractor is contained in a finite-dimensional exponential attractor. This is a joint work with Jan W. Cholewa based on the article [1].

Suitably adapted considerations lead to corresponding results for a modified Swift-Hohenberg equation in $\mathbb{R}^{N}$ as was shown in cooperation with Maria Kania-Błaszczyk in [2].

1. J. W. Cholewa, R. Czaja, On fractal dimension of global and exponential attractors for dissipative higher order parabolic problems in $\mathbb{R}^N$ with general potential, in Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics, Understanding Complex Systems. Springer, 2021, pp. 293-314.
2. R. Czaja, M. Kania-Błaszczyk, Exponential attractors for modified Swift-Hohenberg equation in $\mathbb{R}^{N}$, preprint.
Pedro Duarte

## Stability of the Lyapunov exponents of randomly perturbed quasi-periodic cocycles

The study of linear cocycles and their Lyapunov exponents in Ergodic Theory has an important interface with study of spectral properties of discrete Schrödinger operators in Mathematical Physics, namely via the regularity of the Lyapunov exponents of the Schrödinger cocycles which relate to the spectral properties of the associated Schrödinger operators. Many results have been obtained so far on the continuity of the Lyapunov exponents for two main classes of linear cocycles: the class of quasi-periodic linear cocycles and the class of random linear cocycles. In this talk, I will survey on recent work with Ao Cai and Silvius Klein on a problem posed by Jiangong You about the stability of the Lyapunov exponents for random perturbations of quasi-periodic Schrödinger cocycles.

Teresa Faria

## Periodic solutions for systems of impulsive delay differential equations

For a family of periodic systems of differential equations with (possibly infinite) delay and nonlinear impulses, sufficient conditions for the existence of at least one positive periodic solution are established.

The main technique used here is the Krasnoselskii fixed point theorem on cones.

Although fixed points methods have been extensively employed to show the existence of positive periodic solutions to scalar delay differential equations (DDEs), the literature on $n$-dimensional impulsive DDEs is very scarce.

Our criteria are applied to some classes of mathematical biology models, such as Nicholson-type systems with patch structure.

This a joint work with R. Figueroa (University of Santiago de Compostela) [1].

1. Faria, R. Figueroa, Positive periodic solutions for systems of impulsive delay differential equations, Discrete Contin. Dyn. Syst. Ser. B (to appear).
Edson de Faria

## Rigidity and orbit-flexibility of circle maps

Two given orbits of a minimal circle homeomorphism $f$ are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with $f$. By a well-known theorem due to Herman and Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Swiatek that the same holds if $f$ is a critical circle map with rotation number of bounded type. By contrast, I will show in this talk that the set of geometric equivalence classes is uncountable when $f$ is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0, 1)$. The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map.

This talk is based on joint work with Pablo Guarino.

Rui Loja Fernandes
University of Illinois Urbana-Champaign

## Non-formal deformation quantization

The existence and classification of formal deformation quantizations was solved by Kontsevich some 20 years ago, using his formality theorem. The corresponding non-formal problem seems to be much harder. I will discuss joint work with Alejandro Cabrera (UFRJ) where we introduce a notion of non-formal star product using various concepts from semi-classical analysis and show that their existence is obstructed.

Bernold Fiedler
Freie Universität Berlin

## Three-dimensional Sturm global attractors: orders and progress

Sturm global attractors $\mathcal{A}$ arise, for example, in dissipative scalar reaction-advection-diffusion equations $u_t = u_{xx} + f(x,u,u_x)$ on the unit interval with Neumann boundary.  In the genericity tradition of Jorge Sotomayor, we assume all equilibria to be hyperbolic.  Their unstable manifolds decompose $\mathcal{A}$ into a signed regular cell complex. We call $\mathcal{A}$ three-dimensional, if the maximal Morse index of equilibria $v$=$\mathcal{O}$, alias the maximal cell dimension, is three.

In 1991, Giorgio Fusco and Carlos Rocha explored general Sturm attractors, via meander permutations, and their recursive construction via successive pitchfork bifurcations. Also in 1991, however, Carlos discovered the simplest non-pitchforkable Sturm attractor, which turns out to be planar with 11 equilibria.

For dimension three, we present a geometric recursion of $\mathcal{A}$ which only involves the attachment of saddle-nodes involving Morse indices $i=0$ or 3, and some Fusco-Rocha pitchforks among $i\in{1,2}$.

Tools involve orders induced by the Morse structure and by nodal properties. Progress aspires to become joint work with Carlos Rocha.

Giorgio Fusco
Università degli Studi dellʼAquila

## On the fine structure of minimizers in the Allen-Cahn theory of phase transitions

Let $u^\epsilon$ be a minimizer of the Allen-Cahn energy subjected to Dirichlet condition: $J_\Omega^\epsilon(u^\epsilon)=\min_{u\vert_{\partial\Omega}=v_0^\epsilon}\int_\Omega\Big(\frac{\epsilon}{2}\vert\nabla u\vert^2+\frac{1}{\epsilon}W(u)\Big)dx,$ where $\Omega\subset\mathbb{R}^n$ is a smooth domain, $\epsilon\gt 0$ a small parameter, $v_0^\epsilon:\partial\Omega\rightarrow\mathbb{R}^m$ the boundary datum and $W:\mathbb{R}^m\rightarrow\mathbb{R}$ a smooth nonnegative potential that vanishes on a finite set: $A=\{a_1,\ldots,a_N\}=\{W=0\}.$ The zeros $a_1,\ldots,a_N$ of $W$ represent equally preferred phases.

For $\delta\gt 0$ small we study the diffuse interface $\mathscr{I}^{\epsilon,\delta}=\{x\in\bar{\Omega}:\min_{a\in A}\vert u^\epsilon(x)-a\vert\gt \delta\}.$

We give sufficient conditions on $\Omega$ and $v_0^\epsilon$ ensuring the connectivity of the diffuse interface.

Then we restrict to two space dimensions and show that one can associate a sort of spine to the diffuse interface: a connected network $\mathscr{G}$ which is contained in $\mathscr{I}^{\epsilon,\delta}$ and divides $\Omega$ in $N$ connected subsets corresponding to the $N$ phases. The network $\mathscr{G}$ has a minimality property: it minimizes an energy, a weighted length, that, under a further condition, yields sharp lower bounds for the energy of minimizers and allows for a quantitative description of $\mathscr{G}$.

Ronaldo Garcia

## The mathematical work of Jorge Sotomayor: bifurcation theory and qualitative theory of principal lines.

The goal of this talk is to present, in a historical perspective, the contributions of Jorge Sotomayor (25/03/1942-07/01/2022) in two areas. Namely, his work in bifurcations of codimension one, two and three of vector fields, and the qualitative theory of principal lines on surfaces and hypersurfaces.

Pedro Girão
Instituto Superior Técnico, Universidade de Lisboa

## Higher Order Linear Stability and Instability of Reissner-Nordström

We consider smooth solutions of the wave equation, on a fixed black hole region of a subextremal Reissner-Nordström (asymptotically flat, de Sitter or anti-de Sitter) spacetime, whose restrictions to the event horizon have compact support. We provide criteria, in terms of surface gravities, for the waves to remain in $C^l$, $l≥1$, up to and including the Cauchy horizon. We also provide sufficient conditions for the blow up of solutions in $C^1$ and $H^1$.

Diogo Gomes
KAUST

## Monotone operator techniques in mean-field games

Mean-field games model systems with a large number of competing rational agents. Often these games are determined by a system of a Hamilton-Jacobi equation coupled with a transport or Fokker-Planck equation. In many cases, this system can be regarded as a monotone operator. This structure is at the basis of the uniqueness proof by Lasry and Lions. Using monotone operators techniques, we show how to prove the existence of solutions in a number of cases as well as how they can be used to construct numerical methods.

Gustavo Granja
Instituto Superior Técnico, Universidade de Lisboa

## Topology of almost complex structures in dimension $6$

The space of almost complex structures on the $6$-sphere naturally contains a copy of $7$ dimensional real projective space arising from octonion multiplication. I will describe the relation between this copy of $RP^7$ and the whole space of almost complex structures on $S^6$. The arguments used in this example lead to information about the homotopy type of the space of almost complex structures on closed $6$-manifolds, namely an expression as an $S^1$-quotient and, under some mild restrictions, the rational homotopy type.

In dimension $6$, two orthogonal complex structures regarded as sections of twistor space will generically intersect on a finite set of points. I will give a formula for the intersection number in terms of the Chern classes of the almost complex structures.

This is all joint work with Aleksandar Milivojevic.

José Matias
Instituto Superior Técnico, Universidade de Lisboa

## Upscaling and spatial localization of non-local energies with applications to crystal plasticity

An integral representation result is obtained for the asymptotics of energies including both local and non-local terms, in the context of structured deformations.

Starting from an initial energy featuring a local bulk and interfacial contribution and a non-local measure of the jump discontinuities, an iterated limiting procedure is performed. First, the initial energy is relaxed to structured deformation, and then the measure of non-locality is sent to zero, with the effect of obtaining an explicit local energy in which the non-linear contribution of submacroscopic slips and separations is accounted for. Two terms, different in nature, emerge in the bulk part of the final energy: one coming from the initial bulk energy and one arising from the non-local contribution to the initial energy. This structure turns out to be particularly useful for studying mechanical phenomena such as yielding and hysteresis. Moreover, in the class of invertible structured deformations, applications to crystal plasticity are presented.

This is a joint work with Marco Morandotti, David R. Owen, and Elvira Zappale.

Sérgio Oliva

## A local/nonlocal diffusion model

In this talk, we present some qualitative properties for solutions to an evolution problem that combines local and nonlocal diffusion operators acting in two different subdomains. The coupling takes place at the interface between these two domains in such a way that the resulting evolution problem is the gradient flow of an energy functional. We prove existence and uniqueness results, as well as that the model preserves the total mass of the initial condition. We also study the asymptotic behavior of the solutions. Besides, we show a suitable way to recover the heat equation at the whole domain from taking the limit at the nonlocal rescaled kernel. Finally, we propose a brief discussion about the extension of the problem to higher dimensions.

Jaime Angulo Pava

## The NLS equation on tadpole graphs and beyond

The aim of this lecturer is to study stability properties of standing wave solutions for the nonlinear Schrödinger equation (NLS) with power nonlinearity on tadpole or looping edge graphs (graphs consisting of a ring attached to one or many half-lines at one vertex, respectively). By considering general Neumann-Kirchhoff boundary conditions at the junction, also known as a $\delta$-type condition, we study the stability properties of positive state profiles for every positive power nonlinearity. The main novelty of this research is to give the exact value of the Morse index and the nullity index of a specific linearized operator around the positive state profiles. As application of our theory, we consider the case of the cubic NLS on looping edge graphs and dnoidal profiles on the ring and soliton profiles on the half-lines. The results presented in our investigation have prospect for the study of the stability of other branches of standing wave solutions on looping edge graphs, or more general nonlinearities, as well as, for stationary wave solutions of other nonlinear evolution equations.

Clodoaldo Grotta Ragazzo

## Vortices on surfaces: a history in motion.

In this talk we  outline the history of the equations for point vortices motion on curved surfaces that culminates  with recent  results obtained by Gustafsson. One of the consequences of these equations is the definition of the special  “Steady Vortex Metric” in dimension 2 and its generalization to higher dimensions.

Carlos Rocha
Instituto Superior Técnico, Universidade de Lisboa

## Order Preserving Semiflows Revisited

Dynamical systems generated by scalar reaction-diffusion equations enjoy special properties that lead to a very simple structure for the semiflow. Among these properties, the monotone behavior of the number of zeros of the solutions plays an essential role. This discrete Lyapunov functional, the zero number, contains important information on the spectral behavior of the linearization and leads to the simple description of the dynamical system.

Other systems possess this kind of discrete Lyapunov functional and we review some classes of linear equations that generate semiflows with this property. Moreover, we ask if this property is characteristic of such problems.

This is based on a joint work with Giorgio Fusco.

Anibal Rodríguez-Bernal

## Maximum principle and asymptotic dynamics for some nonlocal diffusion equations in metric measure spaces

We present some results on the asymptotic behavior of linear and nonlinear nonlocal evolution problems in rough media, assimilated to metric measure spaces. We analize weak and strong maximum principles for stationary and evolution linear problems. For nonlinear problems with local reaction we discuss global existence, asymptotic bounds and the existence of extremal solutions.

Diogo Oliveira e Silva
Instituto Superior Técnico, Universidade de Lisboa

## Sharp restriction theory: highlights and future directions

Sharp inequalities have a rich tradition in harmonic analysis, going back to the epoch-making works of Beckner and Lieb for the sharp Hausdorff–Young and Hardy–Littlewood–Sobolev inequalities. Even though the history of sharp restriction theory is considerably shorter, it is moving at an incredible pace. In this talk, we survey selected highlights from the past decade, describe some of our own contributions, and pose a few open problems which lie at the interface of euclidean harmonic analysis and dispersive PDE.

Jorge Drumond Silva
Instituto Superior Técnico, Universidade de Lisboa

## Nonlinear smoothing by the infinite normal form reduction method.

Many nonlinear dispersive partial differential equations exhibit a phenomenon where the nonlinear part of the evolution, in the Duhamel integral formula, is slightly more regular than the purely linear evolution, for the same initial data. This is usually called nonlinear smoothing and is also intimately connected to dispersive blow up. We show how the infinite normal form reduction method, developed in the past few years to prove unconditional uniqueness of initial value problems, can be applied to provide a broad and general approach for establishing nonlinear smoothing of several different dispersive PDEs. This is joint work with Simão Correia.

Fábio Tal

## Rotational Chaos for annular dynamics

We study homeomorphisms of surfaces,with emphasys the annulus, trying to characterize when the associated dynamical system has positive topological entropy and rotational entropy. Using topological techniques, we show (with P. Le Calvez) that the presence of any type of shear in conservative settings imply that either the dynamics is near-integrable or it must have rotational horseshoe. For the general case, we show (with A. Passeggi) that if one can find an invariant circloid with sheared dynamics (having non-trivial rotation set), then the dynamics must also have a rotational horseshoe, solving a well known conjecture in the field dating back to the 80s.

Hugo Tavares
Instituto Superior Técnico, Universidade de Lisboa

## Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number $\ell\geq 2$ of components, for the Yamabe equation on a closed Riemannian manifold $(M,g)$.

To this aim, we study a weakly coupled competitive elliptic system of $\ell$ equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if $\dim M\geq 10$, $(M,g)$ is not locally conformally flat and satisfies an additional geometric assumption whenever $\dim M=10$. Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to $-\infty$, giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition.

For $\ell=2$ the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

Gláucio Terra

## The Smoothing Problem in Chow-Rashevskii

Let $M$ be a finite dimensional paracompact smooth manifold endowed with a smooth linear subbundle $\mathcal{D}$ of $TM$. The well-known Chow-Rashevskii’s connectivity theorem asserts that, if $\mathcal{D}$ is bracket-generating, any two points in the same connected component of $M$ may be connected by an absolutely continuous path tangent to $\mathcal{D}$. This talk concerns the question of whether any two such points may be connected by a smooth immersion tangent to $\mathcal{D}$.

Marcelo Viana
Instituto de Matemática Pura e Aplicada

## Thermodynamic u-formalism

The classical thermodynamic formalism, developed in the 1970s by Sinai, Ruelle, and Bowen in the setting of uniformly hyperbolic systems, provides an elegant and remarkably complete theory of the measures of maximal entropy and, more generally, equilibrium states.

In the presence of some invariant structure, one can sometimes define a partial entropy, which measures the complexity of the dynamics along such a structure. Take, for instance, the strong-unstable foliation of a partially hyperbolic diffeomorphism. The corresponding partial entropy is called u-entropy, and there is also a notion of topological u-entropy.

For partially hyperbolic diffeomorphisms that factor over Anosov, we prove several results on the finiteness of measures of maximal u-entropy, the structure of their supports, fast loss of memory (decay of correlations, large deviations), and transverse invariant measures. We also discuss a series of examples, and the relations between measures of maximal u-entropy and measures of maximal entropy.