Edições Passadas

Abstracts

Luís Salge

Spectrum of elliptic differential operators with constant coefficients on Lps,loc(Ω)

Abstract In this work, we extended the study of spectrum started in [E.R. Aragão Costa and L.M. Salge, Annals of Functional Analysis, 13(2022), No. 4, 1-17] and [Salge, L., and Picon, T. (2024). Matemática Contemporânea, 59(7)] for elliptic homogeneous differential operatators with constant coefficients to the n dimensional case in the real scales of localized Sobolev Spaces Lps,loc(Ω), where 1 ≤ p < ∞ and s ∈ R and Lps,loc(Ω) = {u ∈ D′ (Ω); F −1 [(1 + |ξ|2 F(ϕu)] ∈ Lp (Rn ), ∀ϕ ∈ Cc∞ (Ω)}, where F is the Fourier transform. This is quite different from what we find in the literature, where all the relevant results are concerned with spectrum on Banach spaces. Our aim is to understand the behavior of the spectrum using the clo- sure of the operator. In particular we show that there is no complex number in the resolvent set of such operators, which suggests a new way to define spectrum if we want to reproduce the classical theorems of the Spectral Theory in Fréchet spaces. Joint work with Tiago H. Picon (University of São Paulo).

Asun Jiménez

EDPs com singularidades: uma abordagem geométrica

Algumas equações em derivadas parciais são chamadas de "geométricas" quando modelam algum fenômeno ou superfície que possui propriedades relevantes desde o ponto de vista da geometria. Mesmo no caso elíptico, soluções a problemas que envolvem tais EDPs geométricas podem apresentar singularidades. Nesta palestra veremos como a geometria auxilia na interpretação e resolução de alguns problemas envolvendo  EDPs geométricas de diferente natureza.  Primeiramente nos centraremos na classificação de singularidades isoladas de soluções de equações de Monge-Ampère (em particular, as associadas à superfícies convexas em espaços forma). Também veremos como as técnicas se estendem ao estudo de equações quase-lineares que modelam superfícies de curvatura média prescrita no espaço de Lorentz-Minkowski L3. Finalmente, se o tempo permitir, trataremos de um problema de Neumann geométrico para a equação de Liouville. Através dele estudamos métricas com curvatura de Gauss constante em domínios com fronteira de curvatura geodésica constante (e, eventualmente, pontos singulares).

Alexander Quaas

Mixed local and nonlocal Laplacian without standard critical exponent for Lane-Emden equation

In this paper, we investigate a mixed elliptic equation involving both local and nonlocal Laplacian operators, with a power-type nonlinearity. Specifically, we consider a Lane-Emden type equation of the form −∆u + (−∆)s u = up , in Rn . where the operator combines the classical Laplacian and the fractional Laplacian. We establish the existence of solutions for exponents slightly n+2 below the critical local Sobolev exponent, that is, for p < n−2 , with p n+2 close to n−2 . Our results show that, due to the interaction between the local and nonlocal operators, this mixed Lane-Emden–Fowler equation does not admit a critical exponent in the traditional sense. The existence proof is carried out using a Lyapunov–Schmidt type reduction method and, as far as we know, provide the first example of an elliptic operator for which the duality between critical exponents fails.

Abstracts

Maria Soledad Aronna

Optimal Control of Dynamical Systems with Parametric Uncertainty

Our research focuses on optimal control problems with parameter uncertainty, which involve optimizing systems governed by families of controlled ordinary differential equations. These equations are parameterized on a probability space representing the range of possible parameter values. We develop necessary optimality conditions and numerical algorithms tailored to this problem class. We show applications in fishery management and optimal search strategies, demonstrating the applicability of our methods in real-world scenarios.

Hugo Saraiva 

Some lattice Boltzmann models for a class of multiphysics problems

The Lattice Boltzmann Method (LBM), rooted in kinetic theory with formal links to the Navier-Stokes equations, offers a versatile platform for simulating complex fluid dynamics and multiphysics phenomena. This work explores the application of various LBM models to a specific class of such problems. After reviewing the traditional LBM and efficient simplified variants for momentum and continuity, we focus on extensions necessary for multiphysics coupling. Key among these is the LBM for generalized advection-diffusion equations (ADE). We then demonstrate the utility of these methods through several case studies: analyzing the distinct turbulent mixing characteristics in miscible and immiscible Rayleigh-Taylor instabilities; modeling magnetohydrodynamic (MHD) flows where fluid motion interacts with magnetic fields; capturing the pattern formation in the viscous fingering phenomenon; and simulating complex particle-laden flows using an Eulerian-Eulerian LBM framework, exemplified by systems involving limescale formation. At the end, we provide some concluding remarks offering perspectives on future research directions, including potential algorithmic enhancements, extensions to other coupled physics, and further validation efforts.

Anup Biswas

Pointwise convergence of the solutions to the initial data for the abstract heat equation

In a recent study, Hartzstein, Torrea, and Viviani characterized all the weights $v$  for which the solution to the classical heat equation with initial data $f$, where $f\in L^p_v(\mathbb{R}^n)$, converges to $f$ as $t\to 0$, almost everywhere and for every $f\in L^p_v(\mathbb{R}^n)$. This work is, of course, in the spirit of Carleson’s program, where similar investigations have been conducted for the Schrödinger operators. In this talk, we will extend the results of Hartzstein et al. to a broader class of operators on metric measure spaces with a volume doubling condition, including $\phi$-nonlocal operators, mixed local-nonlocal operators, the Laplacian with a Hardy potential, the Laplacia-Beltrami operators, Laplacian on fractals and many others.

This talk, which is based on a recent joint work with Bhimani and Dalai, is going to be a mix of probability, PDE and harmonic analysis.

Abstracts

Wladimir Neves

Existence of weak solutions to obstacle problems for a quasilinear unidimensional wave equation

In this talk, we prove the existence of a weak solution for the one obstacle problem for a class of quasilinear wave equations in one space dimension, extending previous results obtained in the linear case, as well as for a two obstacle problem, which is novel in the hyperbolic case. In contrast with the linear case, in a strict quasilinear case one has a weak regularity estimate and, by interpolation, we obtain a continuous solution, which satisfies also a weak entropy condition in the free domain where the string is not in contact with the obstacles.

Carlos Guzmán

New results about the biharmonic nonlinear Schrödinger equation

In this talk, we discuss the new results of the inhomogeneous biharmonic equation. We study the local and global well-posedness. To this end, we use the standard fixed point argument. Moreover, we investigate the asymptotic behavior of the solution, assuming the general initial data. We discuss some new techniques to prove that in different cases: intercritical and critical ones. 

Rodrigo Matos

Localização de Anderson como uma interface entre análise, equações diferenciais e probabilidade

Há aproximadamente 70 anos, em seu artigo seminal intitulado "Absence of Diffusion in Certain Random Lattices", o físico P.W. Anderson argumentou que a presença de desordem pode afetar drasticamente as propriedades de transporte em um sistema quântico. Esta descoberta levou ao desenvolvimento de uma teoria matemática que combina técnicas e princípios de teoria espectral, análises complexa e harmônica, equações diferenciais, probabilidade, sistemas dinâmicos e teoria ergódica para entender com rigor este fenômeno, hoje denominado localização de Anderson. Nessa palestra, iremos discutir parte desta teoria focando em seus aspectos analíticos, progresso recente e problemas em aberto.

Abstracts

Julia Domingues Lemos 

Machine learning aided closures of turbulence models

Even though turbulent fluid motion has been the subject of decades of active research, it still lacks an analytic description. Practical applications often require some level of modelling of small scales of motion, which we approach here in the form of a closure problem for a turbulence model. We aim to build a theoretically solid closure for the Sabra model, based on a dynamical rescaling of velocity fluctuations with the aid of machine learning tools. The rescaling will provide us with universal statistics, while a suitable machine learning tool will allow us to learn such statistics from data. This approach allows us to reproduce statistics of the fully resolved model, while including crucial information such as spatial and time correlations. This work is joint with F. Santos.

Reinaldo Resende

Regularity results for area minimizing currents

In this presentation, we’ll explore exciting new results on the interior and boundary regularity of currents T solving the oriented Plateau’s problem, with a special focus on higher codimensions. We will extend well-known estimates concerning the Hausdorff dimension of the interior singular set of T to a broader context, and also share results from an upcoming work that optimally resolves several long-standing open questions on boundary regularity. Additionally, we’ll discuss recent advancements on the rectifiability of the singular set, and, if time allows, showcase examples that highlight the optimality of all these results.

Miguel Soto

On the controllability properties for a higher-order dispersive system

In this talk deals with the boundary controllability of a family of nonlinear Boussinesq systems introduced by J. L. Bona, M. Chen and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. By combining the classical duality approach and a careful spectral analysis of the operator associated with the state equations, we first obtain the exact controllability of the linearized system in suitable Hilbert spaces. Then, by means of a contraction mapping principle, we establish the local exact controllability for the original nonlinear system. This work is joint with Ademir Pazoto and Sorin Micu.

Americo Cunha

The cross-entropy method for optimization in computational science and engineering

The Cross-Entropy (CE) method, a robust stochastic optimization technique, has garnered attention for its efficacy in addressing complex optimization problems across various disciplines. Originating from the field of rare event simulation, the CE method has evolved into a versatile tool for combinatorial optimization, continuous optimization, and machine learning tasks. Its core strategy involves generating sample solutions and iteratively refining probability distributions to hone in on the optimal regions of the solution space. This seminar introduces the CE method and its practical implementation through the CEopt code, a MATLAB-based framework designed to simplify the application of CE techniques. The CEopt code encapsulates the method's adaptability and effectiveness, featuring support for both constrained and unconstrained optimization problems. Its modular architecture, equipped with input validation, adaptive sampling mechanisms, and dynamic parameter adjustment, enables users to tackle a wide array of optimization challenges without deep dives into algorithmic intricacies. Participants from machine learning and computational mechanics backgrounds will find particular interest in how the CEopt code can integrate into their workflows to optimize performance metrics and system designs. This seminar will cover the theoretical foundations, practical considerations, and potential applications of the CE method, demonstrating its utility with real-world examples and discussing future directions in optimization technology.

Abstracts

Felipe Gonçalves

Poisson Summation, Spectroscopy, and the Riemann Hypothesis: A Journey Down the Rabbit Hole

Embark on a journey down the rabbit hole, where Poisson summation, physical diffraction patterns, and the Riemann Hypothesis intertwine in a mathematical tapestry. In this talk we shall explore the connections between these domains, revealing how modern mathematical machinery and insights from physics can shed new light on the elusive Riemann Hypothesis.

Paulo Amorim

An epidemiology model with variable susceptibility and disease awareness using transport equations

We present an epidemiological model considering disease awareness and variable susceptibility. The model uses transport-type partial differential equations in which the susceptible population is described by its distribution in an awareness space and the degree of susceptibility. From these PDEs, a system of ODEs is deduced that represents the solutions. We studied the properties of both the ODE and PDE systems, showing the impact of awareness dynamics on epidemiological results.

Jean-Michel Roquejoffre

Large time dynamics in the Fisher-KPP equation

The Fisher-KPP equation (the acronym KPP standing for Kolmogorov, Petrovskii and Piskunov) is an ubiquitous model that arises in the applied sciences, such as ecology or combustion science, but also in probability theory.  When solved with a a compactly supported initial datum, its level sets advance at asymptotically constant speed, corrected by a term that is logarithmic in time. The correction was discovered by Bramson at the beginning of the 80's, with the aid of probabilistic arguments. A PDE counterpart to this proof was found much later, in 2017, in a joint work with J. Nolen and L. Ryzhik. The goal of the talk is to explain the mechanism leading to this asymptotic behaviour with simple PDE arguments, and to discuss recent models that can be explored with these ideas, such as, for instance, equations with integral diffusion or nonlocal models in epidemiology.

Yulia Petrova

A propagating terrace of two traveling waves in a toy model of Incompressible Porous Medium equation

In the talk we will discuss viscous / gravitational fingering phenomenon - the unstable displacement of miscible liquids in porous media with the speed determined by Darcy's law. Laboratory and numerical experiments show the linear growth of the mixing zone, and we are interested in determining the exact speed of propagation of fingers. We introduce and analyse a "toy" model, that explains the possible mechanism of slowing down the fingers' growth due to convection in the transversal direction. In the simplest setting we show the structure of gravitational fingers - the mixing zone consists of space-time regions of constant intermediate concentration and the profile of propagation is characterized by two consecutive traveling waves which we call a terrace. The talk is based on joint work with S. Tikhomirov and Ya. Efendiev

Abstracts

Débora de Oliveira Medeiros

Lagrangian-finite difference schemes for the upper-convected time derivative

The constitutive equations are reformulated to investigate the reasons for the numerical instabilities that affect the simulations involving high Weissenberg numbers, which present a challenge for the study of viscoelastic fluids. We rewrite the viscoelastic models with the upper-convected time derivative term according to the definition of the generalized Lie derivative and suggest a numerical scheme based on a Lagrangian framework and finite difference method. The

new numerical schemes can reach second-order accuracy in time and space corroborating with the theoretical analysis; making possible a larger time step for the numerical simulations; and can be combined with stabilizing methods, allowing the discussion of High Weissenberg Number Problems.

Sergey Tikhomirov

Free boundary and rattling is systems with hysteresis

Hysteresis naturally appears as a mechanism of self-organization and is often used in control theory. Important features of hysteresis are discontinuity and memory. We consider reaction-diffusion equations with hysteresis. Such equations describe processes in which diffusive and non-diffusive instances interact according to a hysteresis law. Growth of a colony of bacteria in a Petri plate is our prototype model. Due to the discontinuity of hysteresis, questions of well-posedness of such equations are highly non trivial.

For so-called transverse initial data it is possible to establish existence and uniqueness of the solution. Important part of the proof is the free boundary problem.

For non transverse initial data we consider a spatial discretization of the problem and present a new mechanism of pattern formation, which we call rattling. The profile of the solution forms two hills propagating with non-constant velocity. The profile of hysteresis forms a highly oscillating quasiperiodic pattern, which explains mechanism of illposedness of the original problem and suggests a possible regularization. Rattling is very robust and persists in arbitrary dimension and in systems acting on different time scales.

Carlos Guzman Jimenez

Well-posedness and scattering for the generalized Hartree equation in 3D

In this talk, we discuss new results regarding the generalized Hartree equation in 3D. Our study focus on both local and global well-posedness, as well as the scattering problem associated with this equation. We will be considering some models related to the Hartree equations, precisely, the inhomogeneous Hartree equation, and the Hartree equation with a potential. We discuss some new techniques to prove it in different scenarios such as inter-critical and critical cases.

Abstracts

Sergey Sergeev

Whispering gallery-type asymptotics of the eigenfunctions for the Laplacian inside the torus

Whispering galleries have been known for more than a century, and this term came from the acoustics, where it means that sound with certain frequencies propagates along the boundary of some area. Therefore if somebody stands inside of this area he will not hear anything. The very famous example of such construction is the St. Paul's Cathedral in London where Lord Rayleigh studied this effect.
From the mathematical point of view such whispering galleries can be described with the help of the eigenvalues of the Laplace operator. For the large eigenvalues the corresponding eigenfunctions will be localized near the boundary, and are called whispering gallery-type eigenfunctions.
Another important area where whispering gallery-type eigenfunctions appear, is optics, where the toroidal microresonators are used. In the present talk we will consider  the eigenproblem for the Laplacian inside the solid torus with Dirichlet boundary conditions and we pose the asymptotic problem for the large values of the eigenvalues.  We construct explicit asymptotic formulas for whispering gallery-type eigenfunctions which are localized near some part of the boundary of the torus.

Daniel Marroquin

Homogenization of Stochastic Conservation Laws with Multiplicative Noise

We consider the generalized almost periodic homogenization problem for two different types of stochastic conservation laws with oscillatory coefficients and multiplicative noise. In both cases the stochastic perturbations are such that the equation admits special stochastic solutions which play the role of the steady-state solutions in the deterministic case, and are crucial elements in the homogenization analysis. Our homogenization method is based on the notion of stochastic two-scale Young measure, whose existence we establish. This is a joint work with Hermano Frid and Kenneth H. Karlsen.

Aldo Bazan

Innterpolation inequalities in Sobolev spaces with variable exponent

The Sobolev spaces with variable exponent W^{k,p(.)} represent an extension of the well-established Sobolev spaces W^{k,p}. Immersions, an important aspect of the theory,can also be found in this type of spaces . However, in this case it is necessary to consider regularity conditions associated with the involved exponents. In this lecture, we show some interpolation inequalities in these spaces and explore potential applications.