Análise

Daniel Marroquin (UFRJ)

TÍTULO: Weak solutions of the compressible Poisson-Nernst-Planck-Navier-Stokes equations

RESUMO: We consider the compressible Poisson-Nernst-Planck-Navier-Stokes (PNPNS) system of equations, which models the transport of charged particles, under the influence of the self-consistent electrostatic potential, in a compressible fluid. We study the equations posed on a smooth bounded spatial domain of R3 and consider a set of boundary conditions motivated by physical considerations. Then we prove the existence of global weak solutions for the initial/boundary value problem, without restrictions on the size of the initial data. This is a joint work with Dehua Wang.

Disson dos Prazeres (UFS)

TÍTULO: On fractional quasilinear equations with elliptic degeneracy.

RESUMO: In this talk, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive geometric argument that interplays with the uniqueness property for the corresponding homogeneous problem, leading with gradient Hölder regularity estimates. This approach is intrinsically developed for nonlocal scenarios, where uniqueness holds for the local homogeneous problem.

Isadora de Jesus (UFAL)

TÍTULO: Efeitos da Memória infinita na estabilização da Equação Biharmônica de Schrödinger

RESUMO: Nesta palestra trataremos da estabilização da equação linear biharmônica de Schrödinger em um domínio limitado aberto n-dimensional sob condições de fronteira de Dirichlet–Neumann considerando três termos de memória infinita como mecanismos de amortecimento. Nós mostramos que dependendo da suavidade dos dados iniciais e do crescimento arbitrário na infinidade da função kernel, esta classe de solução vai para zero com uma taxa de decaimento polinomial como t^−n dependendo de suposições sobre a função kernel associada aos termos de memória infinita.

Jaime Angulo (USP)

TÍTULO: A splitting eigenvalue method for Schrödinger’s type operators on a tadpole graph

RESUMO: The aim of this lecture is to provide new tools in the mathematical studies associated to the existence and orbital stability of standing wave solutions with positive profiles for the NLS with power nonlinearity on a tadpole graph. This graph is composed of a circle and an infinite half-line attached to a common vertex. The model considers boundary conditions at the graph-vertex of δ-interaction type. It is shown that, via the dynamical system theory for orbits on the plane together with the period map associated to second-order differential equations, one can show the existence of profiles on the circle with Neumann-Kirchhoff boundary conditions. About the stability issue, we establish a splitting eigenvalue method for identifying the Morse and nullity indices of specific linearized operators around standing wave solutions (Schrödinger’s type operators) which are essential in a stability analysis. The theory developed in this investigation has prospects for the study of the NLS on a looping edge graph, as well as for other nonlinear evolution models on a tadpole graph.

João Nariyoshi (USP)

TÍTULO: On the existence of source-solutions to the multidimensional Burgers equation

RESUMO: Recently, D. Serre and L. Silvestre made a profound breakthrough in the study of the multidimensional Burgers equation u_t + uu_{x_1} + ... + u^N u_{x_N} = 0 with the establishment of dispersive estimates for its entropy solutions. This enabled them to show that the multidimensional Burgers equation is well posed in $L^p(R^N)$, thus positively answering a conjecture of M. Crandall. Then, D. Serre investigated the existence of solutions to the multidimensional Burgers equation if the initial data was instead a bounded measure. This is a question motivated by the description of the asymptotic behavior of solutions with integrable initial data. Nonetheless, the results therein were not entirely conclusive: Although one can construct approximate solutions and one can prove their compactness, it was not established that the limit functions fit the initial condition. In this talk, we will explain why this is so by providing negative results regarding the existence of such solutions. In spite of this, we are still able to obtain a depiction of the asymptotic behavior of solutions.

Roberto Capistrano (UFPE)

TÍTULO: Long-time dynamics of water-wave models

RESUMO: Water-wave models play a crucial role in understanding, predicting, and controlling the dynamics of surface water waves across various real-world scenarios, including oceanic waves, waves in lakes and rivers, and those affecting man-made structures. These models integrate mathematical, physical, and numerical frameworks with wide-ranging applications in environmental science, engineering, and maritime industries. In this talk, we will explore key mathematical results for several water-wave models, highlighting their relevance to real-world applications.

Claudia Espitia (UNICAMP)

TÍTULO: On local well-posedness of the stochastic incompressible density-dependent Euler equations

RESUMO: We study the stochastic inhomogeneous incompressible Euler equations in the whole space R3. We show the existence and pathwise uniqueness of local solutions with a multiplicative stochastic noise. Our approach is based on reducing our problem to a random problem and some estimations for type transport equations.

Patrícia Guidolin (UFRGS)

TÍTULO: Resultados de Existência Global para Equações de Advecção-Difusão Conservativas

RESUMO: Nesta palestra vamos mostrar um procedimento de análise que vem sendo aplicado para derivação de varias estimativas básicas importantes para soluções u(·, t) de equações de advecção-difusão conservativas em meios heterogêneos. Tomaremos como protótipo o problema

u_t + (b(x, t)u^{k+1})_x = μ(t)uxx, (1)

u(x, 0) = u_0 ∈ L^∞(R) ∩ L^1(R).

e mostraremos como o método e aplicado na investigação de condições de existência global para soluções de (1). Faremos uma breve revisão do que já se sabe em torno do assunto, bem como, comentários sobre questões em aberto e o que estamos estudando atualmente.

Janielly Araujo (UFERSA)

TÍTULO: Estimativas de Regularidade ótima para Problemas de Evolução Degenerado em Espaços de Orlicz

RESUMO: Nos investigamos estimativas quantitativas de regularidade com a teoria de equações diferenciais parciais parabólicas degeneradas, especificamente a equação no contexto de espaços Orlicz-Sobolev, cujo o protótipo é

u_t+div( g(|\nabla a|)\frac{\nabla u}{|\nabla u|})=f(x)

Empregando uma análise tangencial geométrica adaptadas estruturas de Orlicz-Sobolev e escalas intrínsecas, derivamos estimativas precisas de regularidade Hölder interior para soluções fracas limitadas.

Renan Medrado (UFAL)

TÍTULO: A new class of FBI transforms and applications

RESUMO: In this talk I will present a class of FBI transforms using weight functions (which includes the subclass of Sjostrand’s FBI transforms used by M. Christ in 1997) that is well suited when dealing with ultradifferentiable functions and ultradistributions defined by weight functions in the sense of Braun, Meise and Taylor (BMT). I will show how to characterize BMT local regularity of ultradistributions using this wider class of FBI transform. Also, I will present a characterization of Iterates of Partial

Differential Operators in the sense of BMT classes (called BMT vectors) and, as an application, I will present a relation between BMT local regularity and BMT vectors.

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