José Francisco Rodrigues

55 years of PDEs with João Paulo.

We present a brief overview “à vol d’oiseau”, from 1969-2024, of the works of J. P. Dias with his contributions on the nonlinear theory of partial differential equations, from his original thesis on the regularity of solutions to elliptic and parabolic unilateral problems until his most recent results on solutions of some systems for short wave–long wave interaction models, not forgetting his study of the equations of nematic liquid crystals, of coupled nonlinear systems of Schrödinger and generalised Korteweg-de Vries equations, nonlinear Dirac equations and a scalar discontinuous conservation law in a limit case of phase transitions. 

Maria J. Esteban

Optimal quantitative stability results for the Sobolev and the Gaussian logarithmic Sobolev inequalities.

In this talk I will present recent results concerning optimal quantitative stability properties for the Sobolev and logarithmic-Sobolev inequalities with computable constants. The result for the Gaussian version of the logarithmic Sobolev inequality is actually a corollary of the one for Sobolev. This is done, in an optimal manner, by a limiting argument in high dimensions.

The work presented in this talk is the result of a collaboration with J. Dolbeault, A. Figalli, R. Frank and M. Loss. 

Mário Figueira

Standing Waves for the Ginzburg-Landau Equation.

This talk is concerned with a class of periodic solutions, standing waves, of the Complex Ginzburg-Landau Equation.

Considering the corresponding semilinear elliptic equation, we study the existence of nontrivial solutions as bifurcations starting

from the eigenvalues of the Dirichlet-Laplacian of arbitrary multiplicity. We conclude with the orbital stability of these

solutions under the Ginzburg-Landau flow.

This talk is based on the joint works with Simão Correia (MR 4552143 (2023); MR 4259638 (2021) ). 

Hugo Tavares

A comprehensive study of positive bound states of the NLS equation on single-knot metric graphs.

Over the past decade, there has been significant activity in the study of the Nonlinear Schrödinger Equation (NLS) on metric graphs, namely in how the topology or geometry of the graphs influenciates existence and multiplicity of bound states, existence of both action and energy ground states, along with their stability properties. In this presentation, we will focus on the study of this equation on a particular class of graphs that include the T-graph (comprising two unbounded edges and a terminal edge) and the tadpole graph (a halfline and a loop). These graphs consist of the  two simplest (nontrivial) perturbations of the real and the half line, respectively. We will present a complete classification of all positive symmetric solutions, which turn out to be all positive solutions in some situations. In particular, we will discuss existence and uniqueness of action ground states, determine some situations where the concepts of action and energy ground states do not coincide and where, moreover, energy ground states are not unique for some masses. Finally, we show the surprising fact that there are stable and 

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unstable action ground states both in the L  -subcritical and the L  -supercritical regimes. 

The talk is based on  joint works with Francisco Agostinho and Simão Correia.

Jesús Hernandez

Beyond the Strong Maximum Principle: changing sign forcing and flat solutions.

In this joint work with J. I. Díaz, we show how the classical Strong Maximum Principle for the equation -∆u = f(x) in Ω, with u = 0, on ∂Ω, where Ω is a bounded

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domain in R    , to some cases of f(x) ≤ 0 somewhere in Ω. We study flat solutions, i. e., solutions u > 0 such that ∂u/ ∂n = 0 on ∂Ω. We give some applications to

elliptic and parabolic problems.

Luís Sanchez

Periodic resonant problems, discrete and continuous.

In a short note published in 2000, A. C. Lazer looked back at the first result of Landesman-Lazer type, and provided a simple approach to periodic solutions of a certain ODE with non-invertible linear part. Inspired by Lazer's note, we deal with a related discrete problem, and also a variant of the continuous problem where a bounded restoring force is added to the equation. Variants of Miranda's theorem play a role in the proofs. The results have appeared in joint papers with Luís Ferreira (2020) and with João G. Silva (2021).

Jean-Baptiste Casteras

On a nematic liquid crystal director field equation.

In this talk, I will review recent works [1,2] of Prof. João Paulo Dias on the motion of the director field of a nematic liquid crystal submitted to a magnetic field and to a laser beam. The problem takes the form of a quasilinear wave equation coupled to a Schrödinger equation. I will discuss the existence of solutions to this evolution problem as well as the existence and stability properties of standing wave solutions. 

[1] Amorim, P., Casteras, JB. & Dias, JP. On the Existence and Partial Stability of Standing Waves for a Nematic Liquid Crystal Director Field Equations. Milan J. Math. 92, 143–167 (2024). https://doi.org/10.1007/s00032-024-00395-8

[2] Amorim, P., Dias, JP. & Martins, A.F. On the motion of the director field of a nematic liquid crystal submitted to a magnetic field and a laser beam. Partial Differ. Equ. Appl. 4, 36 (2023). https://doi.org/10.1007/s42985-023-00256-w

Hermano Frid

Short Wave-Long Wave Interactions in the Relativistic Context.

In this talk I review a model introduced in a joint work with J.P. Dias on the relativistic short wave-long wave interaction  where the short waves are described by the massless 1+3-dimensional Thirring model of nonlinear Dirac equation and  the long waves are described by the 1+3-dimensional  relativistic Euler equations.  An important feature of the model is that the Dirac equations are based on the Lagrangian coordinates of the relativistic fluid flow. In particular, an important contribution of this joint work is a clear formulation of  the relativistic Lagrangian transformation.

Filipe Oliveira

On semilinear and quasilinear dispersive systems.

In this talk, we will present several systems involving the coupling of the Nonlinear Schrödinger Equation with other semilinear or quasilinear PDEs of various types. Examples include the Benney system and Schrödinger/gKdV systems, among others. We will discuss results on local and global well-posedness, blow-up phenomena, and the existence and stability of bound and ground states. These findings were obtained between 2011 and 2019 in collaboration with João-Paulo Dias and other colleagues, Mário Figueira, Hugo Tavares, and Stanislav Antontsev.