Titulo: The Caffarelli-Kohn-Nirenberg inequality: a parametric analysis.

Palestrante: Aldo Bazán (UFF)

Data: 29 de agosto de 2019 (quinta-feira)
Horário: 12h
Horário: Instituto de Matemática – Bloco C – Sala C119 – Ilha do Fundão

Resumo: Functional inequalities involving integrals appear quite frequently in estimates and problems of regularity of solutions of partial differential equations, as a consequence of the use of functional spaces that depend on the concept of integral. A simplified version of the inequality presented here appears for the first time in [2], in the analysis of a type of weak solutions of the Navier Stokes equation, and later in its general form in [1]. 

Since it appeared, various modifications and applications have emerged, such as in rigidity problems of differentiable manifolds and in measurement spaces where it is necessary to use alternative definitions to the usual idea of derivative. In this talk, we will give a new proof of the Caffarelli-Kohn-Nirenberg inequality, defining a new real parameter, which is a consequence of the relationships between the original parameters that appear in this inequality.

[1] L.A. Caffarelli, R. Kohn, L. Nirenberg,First order interpolation inequalities with weights, Compositio Math.53(1984), 259–275.
[2] L.A. Caffarelli, R. Kohn, L. Nirenberg,Partial regularity of suitable weak solutions of the Navier-Stokes equa-tions, Comm. Pure Appl. Math.35(1982), 771–831.

Título: Weakly coupled elliptic systems with cooperative or competitive interactions: an overview
Palestrante: Hugo Tavares (Faculdade de Ciências da Universidade de Lisboa)  

Data: 19/07/2019
Horário: 10h
Local: Sala C116

Resumo: In this talk, we will deal with systems of stationary reaction-diffusion equations where the interaction between different components is either cooperative or competitive. Our aim will be to explain some of the relevant questions that can be asked for each type of interaction, explaining as well the motivations for its study. We will survey some of the results proved in the last few years, discussing in general the existence and characterization of positive solutions. Furthermore, we will explain how a strong competition induces a phase separation phenomenon and gives rise to a free boundary problem.

In the last part of the talk, we will consider nonlocal interaction terms between the components. We will highlight some of the similarities and differences between the local and the nonlocal cases, showing some recent results in the nonlocal one.

Data: 20/08/2019
Horário: 12h
Sala: C116

Palestrante: Reinhard Racke (University of Konstanz, Germany)
Titulo:  The Cauchy Problem for Thermoelastic Plates with Two Temperatures
Resumo:We consider the decay rates of solutions to thermoelastic systems in materials where, in contrast to classical thermoelastic models for Kirchhoff type plates, two temperatures are involved, related by an elliptic equation. The arising initial value problems deal with systems of partial differential equations involving Schr¨odinger like equations, hyperbolic and elliptic equations. Depending on the model – with Fourier or with Cattaneo type heat conduction – we obtain polynomial decay rates without or with regularity loss. This way we obtain another example where the loss of regularity in the Cauchy problem corresponds to the loss of exponential stability in bounded domains. The wellposedness is done using semigroup theory in appropriate space reflecting the different regularity compared to the classical single temperature case, and the (optimal) decay estimates are obtained with sophisticated pointwise estimates in Fourier space.

Palestrante: Ludovick Gagnon (INRIA, França)
Titulo: On the link between controllability and integrability
Resumo: The aim of this talk is to make to present the possible applications of the integrability of a dynamical system (ODE or PDE) to its controllability. On one hand, the integrability, in a broad sense, implies that the dynamical system has more rigidity, either by having more conserved quantities or by having a foliation of its phase space. On the other hand, the controllability refers to the notion of being able to drive the initial state of the dynamical system to another target final state by means of external forces. There exists many methods in the literature to study the controllability of linear ODE or PDE but complications arise quickly when one desires to study the small-time controllability of nonlinear PDEs. To motivate the link between controllability and integrability, we shall first revisit the now well known controllability of the linear wave equation on a smooth bounded domain. We shall prove that the integrability of the ellipse yield a surprising result on the controllable regions for the wave equation. We will then move on to challenging open problems of small-time controllability of some nonlinear PDEs, expose limitations of existing methods and give insights of what integrability may provide for the controllability of these equations. 

Titulo: Null controllability of the structurally damped wave equation with a moving control
Palestrante: Lionel Rosier (MINES ParisTech, França)

Data: 10/07/2019 
Horário: 12h 
Local: Instituto de Matemática – Bloco C – Sala C116 – Ilha do Fundão

Resumo: We investigate the null controllability of the wave equation with a Kelvin-Voigt damping. It is well known that the null controllability fails if the control region is fixed and strictly included in the space domain. We consider here a distributed control supported in a moving domain. We shall review recent results obtained by the speaker on this issue: (i) in dimension one (with P. Martin and P. Rouchon); (ii) in dimension N (with F. W. Chaves-Silva and E. Zuazua); (iii) in dimension two, with sharp geometric conditions (with P. Guzman).