Palestra: Controllability properties of anomalous diffusion phenomena
Palestrante: Sorin Micu (University of Craiova and Institute of Statistical Mathematics and Applied Mathematics, Romênia)
Data: 23/01/2020
Horário: 11h
Sala: C116
Resumo: Many physical phenomena are characterized by an anomalous diffusion when the mean square displacement of a particle will grow at a nonlinear rate in time. Some typical examples are the subdiffusional mobility of the proteic macromolecules in overcrowded cellular cytoplasm and the smoke's superdiffusion in turbulent atmosphere. We consider a simple one dimensional linear model which describes an anomalous diffusive behavior, involving a fractional Laplace operator, and we study its controllability property. If the fractional power of the Laplace operator is less or equal than 1/2 we are dealing with a subdiffusion phenomenon and the system is not spectrally controllable. The aim of the paper is twofold. Firstly, to analyze the possibility of controlling a finite number N of eigenmodes of the solution and to find the behavior of the corresponding controls when N tends to infinity. Secondly, to investigate the null-controllability property of the system when the support of the control moves linearly with respect to time.
Titulo: Nonradial blow-up solutions for the Zakharov system
Palestrante: Juan C. Cordero Ceballos (UNAL, Colômbia)
Data: 02/10/2019
Horário: 12:00
Local: Sala - C116
Resumo: We will show that there are nonradial solutions for the Zakharov equations, which have blow-up in finite time in the case of negative energy, due to a virial identity of momentum type. This solutions are standing waves for the Zakharov-Rubenchik system, so we give response to two questions proposed by F. Merle in [1].
References:
[1] F. Merle, Blow-up results of virial type for Zakharov Equations, Communications in Mathematical Physics, 175, 433-455 (1996)
[2] J. C. Cordero, Supersonic limit for the Zakharov-Rubenchik system, Journal Differential Equations, 261 (2016), 5260-5288
[3] J. R. Quintero, J.C. Cordero, Instability of the standing waves for a Benney-Roskes/Zakharov-Rubenchik system and blow-up for the Zakharov equations, Discrete and Continuous Dynamical Systems Series B doi:10.3934/dcdsb.2019217
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.
Titulo: Analysis of the Ericksen-Leslie Equations for Nematic Liquid Crystal Flows
Palestrante: Matthias Hieber (T. U. Darmstadt)
Data: 09 de setembro 2019 (segunda-feira)
Horário: 12h
Local: Instituto de Matemática – Bloco C – Sala C116 – Ilha do Fundão
Resumo: In this talk we discuss various aspects of the analysis of the Ericksen-Leslie equations describing the flow of nematic liquid crystals both in the isothermal and nonisothermal situation. We consider here the case of general Leslie and general Ericksen stress and obtain a rather complete understanding of the dynamics of this system.
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.
