# 24º CBM - Sessões Especiais: EDPs na Indústria e Engenharia

Horário |
16:30 |
17:00 |
17:30 |
17:45 |
18:15 |
18:45 |
19:00 |
19:30 |

Terça-feira29.07.2003 |
Ester Gabetta |
José Antonio Carrillo de la Plata |
Break |
Jorge Salazar |
Claudia Lederman |
Break |
Antonio Leitão |
María Gabriela Armentano |

Quinta-feira31.07.2003 |
Marco Calahorrano |
Fabio Chalub |
Tamara Grava |
Gregorio Falqui |
Fernando Quiros |
Milton Lopes |

**Control over a method for solving Boltzmann equation via probabilistic tools**

- Ester Gabetta – Università di Pavia, Itália

Slides

Data: 29 de julho, terça-feira

Horário: 16:30

Resumo: In a recent paper we have bounded theLerror made when the Wild sum for solutions of the spatially homogeneous Boltzmann equation is truncated at some finite stage N. It was shown that the results obtained there were qualitatively optimal since they let to an exponential bound on the rate at which solutions relax to equilibrium. Then we carried this investigation further for the Kac equation, investigating specially the relation between the rate of relaxation and the rate at which the error in truncation decreases with the level of truncation. We also obtain these results for a broader class of initial data.^{1}The main results are based on a solution of an old McKean’s conjecture.

(co-authors: E.Carlen, M.Carvalho)

**Numerical simulation of the Boltzmann transport equation in semiconductors**

- José Antonio Carrillo de la Plata, Univ. de Granada

Slides

Data: 29 de julho, terça-feira

Horário: 17:00

Resumo: In this talk we review the Boltzmann-Poisson system as a model of transport of charged particles in semiconductors. We propose a deterministic solver by using conservation law numerical methods suitable for dealing with high gradient regions at the same time as being of high order in smooth regions. We show numerical results of these simulations in 2D devices.

**Free boundary regularity for a problem arising in Superconductivity**

- Jorge Salazar – Universidade de Lisboa, Portugal

Slides

Data: 29 de julho, terça-feira

Horário: 17:45

Resumo: This talk concerns regularity properties of a free boundary arising in the mean-field theory of superconductivity. The problem is reminiscent of the one studied earlier by two of the authors and L. Karp in connection with potential theory. The difficulty introduced in this paper is the existence of several patches, where on each patch the solution to the problem may have different constant values. However, using a refined analysis, we reduce the problem to the case of one-patch; at least locally near `regular’ free boundary points. Using a monotonicity formula, due to Georg S. Weiss, we characterize global solutions of a related equation. Hence earlier regularity results apply and we conclude theC-regularity of the free boundary.^{1}This work was developed in sollaboration with L. Caffarelli and H. Shahgholian.

**A free boundary problem from nonlocal combustion**

- Claudia Lederman – Universidad de Buenos Aires, Argentina

Slides

Data: 29 de julho, terça-feira

Horário: 18:15

Resumo: We study a singular perturbation problem for a nonlocal evolution operator. The problem appears in the analysis of the propagation of flames, when the high activation energy limit is considered in a combustion model admitting nonlocal effects.

We obtain uniform estimates and we show that limits are solutions to a free boundary problem in a viscosity sense and in a pointwise sense at regular free boundary points.

**Inverse problems for semiconductor equations**

- Antonio Leitão – Johann Radon Institute for Computational and Applied Mathematics, Áustria

Slides

Data: 29 de julho, terça-feira

Horário: 19:00

Resumo: This talk is devoted to the investigation of inverse problems related to stationary drift-diffusion equations modeling semiconductor devices. In this context we analyze several identification problems corresponding to different types of measurements, where the parameter to be reconstructed is a coefficient (function) in a system of PDE’s.We also consider special scaling limits of the drift-diffusion equations, in which case the inverse problems reduce to classical inverse boundary problems or inverse conductivity problems.

(joint work with H.W.Engl and P.Markowich)

**Nonhomogeneous Neumann problem for the Laplace equation in cuspidal domains**

- María Gabriela Armentano – Universidad de Buenos Aires, Argentina

Slides

Data: 29 de julho, terça-feira

Horário: 19:30

Resumo: nonhomogeneous.pdf

**Soluciones multiples para problemas no lineales con condiciones dirichlet no homogeneas**

- Marco Calahorrano, EPN Quito, Equador

Slides

Data: 31 de julho, quarta-feira

Horário: 16:30

Resumo: marco.pdf

**Kinetic Models for Chemotaxis**

- Fabio A. C. C. Chalub – University of Vienna, Áustria

Slides

Data: 31 de julho, quarta-feira

Horário: 17:00

Resumo: Kinetic models for chemotaxis, nonlinearly coupled to a Poisson equation for the chemo-attractant density, are considered. Under suitable assumptions on the turning kernel (including models introduced by Othmer, Dunbar and Alt), convergence in the macroscopic limit to a drift-diffusion model is proven. The drift-diffusion models derived in this way include the classical Keller-Segel model. Furthermore, sufficient conditions for kinetic models are given such that finite-time-blow-up does not occur. Examples are given satisfying these conditions, whereas the macroscopic limit problem is known to exhibit finite-time-blow-up. The main analytical tools are entropy techniques for the macroscopic limit as well as results from potential theory for the control of the chemo-attractant density. Joint work with Peter Markowich (Vienna), Christian Schmeiser (Vienna) and Benoit Perthame (Paris).

**Riemann-Hilbert problems and algebraic curves**

- Tamara Grava, SISSA Trieste, Itália

Slides

Data: 31 de julho, quarta-feira

Horário: 17:45

Resumo: tamara.pdf

**On Separation of Variables for Hamilton Jacobi Equations**

- Gregorio Falqui, SISSA Trieste, Itália

Slides

Data: 31 de julho, quarta-feira

Horário: 18:15

Resumo: In this talk we will briefly report on a recently introduced method for solving H-J equations by the method of SoV. This method is based on the notionof pencil of Poisson brackets, alias on the so-called bi-Hamiltonian approach to integrable systems. By using the classical Neumann system as a paradigmatic example, we will argue how the classical Levi-Civita separation conditions can be characterized within such a geometrical set-up in a coordinate-free manner. We will show that, under mild conditions, the geometrical datum of the Poisson pencil defines a special class of separation coordinates, and discuss the “tensorial” conditions for establishing the separability of the system.

**Thermal avalanche for blow-up solutions of semilinear heat equations**

- Fernando Quiros, UAM, Spain

- Slides

Data: 31 de julho, quarta-feira

Horário: 19:00

Resumo: thermal_avalanche.pdf

**Renormalized enstrophy defect in 2D turbulence**

- Milton Lopes, UNICAMP, Brasil

Data: 31 de julho, quarta-feira

Horário: 19:30

Resumo: The standard description of 2D turbulence requires an ongoing enstrophy cascade, much in the same way that the Kolmogorov model of 3D turbulence requires an energy cascade. In both cases the rate of loss of the relevant quantity in time, which must stay bounded away from zero and finite as viscosity vanishes, plays a central role in the theory. A subtle paradox arises in the 2D case because weak solutions of the incompressible 2D Euler equations with finite initial enstrophy actually conserve it exactly, so that there is no positive rate of loss of enstrophy in the inviscid limit. Recently, G. Eyink proposed a clever solution for this paradox: turbulent 2D flows could have infinite enstrophy, and the rate of enstrophy loss required for sustained turbulence would take place in a renormalized sense. In fact, Eyink went beyond that, proposing a specific model for renormalized enstrophy defects in the context of weak solutions of the 2D Euler equations and formulating a conjecture regarding the existence and the nature of these defects. In this talk we will present a simplification of Eyink’s model, introduce an example that solves Eyink’s conjecture as it was originally formulated and recast the conjecture in light of this example.