dia das equações
O Centro de Matemática da Universidade do Minho organiza desde 2008 o Dia das Equações.
Este encontro reúne durante um dia especialistas na área das equações diferenciais.
A edição de 2012 realiza-se a 14 de setembro e reunirá investigadores na área das
equações com derivadas parciais e das suas aplicações.
A organização está a cargo de
Assis Azevedo
e
Fernando Miranda.
Oradores convidados
Edições anteriores
Parabolic variational and quasivariational inequalities: regularity and approximation methods
We consider parabolic variational and quasivariational inequalities of the form:
Find $u\in K(u)$ such that
\begin{equation*}
(Lu+\mathcal{A}(u)-f, v-u)\geq 0, \text{ for all }v\in K(u),
\end{equation*}
where $K(v)$ is a subset of a reflexive Banach space $V$, for each $v\in V$ and $-L$ is the infinitesimal generator of a semigroup (of contractions) over $V$. We study conditions on the forcing term $f$, $-L$ and $K(\cdot)$ for which $u$ has higher regularity in time, i.e., such that $u$ belong to the domain of $(-L)^n$ with $n\geq 2$. We also describe approximation methods that lead to constructive proofs of existence and determine schemes for numerical implementation.
Difusão não Fickiana versus difusão Fickiana
Os processos de difusão em certos meios, como por exemplo polímeros,
tecidos vivos, meios porosos heterogéneos, são descritos por modelos
matemáticos estabelecidos a partir da lei de Fick para o fluxo de massa
que são caracterizados por equações parabólicas.
A existência de desvios entre resultados experimentais obtidos em
laboratório e resultados de simulação, levou à necessidade de questionar
a lei de Fick.
Nesta palestra apresentamos alguns modelos estabelecidos a partir de
correcções da lei de Fick e comparamos, quando possível, tais modelos
com os modelos de difusão clássicos.
Stabilized finite element methods for the obstacle problem
We analyse a stabilized FEM scheme for the classical obstacle problem. Based on an
equivalent mixed saddle-point problem resulting from the Lagrange formalism, we
introduce a consistent and stabilized FEM formulation. We derive optimal a priori
error estimates and discuss a posteriori estimates.
This is a joint work with Rolf Stenberg from the Department of Mathematics and
System Analysis at the School of Science of the Aalto University in Finland.
Approximate solutions of some kind of partial differential equation by means of reproducing kernels
We propose new constructions of approximate solutions of arbitrary linear arbitrary of some kind of partial differential equations. This is mainly based on a reproducing kernel Hilbert spaces approach and the realization of reproducing kernel Hilbert spaces by a finite number of points.
(joint work with L. Castro and S. Saitoh)
Linearized stability around a shock for a Schrödinger-Burgers system
We investigate the Schrödinger-Burgers system, which models the interaction of short and long waves,
\[
\begin{cases}
i u_t + u_{xx} = vu - \varepsilon |u|^2 u,\\
v_t + (v^2)_x = \varepsilon(|u^2|)_x,
\end{cases}
\qquad \varepsilon > 0.
\]
Well-posedness for the associated Cauchy problem remains a difficult open problem. We approach this system by a linearized stability approach. Specifically, we prove that the Schrödinger-Burgers system is linearly stable around a particular solution containing a stationary shock.
Our proofs rely on energy techniques and solutions of transport equations with discontinuous coefficients.
This is joint work with João Paulo Dias, Mário Figueira and Philippe LeFloch.
If time permits, I will present some unrelated results on a nonlocal hyperbolic conservation law arising from a gradient constraint problem.
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