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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.

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.

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.

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)

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.