Stabilization arising from PGEM : a review and further developments

The aim of this paper is twofold. First, we review the recent Petrov-Galerkin enriched method (PGEM) to stabilize numerical solutions of BVP's in primal and mixed forms. Then, we extend such enrichment technique to a mixed singularly perturbed problem, namely, the generalized Stokes problem, and focus on a stabilized finite element method arising in a natural way after performing static condensation. The resulting stabilized method is shown to lead to optimal convergences, and afterward, it is numerically validated

On a non-homogeneous and non-linear heat equation

We consider the Cauchy-problem for a parabolic equation of the following type: \begin{equation*} \frac{\partial u}{\partial t}= \Delta u+ f(u,|x|), \end{equation*} where $f=f(u,|x|)$ is supercritical. We supply this equation by the initial condition $u(x,0)=\phi$, and we allow $\phi$ to be either bounded or unbounded in the origin but smaller than stationary singular solutions. We discuss local existence and long time behaviour for the solutions $u(t,x;\phi)$ for a wide class of non-homogeneous non-linearities $f$. We show that in the supercritical case, Ground States with slow decay lie on the threshold between blowing up initial data and the basin of attraction of the null solution. Our results extend previous ones allowing Matukuma-type potential and more generic dependence on $u$. Then, we further explore such a threshold in the subcritical case too. We find two families of initial data $\zeta(x)$ and $\psi(x)$ which are respectively above and below the threshold, and have arbitrarily small distance in $L^{\infty}$ norm, whose existence is new even for $f(u,r)=u^{q-1}$. Quite surprisingly both $\zeta(x)$ and $\psi(x)$ have fast decay (i.e. $\sim |x|^{2-n}$), while the expected critical asymptotic behavior is slow decay (i.e. $\sim |x|^{2/q-2}$).Comment: 2 figure

Exponential decay to equilibrium for a fibre lay-down process on a moving conveyor belt

We show existence and uniqueness of a stationary state for a kinetic Fokker-Planck equation modelling the fibre lay-down process in the production of non-woven textiles. Following a micro-macro decomposition, we use hypocoercivity techniques to show exponential convergence to equilibrium with an explicit rate assuming the conveyor belt moves slow enough. This work is an extension of (Dolbeault et al., 2013), where the authors consider the case of a stationary conveyor belt. Adding the movement of the belt, the global Gibbs state is not known explicitly. We thus derive a more general hypocoercivity estimate from which existence, uniqueness and exponential convergence can be derived. To treat the same class of potentials as in (Dolbeault et al., 2013), we make use of an additional weight function following the Lyapunov functional approach in (Kolb et al., 2013)

Two-dimensional Bose and Fermi gases beyond weak coupling

Using a formalism based on the two-body S-matrix we study two-dimensional Bose and Fermi gases with both attractive and repulsive interactions. Approximate analytic expressions, valid at weak coupling and beyond, are developed and applied to the Berezinskii-Kosterlitz-Thouless (BKT) transition. We successfully recover the correct logarithmic functional form of the critical chemical potential and density for the Bose gas. For fermions, the BKT critical temperature is calculated in BCS and BEC regimes through consideration of Tan's contact.Comment: 20 pages, 8 figures. v2: pubished versio