Title: Rapid boundary stabilization for longitudinal vibrations of a bar

Abstract: Consider the longitudinal vibrations of the cross-sections of a bar fixed at one end, with a mass attached to the other end. This lecture investigates the exponential decay of the solution to the physical problem, with an arbitrary decay rate. To obtain the decay, we use Komornik's work.

Title: Carleman estimates for parabolic equations with super strong degeneracy in a set of positive measure

Abstract: This work is concerned with the obtainment of new Carleman estimates for linear parabolic equations, where the second-order differential operator brings a super strong degeneracy in a positive measure subset of the spatial domain. In order to prove our main result, the control domain must  contain the set of degeneracies.  As a well-known consequence, we achieve a null controllability result in the current context.

Title: On the role of the viscosity parameters in the large time asymptotics of 2D micropolar flows

Abstract: We investigate the role of the four viscosity parameters, in fluids where the particles possess a microstructure (micropolar flows) and are allowed to rotate in a two-dimensional setting. We first establish the existence of global finite energy solutions, satisfying the classical energy equality, for arbitrary initial data in $L^2$, in the case of a spin viscosity $\gamma\ge0$, and we construct the asymptotic profiles of the solution as $t \to +\infty$. We deduce the remarkable fact that the large time behavior only depends on the kinematic viscosity $\mu$, and not on the other parameters $\chi$ (vortex-viscosity), $\gamma$ (spin viscosity) and $\kappa$ (gyroviscosity) of the model. Our primary tool is a new enstrophy-like identity of independent interest, involving the difference between the fluid vorticity and the micro-angular velocity. Another consequence of our analysis is the identification of scenarios where the presence of micro-rotational effects significantly enhances dissipation, thereby slowing down the fluid motion at large times. This is a joint work with L. Brandolese (Institut Camille Jordan, Université Lyon 1), A. V. Busuioc (Université Jean Monnet) and D. Iftimie (Institut Camille Jordan, Université Lyon 1).

Title: Soon

Abstract: Soon.

Title: Pullback attractors for a semilinear parabolic equation with homogeneous Neumann  boundary conditions and time-varying domains

Abstract: In this work, we consider a non-autonomous semilinear parabolic partial differential equation with homogeneous Neumann boundary conditions and time-varying domains. Using a coordinate transformations technique, we will show that the non-autonomous problem in a time-varying domain is equivalent to a non-autonomous problem in a fixed domain, and we will show the existence of pullback attractors.

Title: Soon

Abstract: Soon.

Title: Soon

Abstract: Soon.

Title: Soon

Abstract: Soon.