Summary

My research fits in the field of applied mathematics, with focus on the analysis of partial differential equations (PDEs) and their role in the study of problems arising in fluid mechanics and mathematical physics. I am particularly interested in fluid-solid interaction problems. Interactions of fluids and solids (either rigid or deformable) arise in many natural and physical phenomena. Many studies can be found in the literature concerning dynamics of flight and space technology, and for many geophysical and biological applications. From a mathematical point of view, the equations governing fluid-solid interactions possess all the mathematical features of the nonlinear PDEs describing the motion of viscous fluids (the Navier-Stokes equations). Though the aim is to provide a rigorous description of the behavior of physical systems, the fundamental mathematical questions behind these problems are never overlooked. The lack of a proof for the uniqueness of solutions satisfying the energy balance together with a "chaotic'' behavior of weak solutions for finite time intervals play a fundamental role in our problems. In addition, the coupling of the Navier- Stokes equations with the equations describing the motion of solids features a combination of a dissipative component originating from the fluid, and a conservative or even excited component due the solid counterpart. This dissipative-conservative interplay arises in many other problems characterized by the coupling of parabolic and hyperbolic PDEs. Our investigations aim to answer fundamental questions concerning the existence and regularity properties of solutions to the equations governing the motion of fluid-solid systems, and to provide a detailed description of the stability properties and long-time behaviour of the motions.

I have also worked on some problems in approximation theory related to the behavior of interpolating operators on discontinuous functions and the approximation of cosine functions by iterates of bounded linear operators.

Undergraduate Summer Research Program

• 2020: Linearization principles for systems of ordinary differential equations
  This project focus on the study of the asymptotic stability of solutions to nonlinear systems of ordinary differential equations (ODEs). It is well-known that for systems of ODEs which are "locally linear", information about the stability can be obtained by studying the spectral properties of a (suitably constructed) linear system of ODEs. Whenever this linear approximation is possible, we obtain what is generally known as "linearization principle". General stability principles are proved for systems which are not "locally linear" in the classical sense. For such system, for example, the coefficients matrix is not invertible and equilibria are not isolated, instead they form a (nontrivial) nth-dimensional manifold. The results are applied to the Landau-Lifshitz-Gilbert equation for a single spin.
• 2021: Long-time dynamics of rigid bodies with a damper
  Consider a rigid body with a damper, this is a system constituted by two rigid bodies, one inside the other, and separated by a narrow lubricating layer. The damping mechanism is provided by the coupling of the inner solid and lubricating liquid. A mathematical model describing such mechanical system is given by a system of nonlinear differential equations with an energy functional (Lyapunov function) which is non-increasing along trajectories. It is well known that a permanent rotation (i.e., a rotation with constant angular velocity) of a rigid body with a damper is stable if and only if the axis of rotation is the one corresponding to the largest principal moment of inertia. The objective of this project is to obtain information about the long-time behaviour of generic trajectories (i.e., time dependent motions). We are interested in two types of trajectories: (i) those starting “sufficiently close” to an equilibrium; and (ii) those corresponding to “arbitrary” initial data. The questions that we want to answer are: (1) under which conditions (if any) equilibria are attained by such trajectories; and (2) with what rate (if any) the decay to an equilibrium happens. These problems have many interesting applications in structural mechanics and space dynamics. As a matter of fact, similar types of systems have been widely used to model the passive damping of wobbling structures like buildings, bridges and artificial satellites.