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.
Mher Marc Karakouzian (defended in October 2022): "On the Planar Motion of Rigid Bodies with a Fluid-Filled Gap".
This project focussed on the proof of existence, uniqueness and regularity properties of solutions to the equations governing the planar motion of two rigid bodies, one contained inside the other and separated by a viscous incompressible fluid.
Undergraduate Summer Research Program
2023: On the existence of periodic solutions to differential equations
The aim of this project is to investigate the existence of periodic solutions to (systems of) nonlinear differential equations. It is well-known that there is no "universal" method to solve all possible differential equations. As a matter of fact, there is a relatively small number of differential equations for which a solution method has been developed. Nevertheless, differential equations are ubiquitous as they are mathematical models governing numerous physical and natural systems, and they are used for many engineering applications. Thus, whether it is for mathematical modelling or just numerical validation, it becomes necessary to obtain qualitative information about the solutions to a differential equation without being able to solve it explicitly.
While the existence and uniqueness of solutions to initial value problems is well-understood, less work has been done toward a comprehensive theory for the corresponding problem of finding nontrivial (i.e., non-constant) periodic solutions to nonlinear differential equations. The known results either assume conditions which are difficult to be checked (like finding uniform bounds on the solutions) or they are application specific (that is, they depend on the specific type of differential equation under study). In contrast, non-existence of periodic solutions may have intriguing consequences in the theory as well as in the applications. Take the "phenomenon of resonance", for example. Consider the second-order linear differential equation modelling the undamped motion of a mass-spring system. It is well-known that if the system is subject to an external force which is periodic with the same frequency as the "natural frequency" of the system (i.e., the frequency of oscillations of the undisturbed motion), then the generic motion of the mass is characterized by oscillations with unbounded amplitude. In this case, it is said that the system is working at resonance, and then there is no periodic motion associated to that resonant force. Resonance is at the basis of many energy harvesting techniques utilized nowadays. From the mathematical viewpoint, this phenomenon could be easily described if one was able to write an explicit solution of the corresponding differential equation (like in the case of the mass-spring system, with the method of variation of parameters applied to the corresponding linear differential equation). However, many physical and natural phenomena are described by nonlinear differential equations for which an explicit solution is not available, examples include the occurrence of flutter in aeroelasticity, planetary motions, the modelling of the heartbeat or that of suspensions in wheeled vehicles as well as problems involving a delay. Aim of this project is to find the most general set of conditions (possibly easy to be checked) to determine whether a given differential equation admits a periodic solution. Applications of these results to real-world problems will be given.
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.