PROJECTS

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