Michele Bartuccelli: Research Interests

Dissipative Partial Differential Equations

My interests involve analysis of nonlinear dissipative partial differential equations and their applications to real world phenomena. Examples include the Complex Ginzburg-Landau Equation and the Navier-Stokes Equations. Particular attention is given to regularity, global existence, stability, instability and turbulence. Length scales and patterns involved in the dynamical flows of dissipative partial differential equations are also of fundamental importance in my research. Length scales are arguably one of the most important dynamical concepts for properly understanding the spatio-temporal patterns of dissipative flows, and their estimates are crucial for having an accurate numerical representation of the solutions. The mathematical methods and techniques involve utilising an area of applied mathematics called interpolation inequalities (like for example the famous Gagliardo-Ninerberg inequality), which can give estimates on appropriate norms of the various terms in the governing equations.

Mathematical Biology

Interests centre on partial differential equations which models the interaction of species, such as the so-called reaction-diffusion equations. For such equations, results concerning global stability of steady state solutions and bifurcations from a uniform solution have been proved. Current interests include stabiltiy of travelling fronts and spatio-temporal chaos in solutions of a large class of dissipative partial differential equations.

Higher Order Dissipative Partial Differential Equations And Positivity Preservation of Their Solutions

There are a number of important semilinear parabolic partial differential equations, such as the Cahn-Hilliard, the Swift-Hoenberg and the Kuramoto-Sivashinsky equations, which contain a fourth order spatial derivative. Such equations do not, in general, exhibit positivity preservation but may do so under some further restriction on the initial data and parameters; recent results in this area include the study of positivity and convergence of solutions of such equations, by employing generalized energy methods and sharp interpolation inequalities.

Analysis of Solutions of Nonlinear Oscillators

Autonomous Hamiltonian systems with one degree of freedom are completely understood, mainly because their solution curves in phase space are Hamiltonian contours. When the Hamiltonian depends explicitly on time, the dynamics becomes far more complicated, and a general analysis of the motion does not exist. In fact even for the simplest nonlinear, time-dependent equations of classical dynamics a satisfactory analysis of the solutions is still largely lacking. My interests focus on the dynamics of time-dependent nonlinear oscillators in the presence of dissipation; in fact it is well known that the addition of a dissipative term to the equation of motion can drastically alter the structure of the phase-space of the system.

The problems currently addressed include:

Persistence of orbits

One outstanding question in the theory of dynamical systems is the following: does a periodic solution or quasi-periodic solution of a dissipative system persist in the limit as dissipation goes to zero? Answering this question would shed light on, for example, one of the many mysteries in celestial mechanics, namely why the planets are in the orbits they are in, as opposed to other ones. It is possible that in the process of the creation of solar systems similar to our own, some special orbits self-select because they are in a sense more attracting than others. Notice that generally the answer to the above question is negative: the difference between the presence and the absence of dissipation in a system usually gives qualitatively different dynamics.

Basins of attraction

Another fundamental problem of dissipative systems is that of understanding all their attractors and associated basins of attraction. In a homogeneous linear system with damping, the attracting periodic orbit is independent of the initial conditions. By contrast, the existence of two or more attracting sets for the same parameter values in a nonlinear system indicates that the initial conditions play a critical role in determining the system behaviour. These attracting sets determine the essential dynamical behaviour of the system, and their global stability is determined by constructing their basins of attraction. The problem of classification of attracting sets in dissipative systems for arbitrary damping is still largely open.

Classification of orbits

This is an important problem in many fields of applied mathematics, especially in celestial mechanics where it originated, and it relates to the problem of gaining an understanding of the (generally) complicated orbit structure of the flow of a Hamiltonian system. As is well known, much progress has been made in applying KAM theory, according to which a large part of the phase space of a system close to an integrable one consists of quasi-periodic motion. This result however involves small perturbations of an integrable system; results of a more global nature, for example finding quasi-periodic solutions with prescribed energy are still lacking. Note that in the autonomous case there are many results in the literature, whereas in the time-dependent case the problem is still essentially open.

Other Topics of Interest

Applied Functional Analysis
Blow-up of Solutions of Partial Differential Equations
Chaos and Turbulence
Dissipative Partial Differential Equations
Dynamics of Time-Dependent Nonlinear Oscillators
Ordinary Differential Equations
Hamiltonian Dynamics
Patterns Formation
Mathematical Modelling (Population Dynamics, Ecosystems......)

Last updated 30th October 2007