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.
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
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
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
Mathematical Modelling (Population Dynamics, Ecosystems......)
Last updated 30th October 2007