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 equations.

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.

The problems currently addressed include:

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......)

- Research Interests
- Publications
- Nonlinear PDEs
- Teaching
- Curriculum Vitae (Short Version)
- My Home Page

m.bartuccelli@surrey.ac.uk

Last updated 30th October 2007