Research topics
Mathematical problems in Fluid Mechanics
- Non-Newtonian Fluid flows.
Until 2012 my research work was focused on the mathematical study of non-Newtonian fluid flows of viscoelastic type. I was interested in the problems of mathematical analysis raised by different rheological models. In fact, the systems of partial differential equations modelling the flows of such fluids have both rich and complex structures from a mathematical point of view. I have published with different collaborators about fifteen papers. This field continues to occupy an important place in research, both at the theoretical and applied level, the best evidence of the interest that researchers and industry have for this field is the very important number of high level scientific publications that continue to appear in international journals.
- Asymptotic models in water waves.
I have undertaken a research program on the derivation and full justification of asymptotic models of internal waves in shallow waters, typically in the coastal vicinity. Such models are very important to describe the impact on marine life of internal waves and especially the impact on the problem of pollutant movement in seas and oceans. We have essentially proposed two classes of new models, the first is valid in the so-called Camassa-Holm-Serre regime, while the second family is of Green-Naghdi type.
The second theme deals with Boussinesq type systems and KdV type equations with variable coefficients. This research axis concerns the global existence in time of the solutions as well as the stability of the Boussinesq system with respect to the systems approaching it. We are also interested in the study of a generic class of KdV equations with variable coefficients, in particular the search for solutions with lower regularity, the goal being to establish a global existence result in time for the KdV equation with variable background.
Mathematical problems in biology
- Chemotaxis: Keller-Segel models.
The study of some models involved in chemotaxis, citing for example the famous Keller-Segel model, gives rise to degenerate or non-degenerate parabolic-parabolic systems. In the Keller-Segel model modelling the interaction between cell density and concentration of a chemoattractant, cell diffusion can be modelled by a degenerate operator.
- Electromechanical cardiac modelling of the heart.
We have been interested in the modelling, mathematical and numerical analysis of the electromechanical coupling of the heart. We also derived by homogenisation new macroscopic electrical-physiological models of the electrical activity of the cardiac tissue and studied the well-posedness of these systems. This collaboration has resulted in two theses in cotutelle between the Ecole Centrale de Nantes and the Lebanese University and several articles published in leading specialized journals.