This special issue is devoted to some new trends in Partial Differential Equations and applications, on the occasion of the 60TH birthday of Professor Peter Alexander Markowich. A special focus has been given to fractional PDEs. Fractional calculus is a tremendously growing area because of it is a widespread applications in many domains ranging from Medicine to Engineering. Fractional partial differential equations have become important in modeling of modern technology, e.g. for semiconductor and graphene crystals, which are fundamentally important for photovoltaic and nano-technological applications. They model the flow of glaciers and snow/mud avalanches, the formation of sand dunes as well as the quantum mechanical dynamics of ultra-cold gases, and they are used to describe processes in material science and in the collective motion of biological cells and biological species. Strikingly analogous fractional fluid-type PDE can be used for the modelling of the motion of human crowds, in image processing, in visualization and oil-reservoir modeling, and in water supplies.
Professor Markowich has started working in this area 6 years ago and he has made valuable contributions to this field during this short period. His main contributions address fractional critical problems, Fractional PDEs arising in Quantum Mechanics and Photonics.
7 contributed papers contain most recent contributions in this research field. 4 deal with other aspects of Professor Markowich’s research in modern analysis, this ranges from diffusive wave theory to Nonlinear Schrodinger Equations.
I sincerely thank the authors of this special issue for their contributions.