Global Research Partnership Investigators
2007 Winner Profiles
Dr. Peter A. Markowich
University of Cambridge
Cambridge, England
View Dr. Peter A. Markowich's presentation at the GRP Symposium
Dr. Peter A. Markowich, KAUST Investigator, is a professor of Applied Mathematics at the Centre for Mathematical Sciences of the University of Cambridge in the United Kingdom. Highly distinguished in his field, Dr. Markowich’s work directly aligns with the interests of KAUST’s Institute for Applied Mathematics and Computational Science.
KAUST Investigator Award: Applied and Computational Differential Equations in Life Sciences, Nanoscience and Engineering
Partial differential equations (PDE), based on the differential-integral calculus invented by Leibniz and Newton at the turn of the 17th to the 18th century, relate state variables (such as mass, momentum energy, etc.) to their variations with respect to so-called independent variables (time and/or position). PDEs were used in many important scientific discoveries of the last three centuries (ranging from classical to quantum mechanics and to general relativity). They provide an excellent description for many of the physical, biological, and engineering processes within the KAUST scope of interests.
This project focuses on the modeling, analysis, and numerical simulation of nonlinear partial differential equations appearing in various processes in the life sciences, nanoscience, and engineering. Problems to be treated in the framework of this project include partial differential equations describing particle transport theories in classical and quantum mechanics, such as charge carrier transport in semiconductor crystal lattices, nanostructure modeling (e.g., superlattices, quantum dots, quantum cascade lasers) bosonic and fermionic Bose-Einstein condensates (with possible applications to quantum computing), and in biological processes such as in the modeling of the collective motion of biological cells, driven by chemotactic processes or by self-propelled squirming. These physical and biological processes are, of course, very different in their scientific origins, but all of them can be modeled effectively by nonlinear partial differential equations, which often feature striking structural mathematical similarities. Other examples of particle transport processes admitting similar PDE models are granular material flows and certain socioeconomic processes (like spread of opinions in human societies and Pareto economics). The equations under consideration range from free boundary problems arising as limits of Ginzburg-Landau-type equations (modeling superconducting materials), Boltzmann-type kinetic phase space equations (based on classical or quantum mechanics) featuring long- and short-range binary interactions, to nonlinear Schrödinger equations (e.g., the Gross-Pitaevskii equation describing ultra-cold quantum gases like Bose-Einstein condensates), to macroscopic fluid equations like (quantum) hydrodynamics and drift-diffusion systems. We shall investigate basic mathematical properties like regularity (of solutions and of geometric objects such as free boundaries), existence and uniqueness of solutions, analyze singular scaling limits by asymptotic methods (e.g., by high frequency asymptotics), prove large-time asymptotic results using entropy dissipation and optimal mass transportation techniques (Wasserstein metrics). Also we shall derive, analyze, and adapt numerical discretizations and perform computational experiments, which in many cases will be tested against real physical experiments.
We foresee direct collaborations with KAUST research centers in the areas of material sciences and engineering and applied mathematics and computational sciences. The collaboration will proceed by exchanging short research visits and organizing workshops and conferences (in the UK and KSA), which will expose and embed KAUST researchers to the international community of PDE analysts and modelers, and by engaging KAUST students and faculty in scientific activities at the Centre for Mathematical Sciences of the University of Cambridge.
About the University of Cambridge
As the University of Cambridge approaches its eight hundredth anniversary in 2009, it is looking to the future. Its mission is to contribute to society through the pursuit of education, learning, and research at the highest international levels of excellence. It admits the very best and brightest students, regardless of background, and offers one of the UK’s most generous scholarship programs.
The University of Cambridge’s reputation for excellence is known internationally and reflects the scholastic achievements of its academics and students, as well as the world-class original research carried out by its staff. Some of the most significant scientific breakthroughs occurred at the university, including the splitting of the atom, invention of the jet engine, and the discoveries of stem cells, plate tectonics, pulsars, and the structure of DNA. From Isaac Newton to Stephen Hawking, the university has nurtured some of history’s greatest minds and has produced more Nobel Prize winners than any other UK institution, with over 80 laureates.