"Wigner measures for singular and nonlinear problems: the scalar case"
Professor Agis Athanasoulis, Department of Mathematics
University of Leicester, UK
Tuesday, September 16, 2014, 12:00pm | Room 5-314
Wigner measures (WMs) have been successfully used as a parameter-free tool to provide homogenized descriptions of wave problems. Notable applications are the efficient simulation of large linear wave fields, and the painless resolution of linear caustics. However, their applicability to non-linear problems has been very limited. In this talk we discuss the role of smoothness of the underlying flow as a limiting factor in the applicability of WMs. Non-smooth flows are ill-posed for measures, and new phenomena are possible in that regime. For example, single wavepackets may be "split" cleanly into several new wavepackets. We introduce a modification of the WM approach, and show that it can capture successfully some of these new phenomena. The motivation behind this work is to develop methods applicable to non-linear problems as well. Some first such applications are also explored.
These results include joint work with I. Kyza and Th. Katsaounis.
Agis Athanasoulis graduated from the Math Department of the University of Athens in 2003, and went on to a PhD in Applied and Computational Mathematics in Princeton University, which he obtained in 2007. Subsequently, he worked in France for several years. He was a postdoc in Ecole Normale Superieure and in Ecole polytechnique, working on microlocal and semiclassical analysis. In 2011 he went on to a postdoc in the University of Cambridge, working on nonlinear Schrodinger equations. Currently he is a lecturer at the University of Leicester.