AOS · Asymptotics of Operator Semigroups

Asymptotics of Operator Semigroups

The theory of asymptotic behaviour of solutions of abstract evolution equations is a comparatively new field serving as a common denominator for many other areas of mathematics, such as for instance the theory of partial differential equations, complex analysis, harmonic analysis and topology. The primary interest in this theory comes from the fact that evolution equations on Banach spaces are often abstract reformulations of various phenomena arising in natural sciences, engineering and economics. Knowledge of the asymptotics of associated solution families allows one to understand the character of the long-time evolution of these phenomena. Despite the obvious importance, the theory of asymptotics of strongly continuous semigroups and other solution families was for a very long time a collection of scattered facts rather than an organized area of research. The interest increased in the 1980s and the theory has witnessed a dramatic development over the past thirty years, at least as far as qualitative statements are concerned. The foundational Gearhart-Pruss and Arendt-Batty-Lyubich-Vu (ABLV) theorems provided efficient and elegant criteria for convergence of semigroup orbits to zero in the uniform and strong operator topologies respectively. Through various developments of these results the asymptotic theory has developed a structure and become a mature field with various applications. The input of abstract methods, interconnections and synergies between several branches of mathematics, have proven to be essential for the development of theory. We mention the inputs of complex analysis in tauberian theorems, edge-of-the-wedge theorems and weak maximum principles, of functional analysis and operator theory, and of the theory of Banach algebras and harmonic analysis, e.g. the notions of spectral synthesis and Fourier multipliers. At the same time, the theory has found numerous applications in PDEs; the abstract results from played an important role for example in the study of models from fluid dynamics or various phenomena modeled by damped wave equations and other dispersive systems. The project aimed at achieving a substantial progress in the field of asymptotics of operator semigroups and related areas by means of joint efforts of Institute of Mathematics of the Polish Academy of Sciences, seven European universities including Nicholas Copernicus University of Torun, University of Ulm, Karlsruhe Institute of Technology, Dresden University of Technology, Lille University, University of Marseille, and Tel-Aviv University and their six non-EU partners: University of Missouri, University of Sydney, University of Auckland, University of North Carolina, Northwestern University and College of William and Mary, within the framework of the IRSES staff exchange programme, Marie Curie action. During realization of the programme a number of new international collaborations and intensive cross-countries transfer of knowledge has been realized. Moreover, the realization of the project has led to several essential scientific achievements, have been already published in the first rate mathematical journals (Journal of the European Math. Society, Advances in Mathematics, Journal of Functional Analysis). Several notable problems were solved (e.g. Erdos-de Brujn-Kingman problem by Gomilko and Tomilov) and several small theories have been developed (e.g. of fine scales of decay rates by Batty, Chill and Tomilov, of subordination and its generalizations by Batty, Gomilko and tomilov) More than forty papers have been written within the project framework, and the results have been presented on more than fifty international conferences and workshops. Six workshops and schools were directly related to the realization of the project tasks, and were organized by the project participants. A more detailed information is contained in the corresponding section of the present final report. Overall, the project has been accomplished successfully, and its lasting impact will be seen in continuing collaboration between former partners and project related research outputs appearing instantly. That was also confirmed by evaluators from outside the EU. As the Royal Society of New Zealand has remarked: "[We are] very happy to approve this final report of Professor Tom ter Elst's IRSES project grant. It sounds like the project has been very successful with a large number of both incoming and outgoing research exchanges completed, continued collaborative opportunities and a number of joint research publications. Please pass in my congratulations to Professor Tom ter Elst and his colleagues on this achievement".

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