Eventually positive operator semigroups and their application to evolution equations

• Applicant: Professor Dr. Jochen Glueck
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 515394002

The time evolution of many dynamical systems has a property which is referred to as "positivity" or "preservation of positivity" and which means the following: if the initial value of the system (which can, for instance be a vector with finitely many components, or a function defined on some set) is non-negative in each component or at each point, then this property is also true for the trajectory of the system at each subsequent time. This behaviour occurs naturally in many physical, chemical or probabilistic models (as, for instance, mass distributions or probability distributions are inherently non-negative). In case that such a dynamical system is also linear and autonomous, it can, from a mathematical point of view, be described by a so-called "positive operator semigroup". As of today, a deep and thorough theory of such positive semigroups is available. More recently, the more involved phenomenon of "eventual positivity" has been observed in a number of partial differential equations. This phenomenon means that, for non-negative initial value, the trajectory of the system can change sign for small times, but it again becomes and stays non-negative for all sufficiently large times. Following earlier results in finite dimensions, a general infinite-dimensional theory of eventual positivity was initiated in two articles in 2016 and was from then on developed and applied by various authors. Thus, a large variety of differential equations with eventually positive solutions is nowadays known. Still, the state of the art of the theory is far from comprehensive. It leaves many questions open - some of which have been answered long ago in the case of positive semigroups -, and for many concrete differential equations it is still not possible by means of the currently available theory to determine whether their solutions are eventually positive. This project aims at developing new methods in order to better understand eventual positivity at a theoretical level and to identify eventual positivity in further concrete differential equations. We will employ tools from various fields in functional analysis, in particular strongly continuous operator semigroups and Banach lattices, to prove new characterizations and new sufficient conditions for eventual positivity, and we will apply these results to a variety of differential equations to determine whether their solutions are eventually positive.

• DFG Programme: Research Grants