Evolutionary Equations: Picard's Theorem for Partial Differential Equations, and Applications(287 Operator Theory: Advances and Applications)

This open access book provides a solution theory for time-dependent partial differential equations, which classically have not been accessible by a unified method. Instead of using sophisticated techniques and methods, the approach is elementary in the sense that only Hilbert space methods and some basic theory of complex analysis are required. Nevertheless, key properties of solutions can be recovered in an elegant manner. Moreover, the strength of this method is demonstrated by a large variety of examples, showing the applicability of the approach of evolutionary equations in various fields. Additionally, a quantitative theory for evolutionary equations is developed. The text is self-contained, providing an excellent source for a first study on evolutionary equations and a decent guide to the available literature on this subject, thus bridging the gap to state-of-the-art mathematical research.

Table of Contents:
- Introduction.- Unbounded Operators.- The Time Derivative.- Ordinary Differential Equations.- The Fourier-Laplace Transformation and Material Law Operators.- Solution Theory for Evolutionary Equations.- Examples of Evolutionary Equations.- Causality and a Theorem of Paley and Wiener.- Initial Value Problems and Extrapolation Spaces.- Differential Algebraic Equations.- Exponential Stability of Evolutionary Equations.- Boundary Value Problems and Boundary Value Spaces.- Continuous Dependence on the Coefficients I.- Continuous Dependence on the Coefficients II.

About the Author :
Christian Seifert studied Mathematics with minor in Physics at Technische Universität Dresden and received his Diploma in Mathematics in 2009. He then moved to Technische Universität Chemnitz as research associate and obtained his doctoral degree in Mathematics in 2012. Since 2012 he is senior scientist at Technische Universität Hamburg, where he gained his Habilitation in Mathematics in 2021. Christian Seifert has been deputy professor at various Universities: from 2015 and 2016 at Ludwig-Maximilians-Universität München, at Technische Universität Clausthal from 2019 to 2020, and at Christian-Albrechts-Universität zu Kiel in 2021. Christian Seifert contributed to more than 30 peer-reviewed journal publications mainly on evolution equations and operator theory. Sascha Trostorff studied mathematics with minor in computer sciences at the TU Dresden, Germany. He graduated in 2008. In 2011 he received his PhD providing a general solution concept for evolutionary inclusions, that is, a certain generalisation of nonlinear partial differential equations. In 2018 he completed his habilitation thesis also at the TU Dresden connecting semi-groups methods and evolutionary equations as well as developing the understanding of exponential stability for a large class of evolutionary equations. Since 2019 he works as a lecturer at the Christian-Albrechts-Universität in Kiel, Germany. Sascha Trostorff is author of approximately 40 peer-reviewed research papers and co-authored one book. His main research area is functional analysis and its applications to partial differential equations. Marcus Waurick graduated in Mathematics with minor in Physics at TU Dresden in 2009. During his first employment at the Faculty of Civil Engineering he finished his PhD in 2011 on homogenisation theory and accepted a position at the Institut for Analysis at TU Dresden later this year. 2015 he took up a research post at the University of Bath. 2016 he finished his habilitation thesis. In 2017 he became Chancellor’s Fellow at the University of Strathclyde. 2020 he accepted a position at TU Hamburg as research associate and in November he became deputy professor at TU Bergakademie Freiberg. Since April 2021 he is appointed University Professor at TU Bergakademie Freiberg having the chair for Partial Differential Equations. Marcus Waurick contributed to more than 60 research articles and 2 books. He was guest lecturer at several universities and went on longer research visits around the world. He received the Research Excellence Award 2018 from the University of Strathclyde and was appointed Guest Professor 2020/2021 at TU Graz, Austria.

Review :
“The exposition is well written and clear, and many examples elucidate the theory. The text can be used as a textbook for teaching an advanced course on evolutionary PDEs and it is also suitable for self studying the topic.” (Kai Diethelm, zbMATH 1497.35008, 2022)