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Semigroup applications everywhere.

Rainer Nagel1, Abdelaziz Rhandi2

  • 1Arbeitsbereich Funktionalanalysis, Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|October 19, 2020
PubMed
Summary

The study highlights the power of one-parameter operator semigroups for analyzing dynamical systems derived from partial differential equations (PDEs). This mathematical tool offers insights into system properties across various scientific fields.

Keywords:
Cauchy problemsapplicationsgeneratorssemigroups

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Area of Science:

  • Mathematical Analysis
  • Dynamical Systems Theory
  • Applied Mathematics

Background:

  • Dynamical systems frequently originate from partial differential equations (PDEs).
  • These systems are often formulated as Cauchy problems in abstract function spaces.
  • Understanding system behavior requires robust analytical tools.

Purpose of the Study:

  • To underscore the significance of one-parameter operator semigroups in analyzing dynamical systems.
  • To showcase the broad applicability of semigroup theory in diverse scientific domains.
  • To present recent advancements and applications within this field.

Main Methods:

  • Representation of dynamical systems as abstract evolution equations.
  • Application of Cauchy problem formulation in function spaces.
  • Utilizing the theory of one-parameter operator semigroups for solution analysis.

Main Results:

  • One-parameter operator semigroups provide a powerful framework for studying qualitative and quantitative properties of solutions.
  • This approach is versatile, with applications spanning ordinary and PDEs, nonlinear systems, control theory, and mathematical biology.
  • The methodology facilitates a unified approach to diverse problems in applied mathematics and physics.

Conclusions:

  • The theory of one-parameter operator semigroups is an indispensable tool for the analysis of dynamical systems.
  • Its wide-ranging applicability demonstrates its fundamental importance across multiple scientific disciplines.
  • Continued research in semigroup theory promises further advancements in understanding complex systems.