TCC Course

Operator semigroups in Banach spaces and their applications

This course is a part of the programme organised by the Taught Course Centre.

Outline:

• Preliminaries

  • Cauchy functional equation in finite and infinite dimensions

  • Uniformly continuous semigroups in Banach spaces

  • Examples of semigroups

• Strongly continuous semigroups in Banach spaces

  • Properties

  • Examples

  • Generators and resolvents

  • Hille–Yosida and Lummer–Phillips theorems

• Cauchy problem

  • Well-posedness

  • Mild solutions

  • Special classes of semigroups and regularity of solutions

• Perturbation and approximation of semigroups

  • Bounded perturbations

  • Perturbations in special classes of semigroups

  • Trotter–Kato–Neveu–Sova–Kurtz approximation (convergence) theorems

  • Applications

• Stability and asymptotic of semigroups

  • Outline of the theory

  • Applications

Literature:

• Engel, Klaus-Jochen; Nagel, Rainer. One-parameter semigroups for linear evolution equations. Grad. Texts in Math., 194. Springer-Verlag, New York, 2000. xxii+586 pp. ISBN:0-387-98463-1.

• Pazy, Amnon. Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci., 44. Springer-Verlag, New York, 1983. viii+279 pp. ISBN:0-387-90845-5.

• Bobrowski, Adam. Convergence of one-parameter operator semigroups. In models of mathematical biology and elsewhere. New Math. Monogr., 30. Cambridge University Press, Cambridge, 2016. xiv+438 pp. ISBN:978-1-107-13743-1.

• Arendt, Wolfgang; Batty, Charles J. K.; Hieber, Matthias; Neubrander, Frank. Vector-valued Laplace transforms and Cauchy problems. Second edition. Monogr. Math., 96. Birkhäuser/Springer Basel AG, Basel, 2011. xii+539 pp. ISBN:978-3-0348-0086-0.