Winter 2011/12

Seminar on Infinite-dimensional dynamics (Unendlich-dimensionale Dynamik)

Dr. Stefan Liebscher

Thursday, 10.15-11.45, seminar room 130, Arnimallee 3 (rear building)

Goal

We want to follow the semigroup approch to study partial differential equations.

Wir wollen den Halbgruppenzugang nutzen, um eine dynamische Sicht auf partielle Differentialgleichungen zu entwickeln.

References

• A. Pazy: Semigroups of Linear Operators and Applications to Partia Differential Equations. Springer, 1983
• D. Henry: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840. Springer, 1981
• A. Lunardi: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkäuser, 1995

Target audience

Students of semesters 6-10, students of the BMS (talks can be given in German and/or English)

Prerequisites

Ordinary Differential Equations I or Partial Differential Equations I or Functional Analysis I

Prospects

Bachelor, Master and Diploma theses

Topics

Strongly continuous semigroups I

• References: Pazy, ch. 1.1-1.3, 7.2
• Scope: Strongly continuous semigroup, infinitesimal operator, resolvent, Hille-Yosida theorem, resolvent estimates

Strongly continuous semigroups II

• References: Pazy, ch. 1.4-1.6, 7.2
• Scope: dissipative operators, Lumer-Philips theorem, dissipative operators, groups

Strongly continuous semigroups III

• References: Pazy, ch. 1.7-1.8
• Scope: inverse Laplace transform, exponential formulas

Analytic semigroups [2 talks]

• References: Pazy, ch. 2.3-2.6
• Scope: compact semigroups, analytic semigroups, sectorial operators, fractional powers

Perturbations

• References: Pazy, ch. 3.1-3.3
• Scope: perturbations of infinitesimal generators

Abstract Cauchy problem

• References: Pazy, ch. 4.1-4.3
• Scope: strong solution, mild solution, homogeneous linear equations, inhomogeneous linear equations, regularity for analytic semigroups

Semilinear equations [2 talks]

• References: Pazy, ch. 6.1-6.3
• Scope: mild solution, regularity, stronly continuous vs. analytic semigroups

PDE [n talks]

• References: Pazy, ch. 7-8
• Scope: heat equation, wave equation, Schrödinger equation