On the Well-posedness of Evolutionary Equations on Infinite Graphs

  • Marcus WaurickEmail author
  • Michael Kaliske
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)


The notion of systems with integration by parts is introduced.With this notion, the spatial operator of the transport equation and the spatial operator of the wave or heat equations on graphs can be defined.Th e graphs, which we consider, can consist of arbitrarily many edges and vertices.The respective adjoints of the operators on those graphs can be calculated and skew-selfadjoint operators can be classified via boundary values.Using the work of R.Picard (Math. Meth.App.Sci.32: 1768–1803 [2009]), we can therefore show well-posedness results for the respective evolutionary problems.


Evolutionary equations on graphs (skew-)self-adjoint operators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Felix Ali Mehmeti. Nonlinear waves in networks. Mathematical Research. 80. Berlin: Akademie Verlag. 171 p., 1994.Google Scholar
  2. 2.
    Robert Carlson. Adjoint and self-adjoint differential operators on graphs. Electronic Journal of Differential Equations, 06:1-10, 1998.Google Scholar
  3. 3.
    Valentina Casarino, Klaus-Jochen Engel, Rainer Nagel, and Gregor Nickel. A semigroup approach to boundary feedback systems. Integral Equations Oper. Theory, 47(3):289-306, 2003.zbMATHCrossRefGoogle Scholar
  4. 4.
    Sonja Currie and Bruce A. Watson. Eigenvalue asymptotics for differential operators on graphs. Journal of Computational and Applied Mathematics, 182(1):13-31, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Britta Dorn. Semigroup for flows in infinite networks. Semigroup Forum, 76:341-356, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Sebastian Endres and Frank Steiner. The Berry-Keating operator on L2 (R> ,dx)and on compact quantum graphs with general self-adjoint realizations. J. Phys. A, 43, 2010.Google Scholar
  7. 7.
    Ulrike Kant, Tobias Klauss, Jürgen Voigt, and Matthias Weber. Dirichlet forms for singular one-dimensional operators and on graphs. Journal of Evolution Equations, 9:637-659, 2009.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Takashi Kasuga. On Sobolev-Friedrichs’ Generalisation of Derivatives. Proceedings of the Japan Academy, 33(33):596-599, 1957.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Peter Kuchment. Quantum graphs: an introduction and a brief survey. Exner, Pavel (ed.) et al., Analysis on graphs and its applications. Selected papers based on the Isaac Newton Institute for Mathematical Sciences programme, Cambridge, UK, January 8-June 29, 2007. Providence, RI: American Mathematical Society (AMS). Proceedings of Symposia in Pure Mathematics 77, 291-312 (2008)., 2008.Google Scholar
  10. 10.
    Felix Ali Mehmeti, Joachim von Below, and Serge Nicaise. Partial Differential Equations on Multistructures. Marcel Dekker, 2001.Google Scholar
  11. 11.
    Rainer Picard. Hilbert space approach to some classical transforms. Pitman Research Notes in Mathematics Series. 196., 1989.Google Scholar
  12. 12.
    Rainer Picard. Evolution Equations as operator equations in lattices of Hilbert spaces. Glasnik Matematicki Series III, 35(1):111-136, 2000.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Rainer Picard. A structural observation for linear material laws in classical mathematical physics. Mathematical Methods in the Applied Sciences, 32:1768-1803, 2009.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Rainer Picard and Des McGhee. Partial Differential Equations: A unified Hilbert Space Approach. De Gruyter, 2011.Google Scholar
  15. 15.
    Joachim Weidmann. Linear Operators in Hilbert Spaces. Springer Verlag, New York, 1980.zbMATHGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Institut für Statik und Dynamik der TragwerkeTU Dresden Fakultät BauingenieurwesenDresdenGermany

Personalised recommendations