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The Notion of Energy of Solutions: One of the Most Important Quantities

In: Methods for Partial Differential Equations

Author

Listed:
  • Marcelo R. Ebert

    (University of São Paulo, Department of Computing and Mathematics)

  • Michael Reissig

    (TU Bergakademie Freiberg, Institute of Applied Analysis)

Abstract

The main issue of this chapter is the notion of energy of solutions, a very effective tool for the treatment of nonstationary or evolution models. We introduce energies for different models and explain conservation of energies, if possible. Sometimes, the choice of a suitable energy seems to be a miracle, sometimes one has different choices. Do we have energy conservation? Do we have blow up of the energy for t →∞? Do we have a decay of the energy for t →∞? These and related questions will be answered in this chapter. Moreover, it is shown how lower order terms may influence the choice and the long-time behavior of a suitable energy. Here we restrict ourselves to mass and different damping terms. Finally, we explain for several models the long-time behavior of local energies for solutions of mixed problems with Dirichlet condition in exterior domains.

Suggested Citation

  • Marcelo R. Ebert & Michael Reissig, 2018. "The Notion of Energy of Solutions: One of the Most Important Quantities," Springer Books, in: Methods for Partial Differential Equations, chapter 0, pages 147-170, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-66456-9_11
    DOI: 10.1007/978-3-319-66456-9_11
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