Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Dispersion Phenomena in Partial Differential Equations

  • Piero D'Ancona
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_128

Definition of the Subject

In a very broad sense, dispersion can be defined as the spreading of a fixed amount of matter, or energy, over a volume whichincreases with time. This intuitive picture suggests immediately the most prominent feature of dispersive phenomena: as matter spreads, its size, definedin a suitable sense, decreases at a certain rate. This effect should be contrasted with dissipation, which might be defined as an actual loss ofenergy, transferred to an external system (heat dissipation being the typical example).

This rough idea has been made very precise during the last 30 years for most evolution equations of mathematical physics. For the classical,linear, constant coefficient equations like the wave, Schrödinger, Klein–Gordon and Dirac equations, the decay of solutions can be measured in theLpnorms, and sharp estimates are available. In addition,detailed information on the profile of the solutions can be obtained, producing an accurate description of the...

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Piero D'Ancona
    • 1
  1. 1.Dipartimento di MatematicaUnversità di Roma “La Sapienza”RomaItaly