Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 011, 10 pages      arXiv:1209.1612      https://doi.org/10.3842/SIGMA.2013.011

On the n-Dimensional Porous Medium Diffusion Equation and Global Actions of the Symmetry Group

Jose A. Franco
Department of Mathematics and Statistics, University of North Florida, 1 UNF Drive, Jacksonville, FL 32224 USA

Received September 10, 2012, in final form February 08, 2013; Published online February 12, 2013

Abstract
By restricting to a special class of smooth functions, the local action of the symmetry group is globalized. This special class of functions is constructed using parabolic induction.

Key words: globalization; porous medium equation; Lie group representation; Lorentz group; parabolic induction.

pdf (380 kb)   tex (15 kb)

References

  1. Ames W.F., Anderson R.L., Dorodnitsyn V.A., Ferapontov E.V., Gazizov R.K., Ibragimov N.H., Svirshchevskiĭ S.R., CRC handbook of Lie group analysis of differential equations. Vol. 1, Symmetries, exact solutions and conservation laws, CRC Press, Boca Raton, FL, 1994.
  2. Cowling M., Frenkel E., Kashiwara M., Valette A., Vogan Jr. D.A., Wallach N.R., Representation theory and complex analysis (Lectures from the C.I.M.E. Summer School held in Venice, June 10-17, 2004), Lecture Notes in Mathematics, Vol. 1931, Springer-Verlag, Berlin, 2008.
  3. Craddock M.J., Dooley A.H., On the equivalence of Lie symmetries and group representations, J. Differential Equations 249 (2010), 621-653.
  4. Dorodnitsyn V.A., Knyazeva I.V., Svirshchevskiĭ S.R., Group properties of the anisotropic heat equation with source $T_{t}=\sum_{i}(K_{i}(T)T_{x_{i}})_{x_{i}}+Q(T)$, Akad. Nauk SSSR Inst. Prikl. Mat., Preprint no. 134, 1982, 20 pages.
  5. Dos Santos Cardoso-Bihlo E., Bihlo A., Popovych R.O., Enhanced preliminary group classification of a class of generalized diffusion equations, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 3622-3638, arXiv:1012.0297.
  6. Franco J., Global $\widetilde{{\rm SL}(2,R)}$ representations of the Schrödinger equation with singular potential, Cent. Eur. J. Math. 10 (2012), 927-941, arXiv:1104.3508.
  7. Hunziker M., Sepanski M.R., Stanke R.J., The minimal representation of the conformal group and classical solutions to the wave equation, J. Lie Theory 22 (2012), 301-360, arXiv:0901.2280.
  8. Knapp A.W., Lie groups beyond an introduction, Progress in Mathematics, Vol. 140, Birkhäuser Boston Inc., Boston, MA, 1996.
  9. Knapp A.W., Representation theory of semisimple groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.
  10. Kostant B., Wallach N., Action of the conformal group on steady state solutions to Maxwell's equations and background radiation, arXiv:1109.5745.
  11. Olver P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics, Vol. 107, 2nd ed., Springer-Verlag, New York, 1993.
  12. Ovsjannikov L.V., Group relations of the equation of non-linear heat conductivity, Dokl. Akad. Nauk SSSR 125 (1959), 592-595.
  13. Sepanski M.R., Global actions of Lie symmetries for the nonlinear heat equation, J. Math. Anal. Appl. 360 (2009), 35-46.
  14. Sepanski M.R., Nonlinear potential filtration equation and global actions of Lie symmetries, Electron. J. Differential Equations 2009 (2009), no. 101, 24 pages.
  15. Sepanski M.R., Stanke R.J., Global Lie symmetries of the heat and Schrödinger equation, J. Lie Theory 20 (2010), 543-580.
  16. Vázquez J.L., The porous medium equation. Mathematical theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.
  17. Wu Z., Zhao J., Yin J., Li H., Nonlinear diffusion equations, World Scientific Publishing Co. Inc., River Edge, NJ, 2001.

Previous article  Next article   Contents of Volume 9 (2013)