Renormalization-group theory for the modified porous-medium equation

Lin Yuan Chen, Nigel Goldenfeld, Y. Oono

Research output: Contribution to journalArticlepeer-review


We analyze the long-time behavior of the modified porous-medium equation tu=Du1+n in d dimensions, where n is arbitrary and D=1 for tu>0 and D=1 for tu<0. This equation describes inter alia the height of a groundwater mound during gravity-driven flow in porous media (d=2, n=1) and the propagation of strong thermal waves following an intense explosion (d=3, n=5). Using general renormalization-group (RG) arguments, we show that a radially symmetric mound exists of the form u(r,t)t-(d)f(rt-(), ), where ==1/(2+nd) and and are -dependent anomalous dimensions, obeying the scaling law n(1-nd)=0. We calculate and to O(), for general d and n, using a perturbative RG scheme. In the case of groundwater spreading, our results to O(2) are in good agreement with numerical calculations, with a relative error in the anomalous dimension of about 3% when is 0.5.

Original languageEnglish (US)
Pages (from-to)6544-6550
Number of pages7
JournalPhysical Review A
Issue number10
StatePublished - 1991

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics


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