Renormalization-group theory for the modified porous-medium equation

We analyze the long-time behavior of the modified porous-medium equation ${\mathrm{\partial }}_{\mathit{t}}$u=DΔ${\mathit{u}}^{1+\mathit{n}}$ in d dimensions, where n is arbitrary and D=1 for ${\mathrm{\partial }}_{\mathit{t}}$u>0 and D=1+ε for ${\mathrm{\partial }}_{\mathit{t}}$uinter 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)∼${\mathit{t}}^{\mathrm{-}(\mathit{d}\mathrm{\theta }+\mathrm{\alpha })}$f(${\mathit{rt}}^{\mathrm{-}(\mathrm{\theta }+\mathrm{\beta })}$, ε), where θ==1/(2+nd) and α and β are ε-dependent anomalous dimensions, obeying the scaling law nθα+(1-ndθ)β=0. We calculate α and β to O(ε), for general d and n, using a perturbative RG scheme. In the case of groundwater spreading, our results to O(${\mathrm{\epsilon }}^2$) are in good agreement with numerical calculations, with a relative error in the anomalous dimension α of about 3% when ε is 0.5.