Renormalization-group theory for the modified porous-medium equation

Abstract

We analyze the long-time behavior of the modified porous-medium equation ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$u=D\ensuremath{\Delta}${\mathit{u}}^{1+\mathit{n}}$ in d dimensions, where n is arbitrary and D=1 for ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$u>0 and D=1+\ensuremath{\epsilon} for ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$u0. 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)\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}(\mathit{d}\mathrm{\ensuremath{\theta}}+\mathrm{\ensuremath{\alpha}})}$f(${\mathit{rt}}^{\mathrm{\ensuremath{-}}(\mathrm{\ensuremath{\theta}}+\mathrm{\ensuremath{\beta}})}$, \ensuremath{\epsilon}), where \ensuremath{\theta}==1/(2+nd) and \ensuremath{\alpha} and \ensuremath{\beta} are \ensuremath{\epsilon}-dependent anomalous dimensions, obeying the scaling law n\ensuremath{\theta}\ensuremath{\alpha}+(1-nd\ensuremath{\theta})\ensuremath{\beta}=0. We calculate \ensuremath{\alpha} and \ensuremath{\beta} to O(\ensuremath{\epsilon}), for general d and n, using a perturbative RG scheme. In the case of groundwater spreading, our results to O(${\mathrm{\ensuremath{\epsilon}}}^{2}$) are in good agreement with numerical calculations, with a relative error in the anomalous dimension \ensuremath{\alpha} of about 3% when \ensuremath{\epsilon} is 0.5.