Renormalization-group theory for the modified porous-medium equation

We analyze the long-time behavior of the modified porous-medium equation ∂ₜu=D{Δ}u¹⁺ⁿ in d dimensions, where n is arbitrary and D=1 for ∂ₜu>0 and D=1+{ε} for ∂ₜu0. 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^-(dθ+α)f(rt^-(θ+β), {ε}), where {θ}==1/(2+nd) and {α} and {β} are {ε}-dependent anomalous dimensions, obeying the scaling law n{θ}{α}+(1-nd{θ}){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(ε²) are in good agreement with numerical calculations, with a relative error in the anomalous dimension {α} of about 3% when {ε} is 0.5.