Anomalous dimensions and the renormalization group in a nonlinear diffusion process

We present a renormalization-group (RG) approach to the nonlinear diffusion process ${\mathrm{\partial }}_{\mathit{t}}$u=D ${\mathrm{\partial }}_{\mathit{x}}^2$u, with D=1/2 for ${\mathrm{\partial }}_{\mathit{x}}^2$u>0 and D=(1+ε)/2 for ${\mathrm{\partial }}_{\mathit{x}}^2$u<0, which describes the pressure during the filtration of an elastic fluid in an elastoplastic porous medium. Our approach recovers Barenblatt’s long-time result that, for a localized initial pressure distribution, u(x,t)∼${\mathit{t}}^{\mathrm{-}(\mathrm{\alpha }+1/2)}$f(x/ √t, ε), where f is a scaling function and α=ε(2πe${)}^{1/2}$+O(${\mathrm{\epsilon }}^2$) is an anomalous dimension, which we compute perturbatively using the RG. This is the first application of the RG to a nonlinear partial differential equation in the absence of noise.