Anomalous dimensions and the renormalization group in a nonlinear diffusion process

We present a renormalization-group (RG) approach to the nonlinear diffusion process tu=D x2u, with D=1/2 for x2u>0 and D=(1+)/2 for x2u<0, which describes the pressure during the filtration of an elastic fluid in an elastoplastic porous medium. Our approach recovers Barenblatts long-time result that, for a localized initial pressure distribution, u(x,t)t-(+1/2)f(x/ t), where f is a scaling function and =(2e)1/2+O(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.