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+ε)/2 for ∂_x²u<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)∼t^-(α+1/2)f(x/ √t, ε), where f is a scaling function and α=ε(2πe)^1/2+O(ε²) 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.