Anomalous dimensions and the renormalization group in a nonlinear diffusion process

Abstract

We present a renormalization-group (RG) approach to the nonlinear diffusion process ${\mathrm{\ensuremath{\partial}}}_{\mathit{t}}$u=D ${\mathrm{\ensuremath{\partial}}}_{\mathit{x}}^{2}$u, with D=1/2 for ${\mathrm{\ensuremath{\partial}}}_{\mathit{x}}^{2}$u>0 and D=(1+\ensuremath{\epsilon})/2 for ${\mathrm{\ensuremath{\partial}}}_{\mathit{x}}^{2}$u0, 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)\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}(\mathrm{\ensuremath{\alpha}}+1/2)}$f(x/ \ensuremath{\surd}t, \ensuremath{\epsilon}), where f is a scaling function and \ensuremath{\alpha}=\ensuremath{\epsilon}(2\ensuremath{\pi}e${)}^{1/2}$+O(${\mathrm{\ensuremath{\epsilon}}}^{2}$) 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.