Self-consistent expansion and field-theoretic renormalization group for a singular nonlinear diffusion equation with anomalous scaling

The method of self-consistent expansions is a powerful tool for handling strong coupling problems that might otherwise be beyond the reach of perturbation theory, providing surprisingly accurate approximations even at low order. First applied in its embryonic form to fully-developed turbulence, it has subsequently been successfully applied to a variety of problems that include polymer statistics, interface dynamics, and high-order perturbation theory for the anharmonic oscillator. Here, we show that the self-consistent expansion can be applied to singular perturbation problems arising in the theory of partial differential equations in conjunction with renormalization group methods. We demonstrate its application to Barenblatt's nonlinear diffusion equation for porous media filtration, where the long-time asymptotics exhibits anomalous dimensions that can be systematically calculated using the perturbative renormalization group. We find that even the first-order self-consistent expansion, when combined with the Callan-Symanzik equation, improves the approximation of the anomalous dimension obtained by the first-order perturbative renormalization group, especially in the strong coupling regime. We also develop a field-theoretic framework for deterministic partial differential equations to facilitate the application of self-consistent expansions to other dynamic systems and illustrate its application using the example of Barenblatt's equation. The scope of our results on the application of renormalization group and self-consistent expansions is limited to partial differential equations whose long-time asymptotics is controlled by incomplete similarity. However, our work suggests that these methods could be applied to a broader suite of singular perturbation problems such as boundary layer theory, multiple scales analysis, and matched asymptotic expansions, for which excellent approximations using renormalization group methods alone are already available.