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

TL;DR

This paper introduces a self-consistent expansion method combined with field-theoretic renormalization group techniques to analyze singular nonlinear diffusion equations, improving approximation accuracy for anomalous scaling in long-time asymptotics.

Contribution

It demonstrates the application of self-consistent expansions to PDEs with anomalous scaling and develops a field-theoretic framework to enhance analysis of such systems.

Findings

First-order expansion improves anomalous dimension estimates

Method effectively handles strong coupling regimes

Framework applicable to other singular perturbation problems

Abstract

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. 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 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 combination 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.