Renormalization group analysis of nonlinear diffusion equations with time dependent coefficients: Analytical results

G A Braga, ,Leonardo T Rolla,Frederico Furtado,Jussara M Moreira,,Department Of Mathematics, University Of Wyoming, Laramie, 82071

doi:10.3934/dcdsb.2007.7.699

Abstract

We study the long-time asymptotics of a certain class of nonlinear diffusion equations with time-dependent diffusion coefficients which arise, for instance, in the study of transport by randomly fluctuating velocity fields. Our primary goal is to understand the interplay between anomalous diffusion and nonlinearity in determining the long-time behavior of solutions. The analysis employs the renormalization group method to establish the self-similarity and to uncover universality in the way solutions decay to zero.

Highlights

Theories of transport by a random velocity field are used in a number of important problems in many fields of science and engineering
We shall prove that under hypotheses (H1) − (H3), if the initial data is sufficiently small, problem (27) has a unique solution for each n so that the iterative Renormalization Group (RG) method can be applied to furnish the asymptotic behavior of the solution to IVP (3)
Our analysis focus first on the linear part; the nonlinear part is driven to zero under hypothesis (H3) and, thereby, does not contribute to the asymptotic regime, as we shall prove

Summary

Introduction

Theories of transport by a random velocity field are used in a number of important problems in many fields of science and engineering. Theorem 2 that, given L > 1, there exists an ε > 0 such that the IVP (3) has a unique solution u in Bf for any f ∈ Bq with f < ε. Let g ∈ Bq and for a given n ≥ 0 assume that the IVP (27) with initial condition g has a unique solution un.

Results

Conclusion