Renormalizing Partial Differential Equations

Abstract

In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic equations. We discuss many applications, including the stability of profiles and fronts in the Ginzburg-Landau equation, anomalous scaling laws in reaction-diffusion equations, and the shape of a solution near a blow-up point.

34 pages, Latex; [email protected]; [email protected]

References in corpus (3)

• Renormalization Group Theory for Global Asymptotic Analysis
• Universality in Blow-Up for Nonlinear Heat Equations
• Stability of Moving Fronts in the Ginzburg-Landau Equation

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• Echoing and scaling in Einstein-Yang-Mills critical collapse
• Diffusion in a logarithmic potential: scaling and selection in the approach to equilibrium
• Nonlinear stability of source defects in the complex Ginzburg-Landau equation
• Numerical renormalization group-based approach to secular perturbation theory
• Critical phenomena in gravitational collapse (review paper)
• Renormalization Group approach to Inhomogeneous Cosmology
• Asymptotics for Nonlinear Integral Equations with a Generalized Heat Kernel using Renormalization Group Technique II: Marginal Perturbations and Logarithmic Corrections to the Time Decay of Solutions