A phase transition approach to detecting singularities of partial differential equations

Abstract

We present a mesh refinement algorithm for detecting singularities of time-dependent partial differential equations. The algorithm is inspired by renormalization constructions used in statistical mechanics to evaluate the properties of a system near a critical point, that is, a phase transition. The main idea behind the algorithm is to treat the occurrence of singularities of time-dependent partial differential equations as phase transitions.

The algorithm assumes the knowledge of an accurate reduced model. In particular, we need only assume that we know the functional form of the reduced model, that is, the terms appearing in the reduced model, but not necessarily their coefficients. We provide a way of computing the necessary coefficients on the fly as needed.

We show how the mesh refinement algorithm can be used to calculate the blow-up rate as we approach the singularity. This calculation can be done in three different ways: (i) the direct approach where one monitors the blowing-up quantity as it approaches the singularity and uses the data to calculate the blow-up rate; (ii) the “phase transition” approach (à la Wilson) where one treats the singularity as a fixed point of the renormalization flow equation and proceeds to compute the blow-up rate via an analysis in the vicinity of the fixed point, and (iii) the “scaling” approach (à la Widom–Kadanoff) where one postulates the existence of scaling laws for different quantities close to the singularity, computes the associated exponents and then uses them to estimate the blow-up rate. Our algorithm allows a unified presentation of these three approaches.

The inviscid Burgers and the supercritical focusing Schrödinger equations are used as instructive examples to illustrate the constructions.