Approximate Solutions to Second Order Parabolic Equations I: analytic estimates

TL;DR

This paper introduces a novel local asymptotic formula for Green's functions of parabolic operators, enabling elementary, high-order approximate solutions with precise error estimates in weighted Sobolev spaces.

Contribution

It develops a new method using dilations and Taylor expansions for constructing high-order approximate solutions to parabolic equations, extending classical pseudo-differential approaches.

Findings

Derived a new local asymptotic formula for Green's functions.

Provided an elementary, algorithmic method for high-order approximations.

Established error estimates in weighted Sobolev spaces.

Abstract

We establish a new type of local asymptotic formula for the Green's function ${\mathcal G}_t(x,y)$ of a uniformly parabolic linear operator $\partial_t - L$ with non-constant coefficients using dilations and Taylor expansions at a point $z=z(x,y)$, for a function $z$ with bounded derivatives such that $z(x,x)=x \in {\mathbb R}^N$. For $z(x,y) =x$, we recover the known, classical expansion obtained via pseudo-differential calculus. Our method is based on dilation at $z$, Dyson and Taylor series expansions, and the Baker-Campbell-Hausdorff commutator formula. Our procedure leads to an elementary, algorithmic construction of approximate solutions to parabolic equations which are accurate to arbitrary prescribed order in the short-time limit. We establish mapping properties and precise error estimates in the exponentially weighted, $L^{p}$-type Sobolev spaces $W^{s,p}_a({\mathbb R}^N)$ that appear in practice.