Approximate Solutions to Second Order Parabolic Equations I: analytic estimates

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.